A* search algorithm

A* search algorithm is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The A* search algorithm is an extension of Dijkstra's algorithm useful for finding the lowest cost path between two nodes (aka vertices) of a graph. The path may traverse any number of nodes connected by edges (aka arcs) with each edge having an associated cost. The algorithm uses a heuristic which associates an estimate of the lowest cost path from this node to the goal node, such that this estimate is never greater than the actual cost.

The algorithm should not assume that all edge costs are the same. It should be possible to start and finish on any node, including ones identified as a barrier in the task.

Task

Consider the problem of finding a route across the diagonal of a chess board-like 8x8 grid. The rows are numbered from 0 to 7. The columns are also numbered 0 to 7. The start position is (0, 0) and the end position is (7, 7). Movement is allow by one square in any direction including diagonals, similar to a king in chess. The standard movement cost is 1. To make things slightly harder, there is a barrier that occupy certain positions of the grid. Moving into any of the barrier positions has a cost of 100.

The barrier occupies the positions (2,4), (2,5), (2,6), (3,6), (4,6), (5,6), (5,5), (5,4), (5,3), (5,2), (4,2) and (3,2).

A route with the lowest cost should be found using the A* search algorithm (there are multiple optimal solutions with the same total cost).

Print the optimal route in text format, as well as the total cost of the route.

Optionally, draw the optimal route and the barrier positions.

Note: using a heuristic score of zero is equivalent to Dijkstra's algorithm and that's kind of cheating/not really A*!

Extra Credit

Use this algorithm to solve an 8 puzzle. Each node of the input graph will represent an arrangement of the tiles. The nodes will be connected by 4 edges representing swapping the blank tile up, down, left, or right. The cost of each edge is 1. The heuristic will be the sum of the manhatten distance of each numbered tile from its goal position. An 8 puzzle graph will have 9!/2 (181,440) nodes. The 15 puzzle has over 10 trillion nodes. This algorithm may solve simple 15 puzzles (but there are not many of those).

See also

• Wikipedia webpage:   A* search algorithm.
• An introduction to: Breadth First Search |> Dijkstra’s Algorithm |> A*

Related tasks

• 15 puzzle solver
• Dijkstra's algorithm
• Knapsack problem/0-1

C

<lang c>

1. include <stdlib.h>
2. include <stdio.h>
3. include <string.h>
4. include <float.h>

/* and not not_eq */

1. include <iso646.h>

/* add -lm to command line to compile with this header */

1. include <math.h>

1. define map_size_rows 10
2. define map_size_cols 10

char map[map_size_rows][map_size_cols] = {

   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
   {1, 0, 0, 0, 0, 0, 0, 0, 0, 1},
   {1, 0, 0, 0, 0, 0, 0, 0, 0, 1},
   {1, 0, 0, 0, 0, 1, 1, 1, 0, 1},
   {1, 0, 0, 1, 0, 0, 0, 1, 0, 1},
   {1, 0, 0, 1, 0, 0, 0, 1, 0, 1},
   {1, 0, 0, 1, 1, 1, 1, 1, 0, 1},
   {1, 0, 0, 0, 0, 0, 0, 0, 0, 1},
   {1, 0, 0, 0, 0, 0, 0, 0, 0, 1},
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

};

/* description of graph node */ struct stop {

   double col, row;
   /* array of indexes of routes from this stop to neighbours in array of all routes */
   int * n;
   int n_len;
   double f, g, h;
   int from;

};

int ind[map_size_rows][map_size_cols] = {

   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
   {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1}

};

/* description of route between two nodes */ struct route {

   /* route has only one direction! */
   int x; /* index of stop in array of all stops of src of this route */
   int y; /* intex of stop in array of all stops od dst of this route */
   double d;

};

int main() {

   int i, j, k, l, b, found;
   int p_len = 0;
   int * path = NULL;
   int c_len = 0;
   int * closed = NULL;
   int o_len = 1;
   int * open = (int*)calloc(o_len, sizeof(int));
   double min, tempg;
   int s;
   int e;
   int current;
   int s_len = 0;
   struct stop * stops = NULL;
   int r_len = 0;
   struct route * routes = NULL;

   for (i = 1; i < map_size_rows - 1; i++) {
       for (j = 1; j < map_size_cols - 1; j++) {
           if (!map[i][j]) {
               ++s_len;
               stops = (struct stop *)realloc(stops, s_len * sizeof(struct stop));
               int t = s_len - 1;
               stops[t].col = j;
               stops[t].row = i;
               stops[t].from = -1;
               stops[t].g = DBL_MAX;
               stops[t].n_len = 0;
               stops[t].n = NULL;
               ind[i][j] = t;
           }
       }
   }

   /* index of start stop */
   s = 0;
   /* index of finish stop */
   e = s_len - 1;

   for (i = 0; i < s_len; i++) {
       stops[i].h = sqrt(pow(stops[e].row - stops[i].row, 2) + pow(stops[e].col - stops[i].col, 2));
   }

   for (i = 1; i < map_size_rows - 1; i++) {
       for (j = 1; j < map_size_cols - 1; j++) {
           if (ind[i][j] >= 0) {
               for (k = i - 1; k <= i + 1; k++) {
                   for (l = j - 1; l <= j + 1; l++) {
                       if ((k == i) and (l == j)) {
                           continue;
                       }
                       if (ind[k][l] >= 0) {
                           ++r_len;
                           routes = (struct route *)realloc(routes, r_len * sizeof(struct route));
                           int t = r_len - 1;
                           routes[t].x = ind[i][j];
                           routes[t].y = ind[k][l];
                           routes[t].d = sqrt(pow(stops[routes[t].y].row - stops[routes[t].x].row, 2) + pow(stops[routes[t].y].col - stops[routes[t].x].col, 2));
                           ++stops[routes[t].x].n_len;
                           stops[routes[t].x].n = (int*)realloc(stops[routes[t].x].n, stops[routes[t].x].n_len * sizeof(int));
                           stops[routes[t].x].n[stops[routes[t].x].n_len - 1] = t;
                       }
                   }
               }
           }
       }
   }

   open[0] = s;
   stops[s].g = 0;
   stops[s].f = stops[s].g + stops[s].h;
   found = 0;

   while (o_len and not found) {
       min = DBL_MAX;

       for (i = 0; i < o_len; i++) {
           if (stops[open[i]].f < min) {
               current = open[i];
               min = stops[open[i]].f;
           }
       }

       if (current == e) {
           found = 1;

           ++p_len;
           path = (int*)realloc(path, p_len * sizeof(int));
           path[p_len - 1] = current;
           while (stops[current].from >= 0) {
               current = stops[current].from;
               ++p_len;
               path = (int*)realloc(path, p_len * sizeof(int));
               path[p_len - 1] = current;
           }
       }

       for (i = 0; i < o_len; i++) {
           if (open[i] == current) {
               if (i not_eq (o_len - 1)) {
                   for (j = i; j < (o_len - 1); j++) {
                       open[j] = open[j + 1];
                   }
               }
               --o_len;
               open = (int*)realloc(open, o_len * sizeof(int));
               break;
           }
       }

       ++c_len;
       closed = (int*)realloc(closed, c_len * sizeof(int));
       closed[c_len - 1] = current;

       for (i = 0; i < stops[current].n_len; i++) {
           b = 0;

           for (j = 0; j < c_len; j++) {
               if (routes[stops[current].n[i]].y == closed[j]) {
                   b = 1;
               }
           }

           if (b) {
               continue;
           }

           tempg = stops[current].g + routes[stops[current].n[i]].d;

           b = 1;

           if (o_len > 0) {
               for (j = 0; j < o_len; j++) {
                   if (routes[stops[current].n[i]].y == open[j]) {
                       b = 0;
                   }
               }
           }

           if (b or (tempg < stops[routes[stops[current].n[i]].y].g)) {
               stops[routes[stops[current].n[i]].y].from = current;
               stops[routes[stops[current].n[i]].y].g = tempg;
               stops[routes[stops[current].n[i]].y].f = stops[routes[stops[current].n[i]].y].g + stops[routes[stops[current].n[i]].y].h;

               if (b) {
                   ++o_len;
                   open = (int*)realloc(open, o_len * sizeof(int));
                   open[o_len - 1] = routes[stops[current].n[i]].y;
               }
           }
       }
   }

   for (i = 0; i < map_size_rows; i++) {
       for (j = 0; j < map_size_cols; j++) {
           if (map[i][j]) {
               putchar(0xdb);
           } else {
               b = 0;
               for (k = 0; k < p_len; k++) {
                   if (ind[i][j] == path[k]) {
                       ++b;
                   }
               }
               if (b) {
                   putchar('x');
               } else {
                   putchar('.');
               }
           }
       }
       putchar('\n');
   }

   if (not found) {
       puts("IMPOSSIBLE");
   } else {
       printf("path cost is %d:\n", p_len);
       for (i = p_len - 1; i >= 0; i--) {
           printf("(%1.0f, %1.0f)\n", stops[path[i]].col, stops[path[i]].row);
       }
   }

   for (i = 0; i < s_len; ++i) {
       free(stops[i].n);
   }
   free(stops);
   free(routes);
   free(path);
   free(open);
   free(closed);

   return 0;

} </lang>

Output:

▒▒▒▒▒▒▒▒▒▒
▒x.......▒
▒.x......▒
▒.x..▒▒▒.▒
▒.x▒...▒.▒
▒.x▒...▒.▒
▒.x▒▒▒▒▒.▒
▒..xxxxx.▒
▒.......x▒
▒▒▒▒▒▒▒▒▒▒
path cost is 12:
(1, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 7)
(4, 7)
(5, 7)
(6, 7)
(7, 7)
(8, 8)

C#

<lang csharp> using System; using System.Collections.Generic;

namespace A_star {

   class A_star
   {
       // Coordinates of a cell - implements the method Equals
       public class Coordinates : IEquatable<Coordinates>
       {
           public int row;
           public int col;

           public Coordinates() { this.row = -1; this.col = -1; }
           public Coordinates(int row, int col) { this.row = row; this.col = col; }

           public Boolean Equals(Coordinates c)
           {
               if (this.row == c.row && this.col == c.col)
                   return true;
               else
                   return false;
           }
       }

       // Class Cell, with the cost to reach it, the values g and f, and the coordinates
       // of the cell that precedes it in a possible path
       public class Cell
       {
           public int cost;
           public int g;
           public int f;
           public Coordinates parent;
       }

       // Class Astar, which finds the shortest path
       public class Astar
       {
           // The array of the cells
           public Cell[,] cells = new Cell[8, 8];
           // The possible path found
           public List<Coordinates> path = new List<Coordinates>();
           // The list of the opened cells
           public List<Coordinates> opened = new List<Coordinates>();
           // The list of the closed cells
           public List<Coordinates> closed = new List<Coordinates>();
           // The start of the searched path
           public Coordinates startCell = new Coordinates(0, 0);
           // The end of the searched path
           public Coordinates finishCell = new Coordinates(7, 7);

           // The constructor
           public Astar()
           {
               // Initialization of the cells values
               for (int i = 0; i < 8; i++)
                   for (int j = 0; j < 8; j++)
                   {
                       cells[i, j] = new Cell();
                       cells[i, j].parent = new Coordinates();
                       if (IsAWall(i, j))
                           cells[i, j].cost = 100;
                       else
                           cells[i, j].cost = 1;
                   }

               // Adding the start cell on the list opened
               opened.Add(startCell);

               // Boolean value which indicates if a path is found
               Boolean pathFound = false;

               // Loop until the list opened is empty or a path is found
               do
               {
                   List<Coordinates> neighbors = new List<Coordinates>();
                   // The next cell analyzed
                   Coordinates currentCell = ShorterExpectedPath();
                   // The list of cells reachable from the actual one
                   neighbors = neighborsCells(currentCell);
                   foreach (Coordinates newCell in neighbors)
                   {
                       // If the cell considered is the final one
                       if (newCell.row == finishCell.row && newCell.col == finishCell.col)
                       {
                           cells[newCell.row, newCell.col].g = cells[currentCell.row,
                               currentCell.col].g + cells[newCell.row, newCell.col].cost;
                           cells[newCell.row, newCell.col].parent.row = currentCell.row;
                           cells[newCell.row, newCell.col].parent.col = currentCell.col;
                           pathFound = true;
                           break;
                       }

                       // If the cell considered is not between the open and closed ones
                       else if (!opened.Contains(newCell) && !closed.Contains(newCell))
                       {
                           cells[newCell.row, newCell.col].g = cells[currentCell.row,
                               currentCell.col].g + cells[newCell.row, newCell.col].cost;
                           cells[newCell.row, newCell.col].f =
                               cells[newCell.row, newCell.col].g + Heuristic(newCell);
                           cells[newCell.row, newCell.col].parent.row = currentCell.row;
                           cells[newCell.row, newCell.col].parent.col = currentCell.col;
                           SetCell(newCell, opened);
                       }

                       // If the cost to reach the considered cell from the actual one is
                       // smaller than the previous one
                       else if (cells[newCell.row, newCell.col].g > cells[currentCell.row,
                           currentCell.col].g + cells[newCell.row, newCell.col].cost)
                       {
                           cells[newCell.row, newCell.col].g = cells[currentCell.row,
                               currentCell.col].g + cells[newCell.row, newCell.col].cost;
                           cells[newCell.row, newCell.col].f =
                               cells[newCell.row, newCell.col].g + Heuristic(newCell);
                           cells[newCell.row, newCell.col].parent.row = currentCell.row;
                           cells[newCell.row, newCell.col].parent.col = currentCell.col;
                           SetCell(newCell, opened);
                           ResetCell(newCell, closed);
                       }
                   }
                   SetCell(currentCell, closed);
                   ResetCell(currentCell, opened);
               } while (opened.Count > 0 && pathFound == false);

               if (pathFound)
               {
                   path.Add(finishCell);
                   Coordinates currentCell = new Coordinates(finishCell.row, finishCell.col);
                   // It reconstructs the path starting from the end
                   while (cells[currentCell.row, currentCell.col].parent.row >= 0)
                   {
                       path.Add(cells[currentCell.row, currentCell.col].parent);
                       int tmp_row = cells[currentCell.row, currentCell.col].parent.row;
                       currentCell.col = cells[currentCell.row, currentCell.col].parent.col;
                       currentCell.row = tmp_row;
                   }

                   // Printing on the screen the 'chessboard' and the path found
                   for (int i = 0; i < 8; i++)
                   {
                       for (int j = 0; j < 8; j++)
                       {
                           // Symbol for a cell that doesn't belong to the path and isn't
                           // a wall
                           char gr = '.';
                           // Symbol for a cell that belongs to the path
                           if (path.Contains(new Coordinates(i, j))) { gr = 'X'; }
                           // Symbol for a cell that is a wall
                           else if (cells[i, j].cost > 1) { gr = '\u2588'; }
                           System.Console.Write(gr);
                       }
                       System.Console.WriteLine();
                   }

                   // Printing the coordinates of the cells of the path
                   System.Console.Write("\nPath: ");
                   for (int i = path.Count - 1; i >= 0; i--)
                   {
                       System.Console.Write("({0},{1})", path[i].row, path[i].col);
                   }

                   // Printing the cost of the path
                   System.Console.WriteLine("\nPath cost: {0}", path.Count - 1);

                   // Waiting to the key Enter to be pressed to end the program
                   String wt = System.Console.ReadLine();
               }
           }

           // It select the cell between those in the list opened that have the smaller
           // value of f
           public Coordinates ShorterExpectedPath()
           {
               int sep = 0;
               if (opened.Count > 1)
               {
                   for (int i = 1; i < opened.Count; i++)
                   {
                       if (cells[opened[i].row, opened[i].col].f < cells[opened[sep].row,
                           opened[sep].col].f)
                       {
                           sep = i;
                       }
                   }
               }
               return opened[sep];
           }

           // It finds che cells that could be reached from c
           public List<Coordinates> neighborsCells(Coordinates c)
           {
               List<Coordinates> lc = new List<Coordinates>();
               for (int i = -1; i <= 1; i++)
                   for (int j = -1; j <= 1; j++)
                       if (c.row+i >= 0 && c.row+i < 8 && c.col+j >= 0 && c.col+j < 8 &&
                           (i != 0 || j != 0))
                       {
                           lc.Add(new Coordinates(c.row + i, c.col + j));
                       }
               return lc;
           }

           // It determines if the cell with coordinates (row, col) is a wall
           public bool IsAWall(int row, int col)
           {
               int[,] walls = new int[,] { { 2, 4 }, { 2, 5 }, { 2, 6 }, { 3, 6 }, { 4, 6 },
                   { 5, 6 }, { 5, 5 }, { 5, 4 }, { 5, 3 }, { 5, 2 }, { 4, 2 }, { 3, 2 } };
               bool found = false;
               for (int i = 0; i < walls.GetLength(0); i++)
                   if (walls[i,0] == row && walls[i,1] == col)
                       found = true;
               return found;
           }

           // The function Heuristic, which determines the shortest path that a 'king' can do
           // This is the maximum value between the orizzontal distance and the vertical one
           public int Heuristic(Coordinates cell)
           {
               int dRow = Math.Abs(finishCell.row - cell.row);
               int dCol = Math.Abs(finishCell.col - cell.col);
               return Math.Max(dRow, dCol);
           }

           // It inserts the coordinates of cell in a list, if it's not already present
           public void SetCell(Coordinates cell, List<Coordinates> coordinatesList)
           {
               if (coordinatesList.Contains(cell) == false)
               {
                   coordinatesList.Add(cell);
               }
           }

           // It removes the coordinates of cell from a list, if it's already present
           public void ResetCell(Coordinates cell, List<Coordinates> coordinatesList)
           {
               if (coordinatesList.Contains(cell))
               {
                   coordinatesList.Remove(cell);
               }
           }
       }

       // The main method
       static void Main(string[] args)
       {
           Astar astar = new Astar();
       }
   }

} </lang>

Output:

X.......
.X......
..X.███.
.X█...█.
.X█...█.
.X█████.
..XXXXX.
.......X

Path: (0,0)(1,1)(2,2)(3,1)(4,1)(5,1)(6,2)(6,3)(6,4)(6,5)(6,6)(7,7)
Path cost: 11

C++

<lang cpp>

1. include <list>
2. include <algorithm>
3. include <iostream>

class point { public:

   point( int a = 0, int b = 0 ) { x = a; y = b; }
   bool operator ==( const point& o ) { return o.x == x && o.y == y; }
   point operator +( const point& o ) { return point( o.x + x, o.y + y ); }
   int x, y;

};

class map { public:

   map() {
       char t[8][8] = {
           {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0},
           {0, 0, 0, 0, 1, 1, 1, 0}, {0, 0, 1, 0, 0, 0, 1, 0},
           {0, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 1, 1, 1, 1, 0},
           {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}
       };
       w = h = 8;
       for( int r = 0; r < h; r++ )
           for( int s = 0; s < w; s++ )
               m[s][r] = t[r][s];
   }
   int operator() ( int x, int y ) { return m[x][y]; }
   char m[8][8];
   int w, h;

};

class node { public:

   bool operator == (const node& o ) { return pos == o.pos; }
   bool operator == (const point& o ) { return pos == o; }
   bool operator < (const node& o ) { return dist + cost < o.dist + o.cost; }
   point pos, parent;
   int dist, cost;

};

class aStar { public:

   aStar() {
       neighbours[0] = point( -1, -1 ); neighbours[1] = point(  1, -1 );
       neighbours[2] = point( -1,  1 ); neighbours[3] = point(  1,  1 );
       neighbours[4] = point(  0, -1 ); neighbours[5] = point( -1,  0 );
       neighbours[6] = point(  0,  1 ); neighbours[7] = point(  1,  0 );
   }

   int calcDist( point& p ){
       // need a better heuristic
       int x = end.x - p.x, y = end.y - p.y;
       return( x * x + y * y );
   }

   bool isValid( point& p ) {
       return ( p.x >-1 && p.y > -1 && p.x < m.w && p.y < m.h );
   }

   bool existPoint( point& p, int cost ) {
       std::list<node>::iterator i;
       i = std::find( closed.begin(), closed.end(), p );
       if( i != closed.end() ) {
           if( ( *i ).cost + ( *i ).dist < cost ) return true;
           else { closed.erase( i ); return false; }
       }
       i = std::find( open.begin(), open.end(), p );
       if( i != open.end() ) {
           if( ( *i ).cost + ( *i ).dist < cost ) return true;
           else { open.erase( i ); return false; }
       }
       return false;
   }

   bool fillOpen( node& n ) {
       int stepCost, nc, dist;
       point neighbour;

       for( int x = 0; x < 8; x++ ) {
           // one can make diagonals have different cost
           stepCost = x < 4 ? 1 : 1;
           neighbour = n.pos + neighbours[x];
           if( neighbour == end ) return true;

           if( isValid( neighbour ) && m( neighbour.x, neighbour.y ) != 1 ) {
               nc = stepCost + n.cost;
               dist = calcDist( neighbour );
               if( !existPoint( neighbour, nc + dist ) ) {
                   node m;
                   m.cost = nc; m.dist = dist;
                   m.pos = neighbour;
                   m.parent = n.pos;
                   open.push_back( m );
               }
           }
       }
       return false;
   }

   bool search( point& s, point& e, map& mp ) {
       node n; end = e; start = s; m = mp;
       n.cost = 0; n.pos = s; n.parent = 0; n.dist = calcDist( s );
       open.push_back( n );
       while( !open.empty() ) {
           //open.sort();
           node n = open.front();
           open.pop_front();
           closed.push_back( n );
           if( fillOpen( n ) ) return true;
       }
       return false;
   }

   int path( std::list<point>& path ) {
       path.push_front( end );
       int cost = 1 + closed.back().cost;
       path.push_front( closed.back().pos );
       point parent = closed.back().parent;

       for( std::list<node>::reverse_iterator i = closed.rbegin(); i != closed.rend(); i++ ) {
           if( ( *i ).pos == parent && !( ( *i ).pos == start ) ) {
               path.push_front( ( *i ).pos );
               parent = ( *i ).parent;
           }
       }
       path.push_front( start );
       return cost;
   }

   map m; point end, start;
   point neighbours[8];
   std::list<node> open;
   std::list<node> closed;

};

int main( int argc, char* argv[] ) {

   map m;
   point s, e( 7, 7 );
   aStar as;

   if( as.search( s, e, m ) ) {
       std::list<point> path;
       int c = as.path( path );
       for( int y = -1; y < 9; y++ ) {
           for( int x = -1; x < 9; x++ ) {
               if( x < 0 || y < 0 || x > 7 || y > 7 || m( x, y ) == 1 )
                   std::cout << char(0xdb);
               else {
                   if( std::find( path.begin(), path.end(), point( x, y ) )!= path.end() )
                       std::cout << "x";
                   else std::cout << ".";
               }
           }
           std::cout << "\n";
       }

       std::cout << "\nPath cost " << c << ": ";
       for( std::list<point>::iterator i = path.begin(); i != path.end(); i++ ) {
           std::cout<< "(" << ( *i ).x << ", " << ( *i ).y << ") ";
       }
   }
   std::cout << "\n\n";
   return 0;

} </lang>

Output:

██████████
█x.......█
█x.......█
█x...███.█
█x.█...█.█
█x.█...█.█
█.x█████.█
█..xxxx..█
█......xx█
██████████

Path cost 11: (0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (1, 5) (2, 6) (3, 6) (4, 6) (5, 6) (6, 7) (7, 7)

Common Lisp

<lang lisp>;; * Using external libraries with quicklisp (eval-when (:load-toplevel :compile-toplevel :execute)

 (ql:quickload '("pileup" "iterate")))

* The package definition

(defpackage :a*-search

 (:use :common-lisp :pileup :iterate))

(in-package :a*-search)

* The data

(defvar *size* 8

 "The size of the area.")

I will use simple conses for the positions and directions.

(defvar *barriers*

 '((2 . 4) (2 . 5) (2 . 6) (3 . 6) (4 . 6) (5 . 6) (5 . 5) (5 . 4) (5 . 3) (5 . 2)
   (4 . 2) (3 . 2))
 "The position of the barriers in (X Y) pairs, starting with (0 0) at the lower
 left corner.")

(defvar *barrier-cost* 100 "The costs of a barrier field.")

(defvar *directions* '((0 . -1) (0 . 1) (1 . 0) (-1 . 0) (-1 . -1) (1 . 1))

 "The possible directions left, right, up, down and diagonally.")

* Tha data structure for the node in the search graph

(defstruct (node (:constructor node))

 (pos (cons 0 0) :type cons)
 (path nil)
 (cost 0 :type fixnum)                 ; The costs so far
 (f-value 0 :type fixnum)              ; The value for the heuristics
 )

* The functions

** Printing the final path

(defun print-path (path start end &optional (barriers *barriers*)

                  &aux (size (+ 2 *size*)))
 "Prints the area with the BARRIERS."
 ;; The upper boarder
 (format t "~v@{~A~:*~}~%" size "█")
 ;; The actual area
 ;; The lines
 (iter (for y from (1- *size*) downto 0)
       (format t "█")
       ;; The columns
       (iter (for x from 0 below *size*)
             (format t "~A"
                     (cond ((member (cons y x) barriers :test #'equal) "█")
                           ((equal (cons y x) start) "●")
                           ((equal (cons y x) end) "◆")
                           ((Member (cons y x) path :test #'equal) "▪")
                           (t " "))))
       ;; The last column and jump to the next line
       (format t "█~%"))
 ;; The lower boarder
 (format t "~v@{~A~:*~}~%" size "█")
 (iter
   (for position in path)
   (format t "(~D,~D)" (car position) (cdr position))
   (finally (terpri))))

** Generating the next positions

*** Check if a position is possible

(defun valid-position-p (position)

 "Returns T if POSITION is a valid position."
 (let ((x (car position))
       (y (cdr position))
       (max (1- *size*)))
   (and (<= 0 x max)
        (<= 0 y max))))

*** Move from the current position in direction

(defun move (position direction)

 "Returns a new position after moving from POSITION in DIRECTION assuming only

valid positions."

 (let ((x (car position))
       (y (cdr position))
       (dx (car direction))
       (dy (cdr direction)))
   (cons (+ x dx) (+ y dy))))

*** Generate the possible next positions

(defun next-positions (current-position)

 "Returns a list of conses with possible next positions."
 (remove-if-not #'valid-position-p
                (mapcar (lambda (d) (move current-position d)) *directions*)))

** The heuristics

(defun distance (current-position goal)

 "Returns the Manhattan distance from CURRENT-POSITION to GOAL."
 (+ (abs (- (car goal) (car current-position)))
    (abs (- (cdr goal) (cdr current-position)))))

** The A+ search

(defun a* (start goal heuristics next &optional (information 0))

 "Returns the shortest path from START to GOAL using HEURISTICS, generating the
 next nodes using NEXT."
 (let ((visited (make-hash-table :test #'equalp)))
   (flet ((pick-next-node (queue)
            ;; Get the next node from the queue
            (heap-pop queue))
          (expand-node (node queue)
            ;; Expand the next possible nodes from node and add them to the
            ;; queue if not already visited.
            (iter
              (with costs = (node-cost node))
              (for position in (funcall next (node-pos node)))
              (for cost = (1+ costs))
              (for f-value = (+ cost (funcall heuristics position goal)
                                (if (member position *barriers* :test #'equal)
                                    100
                                    0)))
              ;; Check if this state was already looked at
              (unless (gethash position visited))
              ;; Insert the next node into the queue
              (heap-insert
               (node :pos position :path (cons position (node-path node))
                     :cost cost :f-value f-value)
               queue))))

     ;; The actual A* search
     (iter
       ;; The priority queue
       (with queue = (make-heap #'<= :name "queue" :size 1000 :key #'node-f-value))
       (with initial-cost = (funcall heuristics start goal))
       (initially (heap-insert (node :pos start :path (list start) :cost 0
                                     :f-value initial-cost)
                               queue))
       (for counter from 1)
       (for current-node = (pick-next-node queue))
       (for current-position = (node-pos current-node))
       ;; Output some information each counter or nothing if information
       ;; equals 0.
       (when (and (not (zerop information))
                  (zerop (mod counter information)))
         (format t "~Dth Node, heap size: ~D, current costs: ~D~%"
                 counter (heap-count queue)
                 (node-cost current-node)))

       ;; If the target is not reached continue
       (until (equalp current-position goal))
       ;; Add the current state to the hash of visited states
       (setf (gethash current-node visited) t)
       ;; Expand the current node and continue
       (expand-node current-node queue)
       (finally (return (values (nreverse (node-path current-node))
                                (node-cost current-node)
                                counter)))))))

** The main function

(defun search-path (&key (start '(0 . 0)) (goal '(7 . 7)) (heuristics #'distance))

 "Searches the shortest path from START to GOAL using HEURISTICS."
 (multiple-value-bind (path cost steps)
     (a* start goal heuristics #'next-positions 0)
   (format t "Found the shortest path from Start (●) to Goal (◆) in ~D steps with cost: ~D~%" steps cost)
   (print-path path start goal)))</lang>

Output:

A*-SEARCH> (search-path)
Found the shortest path  from Start (●) to Goal (◆) in 323 steps with cost: 11
██████████
█   ▪▪▪▪◆█
█  ▪     █
█ ▪█████ █
█ ▪█   █ █
█ ▪█   █ █
█ ▪  ███ █
█ ▪      █
█●       █
██████████
(0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,2)(7,3)(7,4)(7,5)(7,6)(7,7)

D

ported from c++ code <lang D>

import std.stdio; import std.algorithm; import std.range; import std.array;

struct Point {

   int x;
   int y;
   Point opBinary(string op = "+")(Point o) { return Point( o.x + x, o.y + y ); }

}

struct Map {

   int w = 8;
   int h = 8;
   bool[][] m = [
           [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 1, 1, 0], [0, 0, 1, 0, 0, 0, 1, 0],
           [0, 0, 1, 0, 0, 0, 1, 0], [0, 0, 1, 1, 1, 1, 1, 0],
           [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]
       ];

}

struct Node {

   Point pos;
   Point parent;
   int dist;
   int cost;
   bool opEquals(const Node n) { return pos == n.pos;  }
   bool opEquals(const Point p) { return pos == p;  }
   int opCmp(ref const Node n) const { return (n.dist + n.cost) - (dist + cost); }

};

struct AStar {

   Map m;
   Point end;
   Point start;
  	Point[8] neighbours = [Point(-1,-1), Point(1,-1), Point(-1,1), Point(1,1), Point(0,-1), Point(-1,0), Point(0,1), Point(1,0)];
   Node[] open;
   Node[] closed;

   int calcDist(Point b) {
       // need a better heuristic
       int x = end.x - b.x, y = end.y - b.y;
       return( x * x + y * y );
   }

   bool isValid(Point b) {
       return ( b.x >-1 && b.y > -1 && b.x < m.w && b.y < m.h );
   }

   bool existPoint(Point b, int cost) {
       auto i = closed.countUntil(b);
       if( i != -1 ) {
           if( closed[i].cost + closed[i].dist < cost ) return true;
           else { closed = closed.remove!(SwapStrategy.stable)(i); return false; }
       }
       i = open.countUntil(b);
       if( i != -1 ) {
           if( open[i].cost + open[i].dist < cost ) return true;
           else { open = open.remove!(SwapStrategy.stable)(i); return false; }
       }
       return false;
   }

   bool fillOpen( ref Node n ) {
       int stepCost;
       int nc;
       int dist;
       Point neighbour;

       for( int x = 0; x < 8; ++x ) {
           // one can make diagonals have different cost
           stepCost = x < 4 ? 1 : 1;
           neighbour = n.pos + neighbours[x];
           if( neighbour == end ) return true;

           if( isValid( neighbour ) && m.m[neighbour.y][neighbour.x] != 1 ) {
               nc = stepCost + n.cost;
               dist = calcDist( neighbour );
               if( !existPoint( neighbour, nc + dist ) ) {
                   Node m;
                   m.cost = nc; m.dist = dist;
                   m.pos = neighbour;
                   m.parent = n.pos;
                   open ~= m;
               }
           }
       }
       return false;
   }

   bool search( ref Point s, ref Point e, ref Map mp ) {
       Node n; end = e; start = s; m = mp;
       n.cost = 0;
       n.pos = s;
       n.parent = Point();
       n.dist = calcDist( s );
       open ~= n ;
       while( !open.empty() ) {
           //open.sort();
           Node nx = open.front();
           open = open.drop(1).array;
           closed ~= nx ;
           if( fillOpen( nx ) ) return true;
       }
       return false;
   }

   int path( ref Point[] path ) {
       path = end ~ path;
       int cost = 1 + closed.back().cost;
       path = closed.back().pos ~ path;
       Point parent = closed.back().parent;

       foreach(ref i ; closed.retro) {
           if( i.pos == parent && !( i.pos == start ) ) {
               path = i.pos ~ path;
               parent = i.parent;
           }
       }
       path = start ~ path;
       return cost;
   }

};

int main(string[] argv) {

   Map m;
   Point s;
   Point e = Point( 7, 7 );
   AStar as;

   if( as.search( s, e, m ) ) {
       Point[] path;
       int c = as.path( path );
       for( int y = -1; y < 9; y++ ) {
           for( int x = -1; x < 9; x++ ) {
               if( x < 0 || y < 0 || x > 7 || y > 7 || m.m[y][x] == 1 )
                   write(cast(char)0xdb);
               else {
                   if( path.canFind(Point(x,y)))
                       write("x");
                   else write(".");
               }
           }
           writeln();
       }

       write("\nPath cost ", c, ": ");
       foreach( i; path ) {
           write("(", i.x, ", ", i.y, ") ");
       }
   }

write("\n\n");

   return 0;

} </lang>

Output:

██████████
█x.......█
█x.......█
█x...███.█
█x.█...█.█
█x.█...█.█
█.x█████.█
█..xxxx..█
█......xx█
██████████

Path cost 11: (0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (1, 5) (2, 6) (3, 6) (4, 6) (5, 6) (6, 7) (7, 7)

FreeBASIC

<lang freebasic> '############################### '### A* search algorithm ### '###############################

'A number big enough to be greater than any possible path cost

1. define MAX_DIST 100000

type coordinates 'coordinates of a cell

   row as integer
   col as integer

end type

type listCoordinates 'list of coordinates

   length as integer
   coord(1 to 64) as coordinates

end type

type cell 'properties of a cell

   cost as integer
   g as integer
   f as integer
   parent as coordinates

end type

sub AddCoordinates(list as listCoordinates, c as coordinates) 'Adds coordinates c to the listCoordinates, checking if it's already present

   dim i as integer, inList as integer = 0
   if (list.length > 0) then
       for i = 1 to list.length
           if (list.coord(i).row = c.row and list.coord(i).col = c.col) then
               inList = i
               exit for
           end if
       next
       if (inList > 0) then
           exit sub
       end if
   end if
   if (list.length < 64) then
       list.length = list.length + 1
       list.coord(list.length).row = c.row
       list.coord(list.length).col = c.col
   end if

end sub

sub RemoveCoordinates(list as listCoordinates, c as coordinates) 'Removes coordinates c from listCoordinates

   dim i as integer, inList as integer = 0
   if (list.length > 0) then
       for i = 1 to list.length
           if (list.coord(i).row = c.row and list.coord(i).col = c.col) then
               inList = i
               exit for
           end if
       next
       if (inList > 0) then
           list.coord(inList).row = list.coord(list.length).row
           list.coord(inList).col = list.coord(list.length).col
           list.length = list.length - 1
       end if
   end if

end sub

function GetOpened(list as listCoordinates, cells() as cell) as coordinates 'Gets the cell between the open ones with the shortest expected cost

   dim i as integer, minf as integer
   dim rv as coordinates
   minf = 1
   if (list.length > 1) then
       for i = 2 to list.length
           if (cells(list.coord(i).row, list.coord(i).col).f < cells(list.coord(minf).row, list.coord(minf).col).f) then
               minf = i
           end if
       next
   end if
   rv.row = list.coord(minf).row
   rv.col = list.coord(minf).col
   return rv

end function

function Heuristic(byval a as coordinates, byval b as coordinates) as integer 'In a chessboard, the shortest path of a king between two cells is the maximum value 'between the orizzontal distance and the vertical one. This could be used as 'heuristic value in the A* algorithm.

   dim dr as integer, dc as integer
   dr = abs(a.row - b.row)
   dc = abs(a.col - b.col)
   if (dr > dc) then
       return dr
   else
       return dc
   end if

end function

function IsACell(r as integer, c as integer) as integer 'It determines if a couple of indeces are inside the chessboard (returns 1) or outside (returns 0)

   dim isCell as integer
   if (r < 0 or r > 7 or c < 0 or c > 7) then
       isCell = 0
   else
       isCell = 1
   end if
   return isCell

end function

sub AppendCell(p as listCoordinates, c as coordinates) 'It appends che coordinates c at the end of the list p

   p.length = p.length + 1
   p.coord(p.length).row = c.row
   p.coord(p.length).col = c.col

end sub

function InList(r as integer, c as integer, p as listCoordinates) as integer 'It determines if the cell with coordinates (r,c) is in the list p

   dim isInPath as integer = 0
   dim i as integer
   for i = 1 to Ubound(p.coord)
       if (p.coord(i).row = r and p.coord(i).col = c) then
           isInPath = 1
           exit for
       end if
   next
   return isInPath

end function

'Variables declaration 'Cost to go to the cell of coordinates (row, column) dim costs(0 to 7, 0 to 7) as integer => { _

   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, _
   {1, 1, 1, 1, 100, 100, 100, 1}, {1, 1, 100, 1, 1, 1, 100, 1}, _
   {1, 1, 100, 1, 1, 1, 100, 1}, {1, 1, 100, 100, 100, 100, 100, 1}, _
   {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}}

dim start as coordinates, finish as coordinates 'the first and the last cell dim opened as listCoordinates, closed as listCoordinates dim aCell as coordinates, nCell as coordinates 'the cell evaluates and the next one dim cells(0 to 7, 0 to 7) as cell 'the cells of the chessboard dim path as listCoordinates 'list used to the path found dim i as integer, j as integer

'MAIN PROCEDURE 'Fixing the starting cell and the finishing one start.row = 0 start.col = 0 finish.row = 7 finish.col = 7 opened.length = 0 closed.length = 0

'Initializing the chessboard for i=0 to 7

   for j=0 to 7
       cells(i, j).cost = costs(i, j)
       cells(i, j).g = MAX_DIST
       cells(i, j).f = MAX_DIST
       cells(i, j).parent.row = -1
       cells(i, j).parent.col = -1
   next

next

cells(start.row, start.col).g = 0 cells(start.row, start.col).f = Heuristic(start, finish) AddCoordinates(opened, start)

do while (opened.length > 0)

   aCell = GetOpened(opened, cells())
   for i = -1 to 1
       for j = -1 to 1
           if ((i <> 0 or j <> 0) and IsACell(aCell.row + i, aCell.col + j)) then
               nCell.row = aCell.row + i
               nCell.col = aCell.col + j
               if (nCell.row = finish.row and nCell.col = finish.col) then
               'The final cell is reached
                   cells(finish.row, finish.col).g = cells(aCell.row, aCell.col).g + cells(finish.row, finish.col).cost
                   cells(finish.row, finish.col).parent.row = aCell.row
                   cells(finish.row, finish.col).parent.col = aCell.col
                   exit do
               end if
               if (InList(nCell.row, nCell.col, opened) = 0 and InList(nCell.row, nCell.col, closed) = 0) then
               'This cell was never visited before
                   cells(nCell.row, nCell.col).g = cells(aCell.row, aCell.col).g + cells(nCell.row, nCell.col).cost
                   cells(nCell.row, nCell.col).f = cells(nCell.row, nCell.col).g + Heuristic(nCell, finish)
                   AddCoordinates(opened, nCell)
                   cells(nCell.row, nCell.col).parent.row = aCell.row
                   cells(nCell.row, nCell.col).parent.col = aCell.col
               else
               'This cell was visited before, it's reopened only if the actual path is shortest of the previous valutation
                   if (cells(aCell.row, aCell.col).g + cells(nCell.row, nCell.col).cost < cells(nCell.row, nCell.col).g) then
                       cells(nCell.row, nCell.col).g = cells(aCell.row, aCell.col).g + cells(nCell.row, nCell.col).cost
                       cells(nCell.row, nCell.col).f = cells(nCell.row, nCell.col).g + Heuristic(nCell, finish)
                       AddCoordinates(opened, nCell)
                       RemoveCoordinates(closed, nCell)
                       cells(nCell.row, nCell.col).parent.row = aCell.row
                       cells(nCell.row, nCell.col).parent.col = aCell.col
                   end if
               end if
           end if
       next
   next
   'The current cell is closed
   AddCoordinates(closed, aCell)
   RemoveCoordinates(opened, aCell)

loop

if (cells(finish.row, finish.col).parent.row >= 0) then 'A possible path was found

   'Add the cells of the shortest path to the list 'path', proceding backward
   path.length = 0
   aCell.row = finish.row
   aCell.col = finish.col
   do while (cells(aCell.row, aCell.col).parent.row >= 0)
       AppendCell(path, aCell)
       nCell.row = cells(aCell.row, aCell.col).parent.row
       aCell.col = cells(aCell.row, aCell.col).parent.col
       aCell.row = nCell.row
   loop

   'Drawing the path
   for i = 0 to 7
       for j = 0 to 7
           if (costs(i,j) > 1) then
               print chr(219);
           elseif (InList(i, j, path)) then
               print "X";
           else
               print ".";
           end if
       next
       print
   next

   'Writing the cells sequence and the path length
   print
   print "Path: "
   for i = path.length to 1 step -1
       print "("; path.coord(i).row; ","; path.coord(i).col; ")";
   next
   print
   print
   print "Path cost: "; cells(finish.row, finish.col).g
   print

else

   print "Path not found"

end if end </lang>

Output:

X.......
.X......
..X.███.
.X█...█.
.X█...█.
.X█████.
..X.....
...XXXXX

Path:
( 1, 1)( 2, 2)( 3, 1)( 4, 1)( 5, 1)( 6, 2)( 7, 3)( 7, 4)( 7, 5)( 7, 6)( 7, 7)

Path cost:  11

Go

<lang go>// Package astar implements the A* search algorithm with minimal constraints // on the graph representation. package astar

import "container/heap"

// Exported node type. type Node interface {

   To() []Arc               // return list of arcs from this node to another
   Heuristic(from Node) int // heuristic cost from another node to this one

}

// An Arc, actually a "half arc", leads to another node with integer cost. type Arc struct {

   To   Node
   Cost int

}

// rNode holds data for a "reached" node type rNode struct {

   n    Node
   from Node
   l    int // route len
   g    int // route cost
   f    int // "g+h", route cost + heuristic estimate
   fx   int // heap.Fix index

}

type openHeap []*rNode // priority queue

// Route computes a route from start to end nodes using the A* algorithm. // // The algorithm is general A*, where the heuristic is not required to be // monotonic. If a route exists, the function will find a route regardless // of the quality of the Heuristic. For an admissiable heuristic, the route // will be optimal. func Route(start, end Node) (route []Node, cost int) {

   // start node initialized with heuristic
   cr := &rNode{n: start, l: 1, f: end.Heuristic(start)}
   // maintain a set of reached nodes.  start is reached initially
   r := map[Node]*rNode{start: cr}
   // oh is a heap of nodes "open" for exploration.  nodes go on the heap
   // when they get an initial or new "g" route distance, and therefore a
   // new "f" which serves as priority for exploration.
   oh := openHeap{cr}
   for len(oh) > 0 {
       bestRoute := heap.Pop(&oh).(*rNode)
       bestNode := bestRoute.n
       if bestNode == end {
           // done.  prepare return values
           cost = bestRoute.g
           route = make([]Node, bestRoute.l)
           for i := len(route) - 1; i >= 0; i-- {
               route[i] = bestRoute.n
               bestRoute = r[bestRoute.from]
           }
           return
       }
       l := bestRoute.l + 1
       for _, to := range bestNode.To() {
           // "g" route distance from start
           g := bestRoute.g + to.Cost
           if alt, ok := r[to.To]; !ok {
               // alt being reached for the first time
               alt = &rNode{n: to.To, from: bestNode, l: l,
                   g: g, f: g + end.Heuristic(to.To)}
               r[to.To] = alt
               heap.Push(&oh, alt)
           } else {
               if g >= alt.g {
                   continue // candidate route no better than existing route
               }
               // it's a better route
               // update data and make sure it's on the heap
               alt.from = bestNode
               alt.l = l
               alt.g = g
               alt.f = end.Heuristic(alt.n)
               if alt.fx < 0 {
                   heap.Push(&oh, alt)
               } else {
                   heap.Fix(&oh, alt.fx)
               }
           }
       }
   }
   return nil, 0

}

// implement container/heap func (h openHeap) Len() int { return len(h) } func (h openHeap) Less(i, j int) bool { return h[i].f < h[j].f } func (h openHeap) Swap(i, j int) {

   h[i], h[j] = h[j], h[i]
   h[i].fx = i
   h[j].fx = j

}

func (p *openHeap) Push(x interface{}) {

   h := *p
   fx := len(h)
   h = append(h, x.(*rNode))
   h[fx].fx = fx
   *p = h

}

func (p *openHeap) Pop() interface{} {

   h := *p
   last := len(h) - 1
   *p = h[:last]
   h[last].fx = -1
   return h[last]

}</lang> <lang go>package main

import (

   "fmt"

   "astar"

)

// rcNode implements the astar.Node interface type rcNode struct{ r, c int }

var barrier = map[rcNode]bool{{2, 4}: true, {2, 5}: true,

   {2, 6}: true, {3, 6}: true, {4, 6}: true, {5, 6}: true, {5, 5}: true,
   {5, 4}: true, {5, 3}: true, {5, 2}: true, {4, 2}: true, {3, 2}: true}

// graph representation is virtual. Arcs from a node are generated when // requested, but there is no static graph representation. func (fr rcNode) To() (a []astar.Arc) {

   for r := fr.r - 1; r <= fr.r+1; r++ {
       for c := fr.c - 1; c <= fr.c+1; c++ {
           if (r == fr.r && c == fr.c) || r < 0 || r > 7 || c < 0 || c > 7 {
               continue
           }
           n := rcNode{r, c}
           cost := 1
           if barrier[n] {
               cost = 100
           }
           a = append(a, astar.Arc{n, cost})
       }
   }
   return a

}

// The heuristic computed is max of row distance and column distance. // This is effectively the cost if there were no barriers. func (n rcNode) Heuristic(fr astar.Node) int {

   dr := n.r - fr.(rcNode).r
   if dr < 0 {
       dr = -dr
   }
   dc := n.c - fr.(rcNode).c
   if dc < 0 {
       dc = -dc
   }
   if dr > dc {
       return dr
   }
   return dc

}

func main() {

   route, cost := astar.Route(rcNode{0, 0}, rcNode{7, 7})
   fmt.Println("Route:", route)
   fmt.Println("Cost:", cost)

}</lang>

Output:

Route: [{0 0} {1 1} {2 2} {3 1} {4 1} {5 1} {6 2} {6 3} {6 4} {6 5} {6 6} {7 7}]
Cost: 11

Haskell

The simplest standalone FIFO priority queue is implemented after Sleator and Tarjan in Louis Wasserman's "Playing with Priority Queues"[1].

<lang haskell>{-# language DeriveFoldable #-}

module PQueue where

data PQueue a = EmptyQueue

             | Node (Int, a) (PQueue a) (PQueue a)
 deriving (Show, Foldable)

instance Ord a => Semigroup (PQueue a) where

 h1@(Node (w1, x1) l1 r1) <> h2@(Node (w2, x2) l2 r2)
   | w1 < w2   = Node (w1, x1) (h2 <> r1) l1
   | otherwise = Node (w2, x2) (h1 <> r2) l2
 EmptyQueue <> h = h
 h <> EmptyQueue = h

entry :: Ord a => a -> Int -> PQueue a entry x w = Node (w, x) EmptyQueue EmptyQueue

enque :: Ord a => PQueue a -> a -> Int -> PQueue a enque q x w = if x `notElem` q

             then entry x w <> q
             else q

deque :: Ord a => PQueue a -> Maybe (a, PQueue a) deque q = case q of

           EmptyQueue -> Nothing
           Node (_, x) l r -> Just (x, l <> r)</lang>

The simple graph combinators:

<lang haskell>import PQueue import Data.Map (Map(..)) import qualified Data.Map as Map import Data.List (unfoldr)

newtype Graph n = Graph { links :: n -> Map n Int }

grid :: Int -> Int -> Graph (Int,Int) grid a b = Graph $ \(x,y) ->

 let links = [((x+dx,y+dy), dx*dx+dy*dy)
             | dx <- [-1..1], dy <- [-1..1]
             , not (dx == 0 && dy == 0)
             , 0 <= x+dx && x+dx <= a
             , 0 <= y+dy && y+dy <= b]
 in Map.fromList links

withHole :: (Foldable t, Ord n) => Graph n -> t n -> Graph n withHole (Graph g) ns = Graph $ \x ->

 if x `elem` ns
 then Map.empty
 else foldr Map.delete (g x) ns </lang>

Finally, the search algorythm, as given in Wikipedia.

<lang haskell>get :: (Ord k, Bounded a) => Map k a -> k -> a get m x = Map.findWithDefault maxBound x m

set :: Ord k => Map k a -> k -> a -> Map k a set m k x = Map.insert k x m

data AstarData n = SetData { cameFrom :: Map n n

                          , gScore   :: Map n Int
                          , openSet  :: PQueue n }

findPath

 :: Ord n => Graph n -> (n -> n -> Int) -> n -> n -> [n]

findPath (Graph links) metric start goal = loop a0

 where
   a0 = SetData
        { cameFrom = mempty
        , gScore   = Map.singleton start 0
        , openSet  = entry start (h start) }
   h = metric goal
   dist = get . links

   loop a = case deque (openSet a) of
     Nothing -> []
     Just (current, q') -> if current == goal
                           then getPath (cameFrom a)
                           else loop a'
       where
         a' = Map.foldlWithKey go a { openSet = q' } neighbours
         neighbours = links current
         go a n _ =
           let g = get $ gScore a
               trial_Score = g current + dist current n
           in if trial_Score >= g n
              then a
              else SetData
                   ( set (cameFrom a) n current )
                   ( set (gScore a) n trial_Score )
                   ( openSet a `enque` n $ trial_Score + h n )

   getPath m = reverse $ goal : unfoldr go goal
     where go n = (\x -> (x,x)) <$> Map.lookup n m</lang>

Example

<lang haskell>distL1 (x,y) (a,b) = max (abs $ x-a) (abs $ y-b)

main = let

 g = grid 9 9 `withHole` wall
 wall = [ (2,4),(2,5),(2,6),(3,6)
        , (4,6),(5,6),(5,5),(5,4)
        , (5,3),(5,2),(3,2),(4,2) ]
 path = shortestPath g distL1 (1,1) (7,7)
 picture = [ [ case (i,j) of
                 p | p `elem` path -> '*'
                   | p `elem` wall -> '#'
                   | otherwise     -> ' '
             | i <- [0..8] ]
           | j <- [0..8] ]
 in do
   print path
   mapM_ putStrLn picture
   putStrLn $ "Path score: " <> show (length path) </lang>

λ> main
[(1,1),(2,1),(3,1),(4,1),(5,1),(6,2),(6,3),(6,4),(6,5),(6,6),(7,7)]

 *****
   ###*
     #*
  #  #*
  #  #*
  ####*
       *

Path score: 11

Java

<lang java> package astar;

import java.util.List; import java.util.ArrayList; import java.util.Collections;

class AStar {

   private final List<Node> open;
   private final List<Node> closed;
   private final List<Node> path;
   private final int[][] maze;
   private Node now;
   private final int xstart;
   private final int ystart;
   private int xend, yend;
   private final boolean diag;

   // Node class for convienience
   static class Node implements Comparable {
       public Node parent;
       public int x, y;
       public double g;
       public double h;
       Node(Node parent, int xpos, int ypos, double g, double h) {
           this.parent = parent;
           this.x = xpos;
           this.y = ypos;
           this.g = g;
           this.h = h;
      }
      // Compare by f value (g + h)
      @Override
      public int compareTo(Object o) {
          Node that = (Node) o;
          return (int)((this.g + this.h) - (that.g + that.h));
      }
  }

   AStar(int[][] maze, int xstart, int ystart, boolean diag) {
       this.open = new ArrayList<>();
       this.closed = new ArrayList<>();
       this.path = new ArrayList<>();
       this.maze = maze;
       this.now = new Node(null, xstart, ystart, 0, 0);
       this.xstart = xstart;
       this.ystart = ystart;
       this.diag = diag;
   }
   /*
   ** Finds path to xend/yend or returns null
   **
   ** @param (int) xend coordinates of the target position
   ** @param (int) yend
   ** @return (List<Node> | null) the path
   */
   public List<Node> findPathTo(int xend, int yend) {
       this.xend = xend;
       this.yend = yend;
       this.closed.add(this.now);
       addNeigborsToOpenList();
       while (this.now.x != this.xend || this.now.y != this.yend) {
           if (this.open.isEmpty()) { // Nothing to examine
               return null;
           }
           this.now = this.open.get(0); // get first node (lowest f score)
           this.open.remove(0); // remove it
           this.closed.add(this.now); // and add to the closed
           addNeigborsToOpenList();
       }
       this.path.add(0, this.now);
       while (this.now.x != this.xstart || this.now.y != this.ystart) {
           this.now = this.now.parent;
           this.path.add(0, this.now);
       }
       return this.path;
   }
   /*
   ** Looks in a given List<> for a node
   **
   ** @return (bool) NeightborInListFound
   */
   private static boolean findNeighborInList(List<Node> array, Node node) {
       return array.stream().anyMatch((n) -> (n.x == node.x && n.y == node.y));
   }
   /*
   ** Calulate distance between this.now and xend/yend
   **
   ** @return (int) distance
   */
   private double distance(int dx, int dy) {
       if (this.diag) { // if diagonal movement is alloweed
           return Math.hypot(this.now.x + dx - this.xend, this.now.y + dy - this.yend); // return hypothenuse
       } else {
           return Math.abs(this.now.x + dx - this.xend) + Math.abs(this.now.y + dy - this.yend); // else return "Manhattan distance"
       }
   }
   private void addNeigborsToOpenList() {
       Node node;
       for (int x = -1; x <= 1; x++) {
           for (int y = -1; y <= 1; y++) {
               if (!this.diag && x != 0 && y != 0) {
                   continue; // skip if diagonal movement is not allowed
               }
               node = new Node(this.now, this.now.x + x, this.now.y + y, this.now.g, this.distance(x, y));
               if ((x != 0 || y != 0) // not this.now
                   && this.now.x + x >= 0 && this.now.x + x < this.maze[0].length // check maze boundaries
                   && this.now.y + y >= 0 && this.now.y + y < this.maze.length
                   && this.maze[this.now.y + y][this.now.x + x] != -1 // check if square is walkable
                   && !findNeighborInList(this.open, node) && !findNeighborInList(this.closed, node)) { // if not already done
                       node.g = node.parent.g + 1.; // Horizontal/vertical cost = 1.0
                       node.g += maze[this.now.y + y][this.now.x + x]; // add movement cost for this square

                       // diagonal cost = sqrt(hor_cost² + vert_cost²)
                       // in this example the cost would be 12.2 instead of 11
                       /*
                       if (diag && x != 0 && y != 0) {
                           node.g += .4;	// Diagonal movement cost = 1.4
                       }
                       */
                       this.open.add(node);
               }
           }
       }
       Collections.sort(this.open);
   }

   public static void main(String[] args) {
       // -1 = blocked
       // 0+ = additional movement cost
       int[][] maze = {
           {  0,  0,  0,  0,  0,  0,  0,  0},
           {  0,  0,  0,  0,  0,  0,  0,  0},
           {  0,  0,  0,100,100,100,  0,  0},
           {  0,  0,  0,  0,  0,100,  0,  0},
           {  0,  0,100,  0,  0,100,  0,  0},
           {  0,  0,100,  0,  0,100,  0,  0},
           {  0,  0,100,100,100,100,  0,  0},
           {  0,  0,  0,  0,  0,  0,  0,  0},
       };
       AStar as = new AStar(maze, 0, 0, true);
       List<Node> path = as.findPathTo(7, 7);
       if (path != null) {
           path.forEach((n) -> {
               System.out.print("[" + n.x + ", " + n.y + "] ");
               maze[n.y][n.x] = -1;
           });
           System.out.printf("\nTotal cost: %.02f\n", path.get(path.size() - 1).g);

           for (int[] maze_row : maze) {
               for (int maze_entry : maze_row) {
                   switch (maze_entry) {
                       case 0:
                           System.out.print("_");
                           break;
                       case -1:
                           System.out.print("*");
                           break;
                       default:
                           System.out.print("#");
                   }
               }
               System.out.println();
           }
       }
   }

} </lang>

Output:

[0, 0] [1, 0] [2, 0] [3, 0] [4, 0] [5, 1] [6, 2] [7, 3] [6, 4] [6, 5] [6, 6] [7, 7]
Total cost: 11,00
*****___
_____*__
___###*_
_____#_*
__#__#*_
__#__#*_
__####*_
_______*

JavaScript

Animated.
To see how it works on a random map go here <lang javascript> var ctx, map, opn = [], clsd = [], start = {x:1, y:1, f:0, g:0}, goal = {x:8, y:8, f:0, g:0}, mw = 10, mh = 10, neighbours, path;

function findNeighbour( arr, n ) {

   var a;
   for( var i = 0; i < arr.length; i++ ) {
       a = arr[i];
       if( n.x === a.x && n.y === a.y ) return i;
   }
   return -1;

} function addNeighbours( cur ) {

   var p;
   for( var i = 0; i < neighbours.length; i++ ) {
       var n = {x: cur.x + neighbours[i].x, y: cur.y + neighbours[i].y, g: 0, h: 0, prt: {x:cur.x, y:cur.y}};
       if( map[n.x][n.y] == 1 || findNeighbour( clsd, n ) > -1 ) continue;
       n.g = cur.g + neighbours[i].c; n.h = Math.abs( goal.x - n.x ) + Math.abs( goal.y - n.y );
       p = findNeighbour( opn, n );
       if( p > -1 && opn[p].g + opn[p].h <= n.g + n.h ) continue;
       opn.push( n );
   }
   opn.sort( function( a, b ) {
       return ( a.g + a.h ) - ( b.g + b.h ); } );

} function createPath() {

   path = [];
   var a, b;
   a = clsd.pop();
   path.push( a );
   while( clsd.length ) {
       b = clsd.pop();
       if( b.x != a.prt.x || b.y != a.prt.y ) continue;
       a = b; path.push( a );
   }
}

function solveMap() {

   drawMap();
   if( opn.length < 1 ) {
       document.body.appendChild( document.createElement( "p" ) ).innerHTML = "Impossible!";
       return;
   }
   var cur = opn.splice( 0, 1 )[0];
   clsd.push( cur );
   if( cur.x == goal.x && cur.y == goal.y ) {
       createPath(); drawMap();
       return;
   }
   addNeighbours( cur );
   requestAnimationFrame( solveMap );

} function drawMap() {

   ctx.fillStyle = "#ee6"; ctx.fillRect( 0, 0, 200, 200 );
   for( var j = 0; j < mh; j++ ) {
       for( var i = 0; i < mw; i++ ) {
           switch( map[i][j] ) {
               case 0: continue;
               case 1: ctx.fillStyle = "#990"; break;
               case 2: ctx.fillStyle = "#090"; break;
               case 3: ctx.fillStyle = "#900"; break;
           }
           ctx.fillRect( i, j, 1, 1 );
       }
   }
   var a;
   if( path.length ) {
       var txt = "Path: " + ( path.length - 1 ) + "
[";
       for( var i = path.length - 1; i > -1; i-- ) {
           a = path[i];
           ctx.fillStyle = "#999";
           ctx.fillRect( a.x, a.y, 1, 1 );
           txt += "(" + a.x + ", " + a.y + ") ";
       }
       document.body.appendChild( document.createElement( "p" ) ).innerHTML = txt + "]";
       return;
   }
   for( var i = 0; i < opn.length; i++ ) {
       a = opn[i];
       ctx.fillStyle = "#909";
       ctx.fillRect( a.x, a.y, 1, 1 );
   }
   for( var i = 0; i < clsd.length; i++ ) {
       a = clsd[i];
       ctx.fillStyle = "#009";
       ctx.fillRect( a.x, a.y, 1, 1 );
   }

} function createMap() {

   map = new Array( mw );
   for( var i = 0; i < mw; i++ ) {
       map[i] = new Array( mh );
       for( var j = 0; j < mh; j++ ) {
           if( !i || !j || i == mw - 1 || j == mh - 1 ) map[i][j] = 1;
           else map[i][j] = 0;
       }
   }
   map[5][3] = map[6][3] = map[7][3] = map[3][4] = map[7][4] = map[3][5] =
   map[7][5] = map[3][6] = map[4][6] = map[5][6] = map[6][6] = map[7][6] = 1;
   //map[start.x][start.y] = 2; map[goal.x][goal.y] = 3;

} function init() {

   var canvas = document.createElement( "canvas" );
   canvas.width = canvas.height = 200;
   ctx = canvas.getContext( "2d" );
   ctx.scale( 20, 20 );
   document.body.appendChild( canvas );
   neighbours = [
       {x:1, y:0, c:1}, {x:-1, y:0, c:1}, {x:0, y:1, c:1}, {x:0, y:-1, c:1},
       {x:1, y:1, c:1.4}, {x:1, y:-1, c:1.4}, {x:-1, y:1, c:1.4}, {x:-1, y:-1, c:1.4}
   ];
   path = []; createMap(); opn.push( start ); solveMap();

} </lang>

Output:

Path: 11
[(1, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 7) (4, 8) (5, 8) (6, 8) (7, 8) (8, 8) ]

Implementation using recursive strategy <lang javascript> function manhattan(x1, y1, x2, y2) {

   return Math.abs(x1 - x2) + Math.abs(y1 - y2);

}

function aStar (board, startx, starty, goalx, goaly,

   open = Array(8 * 8).fill(null),
   closed = Array(8 * 8).fill(null),
   current = {
       "coord": [startx, starty],
       "distance": 0,
       "heuristic": manhattan(startx, starty, goalx, goaly),
       "previous": null
   }) {
   const [x, y] = [...current.coord];

   if (x === goalx && y === goaly) {
       closed[x + y * 8] = current;
       return (lambda = (closed, x, y, startx, starty) => {
           if (x === startx && y === starty) {
               return x, y;
           }
           const [px, py] = closed.filter(e => e !== null)
               .find(({coord: [nx, ny]}) => {
                   return  nx === x && ny === y
               }).previous;
           return lambda(closed, px, py, startx, starty).concat(x,y);
       })(closed, x, y, startx, starty);
   }

   let newOpen = open.slice();
   [
       [x + 1, y + 1], [x - 1, y - 1], [x + 1, y], [x, y + 1],
       [x - 1, y + 1], [x + 1, y - 1], [x - 1, y], [x, y - 1]
   ].filter(([x,y]) => x >= 0 && x < 8 &&
                       y >= 0 && y < 8 &&
                       closed[x + y * 8] === null
   ).forEach(([x,y]) => {
       newOpen[x + y * 8] = {
           "coord": [x,y],
           "distance": current.distance + (board[x + y * 8] === 1 ? 100 : 1),
           "heuristic": manhattan(x, y, goalx, goaly),
           "previous": [...current.coord]
       };
   });

   let newClosed = closed.slice();
   newClosed[x + y * 8] = current;

   const [newCurrent,] = newOpen.filter(e => e !== null)
       .sort((a, b) => {
           return (a.distance + a.heuristic) - (b.distance + b.heuristic);
       });

   const [newx, newy] = [...newCurrent.coord];
   newOpen[newx + newy * 8] = null;

   return aStar(board, startx, starty, goalx, goaly,
       newOpen, newClosed, newCurrent);

}

const board = [

   0,0,0,0,0,0,0,0,
   0,0,0,0,0,0,0,0,
   0,0,0,0,1,1,1,0,
   0,0,1,0,0,0,1,0,
   0,0,1,0,0,0,1,0,
   0,0,1,1,1,1,1,0,
   0,0,0,0,0,0,0,0,
   0,0,0,0,0,0,0,0

];

console.log(aStar(board, 0,0, 7,7)); </lang>

Output:

[ [ 0, 0 ], [ 1, 1 ], [ 2, 2 ], [ 3, 1 ], [ 4, 1 ], [ 5, 1 ], [ 6, 1 ], [ 7, 2 ], [ 7, 3 ], [ 7, 4 ], [ 7, 5 ], [ 7, 6 ], [ 7, 7 ] ]

Julia

The graphic in this solution is displayed in the more standard orientation of origin at bottom left and goal at top right. <lang Julia>using LightGraphs, SimpleWeightedGraphs

const chessboardsize = 8 const givenobstacles = [(2,4), (2,5), (2,6), (3,6), (4,6), (5,6), (5,5), (5,4), (5,3), (5,2), (4,2), (3,2)] vfromcart(p, n) = (p[1] - 1) * n + p[2] const obstacles = [vfromcart(o .+ 1, chessboardsize) for o in givenobstacles] zbasedpath(path, n) = [(div(v - 1, n), (v - 1) % n) for v in path] pathcost(path) = sum(map(x -> x in obstacles ? 100 : 1, path[2:end]))

function surround(x, y, n)

   bottomx = x > 1 ? x -1 : x
   topx = x < n ? x + 1 : x
   bottomy = y > 1 ? y - 1 : y
   topy = y < n ? y + 1 : y
   [CartesianIndex(x,y) for x in bottomx:topx for y in bottomy:topy]

end

function kinggraph(N)

   graph = SimpleWeightedGraph(N*N)
   for row in 1:N, col in 1:N, p in surround(row, col, N)
       origin = vfromcart(CartesianIndex(row, col), N)
       targ = vfromcart(p, N)
       hcost = (targ in obstacles || origin in obstacles) ? 100 : 1
       add_edge!(graph, origin, targ, hcost)
   end
   graph

end

kgraph = kinggraph(chessboardsize) path = enumerate_paths(dijkstra_shortest_paths(kgraph, 1), 64) println("Solution has cost $(pathcost(path)):\n", zbasedpath(path, chessboardsize))

path2graphic(x, path) = (x in obstacles ? '█' : x in path ? 'x' : '.') for row in 8:-1:1, col in 7:-1:0

   print(path2graphic(row*8 - col, path))
   if col == 0
       println()
   end

end</lang>

Output:

Solution has cost 11:
Tuple{Int64,Int64}[(0, 0), (1, 1), (2, 2), (3, 1), (4, 1), (5, 1), (6, 2), (7, 3), (7, 4), (6, 5), (6, 6), (7, 7)]
...xx..x
..x..xx.
.x█████.
.x█...█.
.x█...█.
..x.███.
.x......
x.......

Kotlin

<lang kotlin> import java.lang.Math.abs

typealias GridPosition = Pair<Int, Int> typealias Barrier = Set<GridPosition>

const val MAX_SCORE = 99999999

abstract class Grid(private val barriers: List<Barrier>) {

   open fun heuristicDistance(start: GridPosition, finish: GridPosition): Int {
       val dx = abs(start.first - finish.first)
       val dy = abs(start.second - finish.second)
       return (dx + dy) + (-2) * minOf(dx, dy)
   }

   fun inBarrier(position: GridPosition) = barriers.any { it.contains(position) }

   abstract fun getNeighbours(position: GridPosition): List<GridPosition>

   open fun moveCost(from: GridPosition, to: GridPosition) = if (inBarrier(to)) MAX_SCORE else 1

}

class SquareGrid(width: Int, height: Int, barriers: List<Barrier>) : Grid(barriers) {

   private val heightRange: IntRange = (0 until height)
   private val widthRange: IntRange = (0 until width)

   private val validMoves = listOf(Pair(1, 0), Pair(-1, 0), Pair(0, 1), Pair(0, -1), Pair(1, 1), Pair(-1, 1), Pair(1, -1), Pair(-1, -1))

   override fun getNeighbours(position: GridPosition): List<GridPosition> = validMoves
           .map { GridPosition(position.first + it.first, position.second + it.second) }
           .filter { inGrid(it) }

   private fun inGrid(it: GridPosition) = (it.first in widthRange) && (it.second in heightRange)

}

/**

* Implementation of the A* Search Algorithm to find the optimum path between 2 points on a grid.
*
* The Grid contains the details of the barriers and methods which supply the neighboring vertices and the
* cost of movement between 2 cells.  Examples use a standard Grid which allows movement in 8 directions
* (i.e. includes diagonals) but alternative implementation of Grid can be supplied.
*
*/

fun aStarSearch(start: GridPosition, finish: GridPosition, grid: Grid): Pair<List<GridPosition>, Int> {

   /**
    * Use the cameFrom values to Backtrack to the start position to generate the path
    */
   fun generatePath(currentPos: GridPosition, cameFrom: Map<GridPosition, GridPosition>): List<GridPosition> {
       val path = mutableListOf(currentPos)
       var current = currentPos
       while (cameFrom.containsKey(current)) {
           current = cameFrom.getValue(current)
           path.add(0, current)
       }
       return path.toList()
   }

   val openVertices = mutableSetOf(start)
   val closedVertices = mutableSetOf<GridPosition>()
   val costFromStart = mutableMapOf(start to 0)
   val estimatedTotalCost = mutableMapOf(start to grid.heuristicDistance(start, finish))

   val cameFrom = mutableMapOf<GridPosition, GridPosition>()  // Used to generate path by back tracking

   while (openVertices.size > 0) {

       val currentPos = openVertices.minBy { estimatedTotalCost.getValue(it) }!!

       // Check if we have reached the finish
       if (currentPos == finish) {
           // Backtrack to generate the most efficient path
           val path = generatePath(currentPos, cameFrom)
           return Pair(path, estimatedTotalCost.getValue(finish)) // First Route to finish will be optimum route
       }

       // Mark the current vertex as closed
       openVertices.remove(currentPos)
       closedVertices.add(currentPos)

       grid.getNeighbours(currentPos)
               .filterNot { closedVertices.contains(it) }  // Exclude previous visited vertices
               .forEach { neighbour ->
                   val score = costFromStart.getValue(currentPos) + grid.moveCost(currentPos, neighbour)
                   if (score < costFromStart.getOrDefault(neighbour, MAX_SCORE)) {
                       if (!openVertices.contains(neighbour)) {
                           openVertices.add(neighbour)
                       }
                       cameFrom.put(neighbour, currentPos)
                       costFromStart.put(neighbour, score)
                       estimatedTotalCost.put(neighbour, score + grid.heuristicDistance(neighbour, finish))
                   }
               }

   }

   throw IllegalArgumentException("No Path from Start $start to Finish $finish")

}

fun main(args: Array<String>) {

   val barriers = listOf(setOf( Pair(2,4), Pair(2,5), Pair(2,6), Pair(3,6), Pair(4,6), Pair(5,6), Pair(5,5),
                Pair(5,4), Pair(5,3), Pair(5,2), Pair(4,2), Pair(3,2)))

   val (path, cost) = aStarSearch(GridPosition(0,0), GridPosition(7,7), SquareGrid(8,8, barriers))

   println("Cost: $cost  Path: $path")

} </lang>

Output:

Cost: 11
Path: [(0, 0), (1, 1), (2, 2), (3, 1), (4, 1), (5, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), (7, 7)]

Lua

<lang lua> -- QUEUE ----------------------------------------------------------------------- Queue = {} function Queue:new()

   local q = {}
   self.__index = self
   return setmetatable( q, self )

end function Queue:push( v )

   table.insert( self, v )

end function Queue:pop()

   return table.remove( self, 1 )

end function Queue:getSmallestF()

   local s, i = nil, 2
   while( self[i] ~= nil and self[1] ~= nil ) do
       if self[i]:F() < self[1]:F() then
           s = self[1]
           self[1] = self[i]
           self[i] = s
       end
       i = i + 1
   end
   return self:pop()

end

-- LIST ------------------------------------------------------------------------ List = {} function List:new()

   local l = {}
   self.__index = self
   return setmetatable( l, self )

end function List:push( v )

 table.insert( self, v )

end function List:pop()

   return table.remove( self )

end

-- POINT ----------------------------------------------------------------------- Point = {} function Point:new()

   local p = { y = 0, x = 0 }
   self.__index = self
   return setmetatable( p, self )

end function Point:set( x, y )

   self.x, self.y = x, y

end function Point:equals( o )

   return (o.x == self.x and o.y == self.y)

end function Point:print()

   print( self.x, self.y )

end

-- NODE ------------------------------------------------------------------------ Node = {} function Node:new()

   local n = { pos = Point:new(), parent = Point:new(), dist = 0, cost = 0 }
   self.__index = self
   return setmetatable( n, self )

end function Node:set( pt, parent, dist, cost )

   self.pos = pt
   self.parent = parent
   self.dist = dist
   self.cost = cost

end function Node:F()

   return ( self.dist + self.cost )

end

-- A-STAR ---------------------------------------------------------------------- local nbours = {

   {  1,  0, 1 }, {  0,  1, 1 }, {  1,  1, 1.4 }, {  1, -1, 1.4 },
   { -1, -1, 1.4 }, { -1,  1, 1.4 }, {  0, -1, 1 }, { -1,  0, 1 }

} local map = {

       1,1,1,1,1,1,1,1,1,1,
       1,0,0,0,0,0,0,0,0,1,
       1,0,0,0,0,0,0,0,0,1,
       1,0,0,0,0,1,1,1,0,1,
       1,0,0,1,0,0,0,1,0,1,
       1,0,0,1,0,0,0,1,0,1,
       1,0,0,1,1,1,1,1,0,1,
       1,0,0,0,0,0,0,0,0,1,
       1,0,0,0,0,0,0,0,0,1,
       1,1,1,1,1,1,1,1,1,1

} local open, closed, start, goal,

     mapW, mapH = Queue:new(), List:new(), Point:new(), Point:new(), 10, 10

start:set( 2, 2 ); goal:set( 9, 9 )

function hasNode( arr, pos )

   for nx, val in ipairs( arr ) do
       if val.pos:equals( pos ) then
           return nx
       end
   end
   return -1

end function isValid( pos )

   return pos.x > 0 and pos.x <= mapW
          and pos.y > 0 and pos.y <= mapH
          and map[pos.x + mapW * pos.y - mapW] == 0

end function calcDist( p1 )

   local x, y = goal.x - p1.x, goal.y - p1.y
   return math.abs( x ) + math.abs( y )

end function addToOpen( node )

   local nx
   for n = 1, 8 do
       nNode = Node:new()
       nNode.parent:set( node.pos.x, node.pos.y )
       nNode.pos:set( node.pos.x + nbours[n][1], node.pos.y + nbours[n][2] )
       nNode.cost = node.cost + nbours[n][3]
       nNode.dist = calcDist( nNode.pos )

       if isValid( nNode.pos ) then
           if nNode.pos:equals( goal ) then
               closed:push( nNode )
               return true
           end
           nx = hasNode( closed, nNode.pos )
           if nx < 0 then
               nx = hasNode( open, nNode.pos )
               if( nx < 0 ) or ( nx > 0 and nNode:F() < open[nx]:F() ) then
                   if( nx > 0 ) then
                       table.remove( open, nx )
                   end
                   open:push( nNode )
               else
                   nNode = nil
               end
           end
       end
   end
   return false

end function makePath()

   local i, l = #closed, List:new()
   local node, parent = closed[i], nil

   l:push( node.pos )
   parent = node.parent
   while( i > 0 ) do
       i = i - 1
       node = closed[i]
       if node ~= nil and node.pos:equals( parent ) then
           l:push( node.pos )
           parent = node.parent
       end
   end
   print( string.format( "Cost: %d", #l - 1 ) )
   io.write( "Path: " )
   for i = #l, 1, -1 do
       map[l[i].x + mapW * l[i].y - mapW] = 2
       io.write( string.format( "(%d, %d) ", l[i].x, l[i].y ) )
   end
   print( "" )

end function aStar()

   local n = Node:new()
   n.dist = calcDist( start )
   n.pos:set( start.x, start.y )
   open:push( n )
   while( true ) do
       local node = open:getSmallestF()
       if node == nil then break end
       closed:push( node )
       if addToOpen( node ) == true then
           makePath()
           return true
       end
   end
   return false

end -- ENTRY POINT ----------------------------------------------------------------- if true == aStar() then

   local m
   for j = 1, mapH do
       for i = 1, mapW do
           m = map[i + mapW * j - mapW]
           if m == 0 then
               io.write( "." )
           elseif m == 1 then
               io.write( string.char(0xdb) )
           else
               io.write( "x" )
           end
       end
       io.write( "\n" )
   end

else

   print( "can not find a path!" )

end </lang>

Output:

Cost: 11
Path: (2, 2) (3, 3) (3, 4) (3, 5) (3, 6) (3, 7) (4, 8) (5, 9) (6, 9) (7, 9) (8, 9) (9, 9)
██████████
█x.......█
█.x......█
█.x..███.█
█.x█...█.█
█.x█...█.█
█.x█████.█
█..x.....█
█...xxxxx█
██████████

Nim

Implementation of the Wikipedia pseudocode. <lang Nim>

1. A* search algorithm.

from algorithm import reverse import sets import strformat import tables

const Infinity = 1_000_000_000

type Cell = tuple[row, col: int]

const

 Barriers = [(2, 4), (2, 5), (2, 6), (3, 6), (4, 6), (5, 6),
             (5, 5), (5, 4), (5, 3), (5, 2), (4, 2), (3, 2)].toHashSet
 Moves = [(-1, -1), (-1, 0), (-1, 1), (0, 1), (1, 1), (1, 0), (1, -1), (0, -1)]

1. ---------------------------------------------------------------------------------------------------

iterator neighbors(cell: Cell): Cell =

 ## Yield the neighbors of "cell".
 for move in Moves:
   let next = (row: cell.row + move[0], col: cell.col + move[1])
   if next.row in 0..7 and next.col in 0..7:
     yield next

1. ---------------------------------------------------------------------------------------------------

func d(current, neighbor: Cell): int =

 ## Return the cost for the move from "current" to "neighbor".
 if neighbor in Barriers: 100 else: 1

1. ---------------------------------------------------------------------------------------------------

func h(cell, goal: Cell): int =

   ## Compute the heuristic cost for a move form the cell to the goal.
   ## We use the Chebychev distance as appropriate for this kind of move.
   max(abs(goal.row - cell.row), abs(goal.col - cell.col))

1. ---------------------------------------------------------------------------------------------------

func reconstructedPath(cameFrom: Table[Cell, Cell]; current: Cell): seq[Cell] =

 ## Return the shortest path from the start to the goal.
 var cell = current
 result = @[cell]
 while cell in cameFrom:
   cell = cameFrom[cell]
   result.add(cell)
 result.reverse()

1. ---------------------------------------------------------------------------------------------------

func search(start, goal: Cell): tuple[path: seq[Cell], cost: int] =

 ## Search the shortest path from "start" to "goal" using A* algorithm.
 ## Return the path and the cost.

 var openSet = [start].toHashSet()
 var visited: HashSet[Cell]
 var cameFrom: Table[Cell, Cell]
 var gScore, fScore: Table[Cell, int]
 gscore[start] = 0
 fScore[start] = h(start, goal)

 while openSet.len != 0:

   # Find cell in "openset" with minimal "fScore".
   var current: Cell
   var minScore = Infinity
   for cell in openSet:
     let score = fScore.getOrDefault(cell, Infinity)
     if score < minScore:
       current = cell
       minScore = score

   if current == goal:
     # Return the path and cost.
     return (reconstructedPath(cameFrom, current), fScore[goal])

   openSet.excl(current)
   visited.incl(current)

   # Update scores for neighbors.
   for neighbor in current.neighbors():
     if neighbor in visited:
       # Already processed.
       continue
     let tentative = gScore[current] + d(current, neighbor)
     if tentative < gScore.getOrDefault(neighbor, Infinity):
       cameFrom[neighbor]= current
       gScore[neighbor] = tentative
       fScore[neighbor] = tentative + h(neighbor, goal)
       openSet.incl(neighbor)

1. ---------------------------------------------------------------------------------------------------

proc drawBoard(path: seq[Cell]) =

 ## Draw the board and the path.

 func `$`(cell: Cell): string =
   ## Return the Unicode string to use for a cell.
   if cell in Barriers: "██" else: (if cell in path: " #" else: " ·")

 echo "████████████████████"
 for row in 0..7:
   stdout.write("██")
   for col in 0..7:
     stdout.write((row, col))
   stdout.write("██\n")
 echo "████████████████████"
 echo '\n'

1. ---------------------------------------------------------------------------------------------------

proc printSolution(path: seq[Cell]; cost: int) =

 ## Print the solution.
 var pathLine = "Path: "
 let start = pathLine.len
 for cell in path:
   pathLine.addSep(" → ", start)
   pathLine.add(&"({cell.row}, {cell.col})")
 echo pathLine
 echo(&"Cost: {cost}\n\n")
 drawBoard(path)

1. ---------------------------------------------------------------------------------------------------

let (path, cost) = search((0, 0), (7, 7)) if cost == 0:

 echo "No solution found"

else:

 printSolution(path, cost)

</lang>

Output:

Path: (0, 0) → (1, 1) → (2, 2) → (3, 1) → (4, 1) → (5, 1) → (6, 2) → (7, 3) → (7, 4) → (6, 5) → (7, 6) → (7, 7)
Cost: 11


████████████████████
██ # · · · · · · ·██
██ · # · · · · · ·██
██ · · # ·██████ ·██
██ · #██ · · ·██ ·██
██ · #██ · · ·██ ·██
██ · #██████████ ·██
██ · · # · · # · ·██
██ · · · # # · # #██
████████████████████

OCaml

A very close/straightforward implementation of the Wikipedia pseudocode.

<lang ocaml> module IntPairs =

 struct
   type t = int * int
   let compare (x0,y0) (x1,y1) =
     match Stdlib.compare x0 x1 with
     | 0 -> Stdlib.compare y0 y1
     | c -> c
 end

module PairsMap = Map.Make(IntPairs) module PairsSet = Set.Make(IntPairs)

let find_path start goal board =

 let max_y = Array.length board in
 let max_x = Array.length board.(0) in

 let get_neighbors (x, y) =
   let moves = [(0, 1); (0, -1); (1, 0); (-1, 0);
                (1, 1); (1, -1); (-1, 1); (-1, -1)] in
   let ms = List.map (fun (_x, _y) -> x+_x, y+_y) moves in
   let ms = List.filter (fun (x, y) ->
       x >= 0 && x < max_x && y >= 0 && y < max_y
       && board.(y).(x) <> 0
     ) ms in
   (ms)
 in
 let h (x0, y0) (x1, y1) =
   abs (x0 - x1) + abs (y0 - y1)
 in
 let openSet = PairsSet.add start PairsSet.empty in
 let closedSet = PairsSet.empty in

 let fScore = PairsMap.add start (h goal start) PairsMap.empty in
 let gScore = PairsMap.add start 0 PairsMap.empty in

 let cameFrom = PairsMap.empty in

 let reconstruct_path cameFrom current =
   let rec aux acc current =
     let from = PairsMap.find current cameFrom in
     if from = start then (from::acc)
     else aux (from::acc) from
   in
   aux [current] current
 in
 let d current neighbor =
   let x, y = neighbor in
   board.(y).(x)
 in
 let g gScore cell =
   match PairsMap.find_opt cell gScore with
   | Some v -> v | None -> max_int
 in

 let rec _find_path (openSet, closedSet, fScore, gScore, cameFrom) =
   if PairsSet.is_empty openSet then None else
   let current =
     PairsSet.fold (fun p1 p2 ->
       if p2 = (-1, -1) then p1 else
         let s1 = PairsMap.find p1 fScore
         and s2 = PairsMap.find p2 fScore in
         if s1 < s2 then p1 else p2
     ) openSet (-1, -1)
   in
   if current = goal then Some (reconstruct_path cameFrom current) else
   let openSet = PairsSet.remove current openSet in
   let closedSet = PairsSet.add current closedSet in
   let neighbors = get_neighbors current in
   neighbors |>
     List.fold_left
       (fun ((openSet, closedSet, fScore, gScore, cameFrom) as v) neighbor ->
         if PairsSet.mem neighbor closedSet then (v) else
           let tentative_gScore = (g gScore current) + (d current neighbor) in
           if tentative_gScore < (g gScore neighbor) then
             let cameFrom = PairsMap.add neighbor current cameFrom in
             let gScore = PairsMap.add neighbor tentative_gScore gScore in
             let f = (g gScore neighbor) + (h neighbor goal) in
             let fScore = PairsMap.add neighbor f fScore in
             let openSet =
               if not (PairsSet.mem neighbor openSet)
               then PairsSet.add neighbor openSet else openSet
             in
             (openSet, closedSet, fScore, gScore, cameFrom)
           else (v)
       ) (openSet, closedSet, fScore, gScore, cameFrom)
   |> _find_path
 in
 _find_path (openSet, closedSet, fScore, gScore, cameFrom)

let () =

 let board = [|
   [| 1; 1; 1; 1; 1; 1; 1; 1; |];
   [| 1; 1; 1; 1; 1; 1; 1; 1; |];
   [| 1; 1; 1; 0; 0; 0; 1; 1; |];
   [| 1; 1; 1; 1; 1; 0; 1; 1; |];
   [| 1; 1; 0; 1; 1; 0; 1; 1; |];
   [| 1; 1; 0; 1; 1; 0; 1; 1; |];
   [| 1; 1; 0; 0; 0; 0; 1; 1; |];
   [| 1; 1; 1; 1; 1; 1; 1; 1; |];
 |] in
 let start = (0, 0) in
 let goal = (7, 7) in

 let dim_x = Array.length board.(0) in
 let dim_y = Array.length board in

 let r = find_path start goal board in

 match r with
 | None -> failwith "path not found"
 | Some p ->
     List.iter (fun (x, y) -> Printf.printf " (%d, %d)\n" x y) p;
     print_newline ();
     let _board =
       Array.init dim_y (fun y ->
         Array.init dim_x (fun x ->
           if board.(y).(x) = 0 then '#' else '.'))
     in
     List.iter (fun (x, y) -> _board.(y).(x) <- '*') p;

     Array.iter (fun line ->
       Array.iter (fun c ->
         Printf.printf " %c" c;
       ) line;
       print_newline ()
     ) _board;
     print_newline ()</lang>

Output:

 (0, 0)
 (1, 1)
 (2, 2)
 (2, 3)
 (1, 4)
 (1, 5)
 (1, 6)
 (2, 7)
 (3, 7)
 (4, 7)
 (5, 7)
 (6, 7)
 (7, 7)

 * . . . . . . .
 . * . . . . . .
 . . * # # # . .
 . . * . . # . .
 . * # . . # . .
 . * # . . # . .
 . * # # # # . .
 . . * * * * * *

Ol

<lang scheme>

level

list of lists, any except 1 means the cell is empty

from

start cell in (x . y) mean

to

destination cell in (x . y) mean

(define (A* level from to)

  (define (hash xy) ; internal hash
     (+ (<< (car xy) 16) (cdr xy)))

  ; "is the cell is empty?"
  (define (floor? x y)
     (let ((line (list-ref level y)))
        (if line (not (eq? (list-ref line x) 1)))))

  (unless (equal? from to) ; search not finished yet
     (let step1 ((n 999) ; maximal count of search steps
                 (c-list-set #empty)
                 (o-list-set (put #empty (hash from)  [from #f  0 0 0])))
        (unless (empty? o-list-set) ; do we have a space to move?
           ; no. let's find cell with minimal const
           (let*((f (ff-fold (lambda (s key value)
                                (if (< (ref value 5) (car s))
                                   (cons (ref value 5) value)
                                   s))
                       (cons 9999 #f) o-list-set))
                 (xy (ref (cdr f) 1))
                 ; move the cell from "open" to "closed" list
                 (o-list-set (del o-list-set (hash xy)))
                 (c-list-set (put c-list-set (hash xy) (cdr f))))

              ;
              (if (or (eq? n 0)
                      (equal? xy to))
                 (let rev ((xy xy))
                    ; let's unroll the math and return only first step
                    (let*((parent (ref (get c-list-set (hash xy) #f) 2))
                          (parent-of-parent (ref (get c-list-set (hash parent) #f) 2)))
                       (if parent-of-parent (rev parent)
                          (cons
                             (- (car xy) (car parent))
                             (- (cdr xy) (cdr parent))))))

                 (let*((x (car xy))
                       (y (cdr xy))
                       (o-list-set (fold (lambda (n v)
                                      (if (and
                                            (floor? (car v) (cdr v))
                                            (eq? #f (get c-list-set (hash v) #f)))
                                         (let ((G (+ (ref (get c-list-set (hash xy) #f) 3) 1)); G of parent + 1
                                               ; H calculated by "Manhattan method"
                                               (H (* (+ (abs (- (car v) (car to)))
                                                        (abs (- (cdr v) (cdr to))))
                                                     2))
                                               (got (get o-list-set (hash v) #f)))

                                            (if got
                                               (if (< G (ref got 3))
                                                  (put n (hash v)  [v xy  G H (+ G H)])
                                                  n)
                                               (put n (hash v)  [v xy  G H (+ G H)])))
                                         n))
                                      o-list-set (list
                                                     (cons x (- y 1))
                                                     (cons x (+ y 1))
                                                     (cons (- x 1) y)
                                                     (cons (+ x 1) y)))))
                    (step1 (- n 1) c-list-set o-list-set))))))))

</lang>

Output:

<lang scheme> (define level '(

  (1 1 1 1 1 1 1 1 1 1)
  (1 A 0 0 0 0 0 0 0 1)
  (1 0 0 0 0 0 0 0 0 1)
  (1 0 0 0 0 1 1 1 0 1)
  (1 1 0 0 0 0 0 1 0 1)
  (1 0 0 1 0 0 0 1 0 1)
  (1 0 0 1 1 1 1 1 0 1)
  (1 0 0 0 0 0 0 0 0 1)
  (1 0 0 0 1 0 0 0 B 1)
  (1 1 1 1 1 1 1 1 1 1)

)) (for-each print level)

let's check that we can't move to (into wall)

(print (A* level '(1 . 1) '(9 . 9)))

(define to '(8 . 8)) (define (plus a b) (cons (+ (car a) (car b)) (+ (cdr a) (cdr b)))) ; helper

(define path (let loop ((me '(1 . 1)) (path '()))

  (if (equal? me to)
     (begin
        (print "here I am!")
        (cons to path))
  (let ((move (A* level me to)))
     (unless move
        (begin
           (print "no way, sorry :(")
           #false)
        (let ((step (plus me move)))
           (print me " + " move " -> " step)
           (loop step (cons me path))))))))

let's draw the path?

(define (has? lst x) ; helper

  (cond
     ((null? lst) #false)
     ((equal? (car lst) x) lst)
     (else (has? (cdr lst) x))))

(define solved

  (map (lambda (row y)
        (map (lambda (cell x)
              (cond
                 ((equal? (cons x y) '(1 . 1)) "A")
                 ((equal? (cons x y) '(8 . 8)) "B")
                 ((has? path (cons x y)) "*")
                 (else cell)))
           row (iota 10)))
     level (iota 10)))

(for-each print solved) </lang>

the map:
(1 1 1 1 1 1 1 1 1 1)
(1 A 0 0 0 0 0 0 0 1)
(1 0 0 0 0 0 0 0 0 1)
(1 0 0 0 0 1 1 1 0 1)
(1 1 0 0 0 0 0 1 0 1)
(1 0 0 1 0 0 0 1 0 1)
(1 0 0 1 1 1 1 1 0 1)
(1 0 0 0 0 0 0 0 0 1)
(1 0 0 0 1 0 0 0 B 1)
(1 1 1 1 1 1 1 1 1 1)
we should not reach the '(9 . 9) cell:
#false
ok, we got #false, so really can't.
now try to reach cell '(8 . 8) - the 'B' point:
(1 . 1) + (0 . 1) -> (1 . 2)
(1 . 2) + (0 . 1) -> (1 . 3)
(1 . 3) + (1 . 0) -> (2 . 3)
(2 . 3) + (0 . 1) -> (2 . 4)
(2 . 4) + (0 . 1) -> (2 . 5)
(2 . 5) + (0 . 1) -> (2 . 6)
(2 . 6) + (0 . 1) -> (2 . 7)
(2 . 7) + (1 . 0) -> (3 . 7)
(3 . 7) + (1 . 0) -> (4 . 7)
(4 . 7) + (1 . 0) -> (5 . 7)
(5 . 7) + (0 . 1) -> (5 . 8)
(5 . 8) + (1 . 0) -> (6 . 8)
(6 . 8) + (1 . 0) -> (7 . 8)
(7 . 8) + (1 . 0) -> (8 . 8)
here I am!
(1 1 1 1 1 1 1 1 1 1)
(1 A 0 0 0 0 0 0 0 1)
(1 * 0 0 0 0 0 0 0 1)
(1 * * 0 0 1 1 1 0 1)
(1 1 * 0 0 0 0 1 0 1)
(1 0 * 1 0 0 0 1 0 1)
(1 0 * 1 1 1 1 1 0 1)
(1 0 * * * * 0 0 0 1)
(1 0 0 0 1 * * * B 1)
(1 1 1 1 1 1 1 1 1 1)

Phix

rows and columns are numbered 1 to 8. start position is {1,1} and end position is {8,8}. barriers are simply avoided, rather than costed at 100. Note that the 23 visited nodes does not count walls, but with them this algorithm exactly matches the 35 of Racket.

sequence grid = split("""
x:::::::
::::::::
::::###:
::#:::#:
::#:::#:
::#####:
::::::::
::::::::
""",'\n')
constant permitted = {{-1,-1},{0,-1},{1,-1},
                      {-1, 0},       {1, 0},
                      {-1, 1},{0,+1},{1,+1}}
sequence key = {7,0},   -- chebyshev, cost
         moves = {{1,1}},
         data = {moves},
         acta = {}      -- actually analysed set
setd(key,data)
bool found = false
integer count = 0
while not found do
    if dict_size()=0 then ?"impossible" exit end if
    key = getd_partial_key(0)
    data = getd(key)
    moves = data[$]
    if length(data)=1 then
        deld(key)
    else
        data = data[1..$-1]
        putd(key,data)
    end if
    count += 1
    acta = append(acta,moves[$])
    for i=1 to length(permitted) do
        sequence newpos = sq_add(moves[$],permitted[i])
        integer {nx,ny} = newpos
        if nx>=1 and nx<=8
        and ny>=1 and ny<=8
        and grid[nx,ny] = ':' then -- (unvisited)
            grid[nx,ny] = '.'
            sequence newkey = {max(8-nx,8-ny),key[2]+1},
                     newmoves = deep_copy(moves)
            newmoves = append(newmoves,newpos)
            if newpos = {8,8} then
                moves = newmoves
                found = true
                exit
            end if
            integer k = getd_index(newkey)
            if k=0 then
                data = {}
            else
                data = deep_copy(getd_by_index(k))
            end if
            data = append(data,newmoves)
            putd(newkey,data)
        end if
    end for
end while
if found then
    printf(1,"visited %d nodes\ncost:%d\npath:%v\n",{count,length(moves)-1,moves})
    for i=1 to length(acta) do
        integer {x,y} = acta[i]
        grid[x,y] = '_'
    end for
    for i=1 to length(moves) do
        integer {x,y} = moves[i]
        grid[x,y] = 'x'
    end for
    puts(1,join(grid,'\n'))
end if

Output:

visited 23 nodes
cost:11
path:{{1,1},{2,2},{3,3},{4,2},{5,2},{6,2},{7,3},{8,4},{8,5},{8,6},{8,7},{8,8}}
x......:
.x____.:
._x_###:
.x#___#:
.x#___#:
.x#####:
..x.....
:..xxxxx

The : represent nodes it did not even look at, the . those added but never gone back to, obviously x represent the path, and together _ and x all nodes actually analysed.

Extra credit

Well, why not. Note this does not reuse/share any code with the above, although I presume the task author assumed it would, instead the main loop uses a priority queue to obtain the next lowest cost and a simple dictionary to avoid re-examination/inifinte recursion.

--set_rand(3)    -- (for consistent output)
constant optimal = false,
         mtm = true,                    -- mutli-tile metrics
         target = {1,2,3,4,5,6,7,8,0},
             --   <-tile found 0..8->
         mcost = {{0,0,1,2,1,2,3,2,3},  -- position 1
                  {0,1,0,1,2,1,2,3,2},
                  {0,2,1,0,3,2,1,4,3},
                  {0,1,2,3,0,1,2,1,2},
                  {0,2,1,2,1,0,1,2,1},  -- ...
                  {0,3,2,1,2,1,0,3,2},
                  {0,2,3,4,1,2,3,0,1},
                  {0,3,2,3,2,1,2,1,0},
                  {0,4,3,2,3,2,1,2,1}}, -- position 9
         udlr = "udlr",
         dirs = {+3,-3,+1,-1},          -- udlr
         lims = {{9,9,9,9,9,9,9,9,9},   -- up
                 {1,1,1,1,1,1,1,1,1},   -- down
                 {3,3,3,6,6,6,9,9,9},   -- left
                 {1,1,1,4,4,4,7,7,7}}   -- right

function get_moves(sequence grid, bool mtm)
    sequence valid = {}
    integer p0 = find(0,grid)
    for dx=1 to length(dirs) do
        integer step = dirs[dx],
                lim = lims[dx][p0],
                count = 1
        integer i = p0+step
        while true do
            if step<0 then
                if i<lim then exit end if
            else
                if i>lim then exit end if
            end if
            valid = append(valid,{step,i,udlr[dx],count})
            if not mtm then exit end if
            count += 1
            i += step
        end while
    end for
    return valid
end function

function make_move(sequence grid, move)
    integer p0 = find(0,grid),
            {step,lim} = move
    grid = deep_copy(grid)
    integer i = p0+step
    while true do
        if step<0 then
            if i<lim then exit end if
        else
            if i>lim then exit end if
        end if
        grid[p0] = grid[i]
        grid[i] = 0
        p0 = i
        i += step
    end while
    return grid
end function

function manhattan(sequence grid)
    integer res = 0
    for i=1 to 9 do
        res += mcost[i][grid[i]+1]
    end for
    return res
end function

sequence problem, grid, new_grid,
         moves, next_moves, move

procedure show_grid()
    printf(1,"%s\n",join_by(sq_add(grid,'0'),1,3,""))
end procedure

grid = target
for i=1 to 1000 do
    -- (initially shuffle as if mtm==true, otherwise
    --  output compares answers to different puzzles)
    moves = get_moves(grid,true)
    move = moves[rand(length(moves))]
    grid = make_move(grid,move)
end for
problem = grid
printf(1,"problem (manhattan cost is %d):\n",manhattan(grid))
show_grid()
integer todo = pq_new(),
        seen = new_dict()
pq_add({{grid,{}},iff(optimal?0:manhattan(grid))},todo)
setd(grid,true,seen)
atom t1 = time()+1
bool found = false
integer count = 0, mc
while not found do
    if pq_size(todo)=0 then ?"impossible" exit end if
    {{grid,moves},mc} = pq_pop(todo)
    if time()>t1 then
        string m = iff(optimal?"moves":"manhattan")
        printf(1,"searching (count=%d, %s=%d)\r",{count,m,mc})
        t1 = time()+1
    end if
    next_moves = get_moves(grid,mtm)
    count += length(next_moves)
    integer l = length(moves)
    for i=1 to length(next_moves) do
        move = next_moves[i]
        new_grid = make_move(grid,move)
        mc = manhattan(new_grid)
        if mc=0 then
            if new_grid!=target then ?9/0 end if
            moves = append(moves,move)
            found = true
            exit
        end if
        if getd_index(new_grid,seen)=NULL then
            if optimal then mc = l+1 end if
            pq_add({{new_grid,append(deep_copy(moves),move)},mc},todo)
            setd(new_grid,true,seen)
        end if
    end for
end while
if found then
    string s = iff(length(moves)=1?"":"s")
    if optimal then
        s &= sprintf(" (max shd be %d)",iff(mtm?24:31))
    end if
    grid = problem
    string soln = ""
    for i=1 to length(moves) do
        move = moves[i]
        grid = make_move(grid,move)
        integer {{},{},ch,c} = move
        soln &= ch
        if c>1 then soln&='0'+c end if
--      show_grid() -- (set the initial shuffle to eg 5 first!)
    end for
--  show_grid() -- (not very educational!)
    if grid!=target then ?9/0 end if
    printf(1,"solved in %d move%s:%s\n",{length(moves),s,soln})
end if
printf(1,"count:%d, seen:%d, queue:%d\n",{count,dict_size(seen),pq_size(todo)})

Output:

Note: The solutions are non-optimal (far from it, in fact), since it searches lowest manhattan() first.
In fact that set_rand(3), used for all the results below, is somewhat worse than 0, 1, and 2, and the first to breach optimal limits, ie 31/24, but obviously only when the optimal flag is set to false, as well as being the first to hint at the potential thousand-fold-or-more performance gains on offer.
An optimal solution can instead be found by searching fewest moves first, albeit significantly slower! Note this approach is not really suitable for solving 15-puzzles (or larger).
with optimal false and mtm false:

problem (manhattan cost is 20):
546
807
321

solved in 88 moves:ulddruurdluldrdluurrddlurulldrrdlulurrddlurulldrdlururdllurrdlulddrurdlurdlulurrddlurull
count:592, seen:371, queue:155

with optimal false and mtm true:

solved in 45 moves:uld2r2u2l2d2r2u2ld2rul2dru2rdl2urdrdlu2rd2luruld2ru2l2dr2uldlu
count:328, seen:164, queue:82

with optimal true and mtm false:

solved in 26 moves (max shd be 31):rulldrdruulddruullddrruull
count:399996, seen:163976, queue:13728

with optimal true and mtm true:

solved in 17 moves (max shd be 24):rul2drdru2ld2ru2l2d2r2u2l2
count:298400, seen:106034, queue:31434

PowerShell

<lang powershell>function CreateGrid($h, $w, $fill) {

   $grid = 0..($h - 1) | ForEach-Object { , (, $fill * $w) }
   return $grid

}

function EstimateCost($a, $b) {

   $xd = [Math]::Abs($a.Item1 - $b.Item1)
   $yd = [Math]::Abs($a.Item2 - $b.Item2)
   return [Math]::Max($xd, $yd)

}

function AStar($costs, $start, $goal) {

   # ValueTuples can be used to index a Hashtable:
   $start = [ValueTuple]::Create($start[0], $start[1])
   $goal = [ValueTuple]::Create($goal[0], $goal[1])

   $rows = $costs.Length
   $cols = $costs[0].Length

   $cameFrom = CreateGrid $rows $cols $null
   $openSet = @{$start = (EstimateCost $start $goal), 0}
   $closedSet = @{}

   while ($openSet.Count -gt 0) {
       # find the value in openSet with the lowest fScore
       $curFScore = [int]::MaxValue

       foreach ($p in $openSet.Keys) {
           $fScore, $gScore = $openSet[$p]
           if ($fScore -lt $curFScore) {
               $curFScore = $fScore
               $curGScore = $gScore
               $cur = $p
           }
       }

       if ($cur -eq $goal) {
           $totalCost = $curGScore
           break
       }

       $openSet.Remove($cur)
       $closedSet.Add($cur, 0)
       $r, $c = $cur.Item1, $cur.Item2

       # iterate over each cell in the 3x3 neighborhood
       foreach ($i in [Math]::Max($r - 1, 0)..[Math]::Min($r + 1, $rows - 1)) {
           foreach ($j in [Math]::Max($c - 1, 0)..[Math]::Min($c + 1, $cols - 1)) {
               $neighbor = [ValueTuple]::Create($i, $j)
               if ($closedSet.ContainsKey($neighbor)) { continue }

               $newGScore = $curGScore + $costs[$i][$j]
               $newFScore = $newGScore + (EstimateCost $neighbor $goal)

               if (-not $openSet.ContainsKey($neighbor)) {
                   $openSet[$neighbor] = $newFScore, $newGScore
               }
               else {
                   $fs, $gs = $openSet[$neighbor]
                   if ($newGScore -ge $gs) { continue }
               }

               $cameFrom[$i][$j] = $cur
           }
       }
   }

   # Walk back from the goal
   $route = @(, ($goal.Item1, $goal.Item2))
   $cur = $goal

   while ($cur -ne $start) {
       $cur = $cameFrom[$cur.Item1][$cur.Item2]
       $route += , ($cur.Item1, $cur.Item2)
   }

   [array]::Reverse($route)
   return $route, $totalCost

}

$grid = CreateGrid 8 8 1 $grid[2][4] = 100 $grid[2][5] = 100 $grid[2][6] = 100 $grid[3][6] = 100 $grid[4][6] = 100 $grid[5][6] = 100 $grid[5][5] = 100 $grid[5][4] = 100 $grid[5][3] = 100 $grid[5][2] = 100 $grid[4][2] = 100 $grid[3][2] = 100

$route, $cost = AStar $grid (0, 0) (7, 7) $displayGrid = CreateGrid 8 8 '.'

foreach ($i in 0..7) {

   foreach ($j in 0..7) {
       if ($grid[$i][$j] -gt 1) {
           $displayGrid[$i][$j] = '#'
       }
   }

}

foreach ($step in $route) {

   $displayGrid[$step[0]][$step[1]] = 'x'

}

Write-Output ($displayGrid | ForEach-Object { $_ -join }) Write-Output "Cost: $cost" $routeString = ($route | ForEach-Object { "($($_[0]), $($_[1]))" }) -join ', ' Write-Output "Route: $routeString"</lang>

Output:

x.......
.x......
..x.###.
.x#...#.
.x#...#.
.x#####.
..x.x.x.
...x.x.x
Cost: 11
Route: (0, 0), (1, 1), (2, 2), (3, 1), (4, 1), (5, 1), (6, 2), (7, 3), (6, 4), (7, 5), (6, 6), (7, 7)

Python

<lang python>from __future__ import print_function import matplotlib.pyplot as plt

class AStarGraph(object): #Define a class board like grid with two barriers

def __init__(self): self.barriers = [] self.barriers.append([(2,4),(2,5),(2,6),(3,6),(4,6),(5,6),(5,5),(5,4),(5,3),(5,2),(4,2),(3,2)])

def heuristic(self, start, goal): #Use Chebyshev distance heuristic if we can move one square either #adjacent or diagonal D = 1 D2 = 1 dx = abs(start[0] - goal[0]) dy = abs(start[1] - goal[1]) return D * (dx + dy) + (D2 - 2 * D) * min(dx, dy)

def get_vertex_neighbours(self, pos): n = [] #Moves allow link a chess king for dx, dy in [(1,0),(-1,0),(0,1),(0,-1),(1,1),(-1,1),(1,-1),(-1,-1)]: x2 = pos[0] + dx y2 = pos[1] + dy if x2 < 0 or x2 > 7 or y2 < 0 or y2 > 7: continue n.append((x2, y2)) return n

def move_cost(self, a, b): for barrier in self.barriers: if b in barrier: return 100 #Extremely high cost to enter barrier squares return 1 #Normal movement cost

def AStarSearch(start, end, graph):

G = {} #Actual movement cost to each position from the start position F = {} #Estimated movement cost of start to end going via this position

#Initialize starting values G[start] = 0 F[start] = graph.heuristic(start, end)

closedVertices = set() openVertices = set([start]) cameFrom = {}

while len(openVertices) > 0: #Get the vertex in the open list with the lowest F score current = None currentFscore = None for pos in openVertices: if current is None or F[pos] < currentFscore: currentFscore = F[pos] current = pos

#Check if we have reached the goal if current == end: #Retrace our route backward path = [current] while current in cameFrom: current = cameFrom[current] path.append(current) path.reverse() return path, F[end] #Done!

#Mark the current vertex as closed openVertices.remove(current) closedVertices.add(current)

#Update scores for vertices near the current position for neighbour in graph.get_vertex_neighbours(current): if neighbour in closedVertices: continue #We have already processed this node exhaustively candidateG = G[current] + graph.move_cost(current, neighbour)

if neighbour not in openVertices: openVertices.add(neighbour) #Discovered a new vertex elif candidateG >= G[neighbour]: continue #This G score is worse than previously found

#Adopt this G score cameFrom[neighbour] = current G[neighbour] = candidateG H = graph.heuristic(neighbour, end) F[neighbour] = G[neighbour] + H

raise RuntimeError("A* failed to find a solution")

if __name__=="__main__": graph = AStarGraph() result, cost = AStarSearch((0,0), (7,7), graph) print ("route", result) print ("cost", cost) plt.plot([v[0] for v in result], [v[1] for v in result]) for barrier in graph.barriers: plt.plot([v[0] for v in barrier], [v[1] for v in barrier]) plt.xlim(-1,8) plt.ylim(-1,8) plt.show()</lang>

Output:

route [(0, 0), (1, 1), (2, 2), (3, 1), (4, 1), (5, 1), (6, 2), (7, 3), (6, 4), (7, 5), (6, 6), (7, 7)]
cost 11

Racket

This code is lifted from: this blog post. Read it, it's very good.

<lang racket>#lang scribble/lp @(chunk

 <graph-sig>
 (define-signature graph^
   (node? edge? node-edges edge-src edge-cost edge-dest)))

@(chunk

 <map-generation>
 (define (make-map N)
   ;; Jay's random algorithm
   ;; (build-matrix N N (λ (x y) (random 3)))
   ;; RC version
   (matrix [[0 0 0 0 0 0 0 0]
            [0 0 0 0 0 0 0 0]
            [0 0 0 0 1 1 1 0]
            [0 0 1 0 0 0 1 0]
            [0 0 1 0 0 0 1 0]
            [0 0 1 1 1 1 1 0]
            [0 0 0 0 0 0 0 0]
            [0 0 0 0 0 0 0 0]])))

@(chunk

 <map-graph-rep>
 (struct map-node (M x y) #:transparent)
 (struct map-edge (src dx dy dest)))

@(chunk

 <map-graph-cost>
 (define (edge-cost e)
   (match-define (map-edge _ _ _ (map-node M x y)) e)
   (match (matrix-ref M x y)
     [0  1]
     [1  100]
     [2 1000])))

@(chunk

 <map-graph-edges>
 (define (node-edges n)
   (match-define (map-node M x y) n)
   (append*
    (for*/list ([dx (in-list '(1 0 -1))]
                [dy (in-list '(1 0 -1))]
                #:when
                (and (not (and (zero? dx) (zero? dy)))
                     ;; RC -- allowed to move diagonally, so not this clause
                     ;;(or (zero? dx) (zero? dy))
                     ))
      (cond
        [(and (<= 0 (+ dx x) (sub1 (matrix-num-cols M)))
              (<= 0 (+ dy y) (sub1 (matrix-num-rows M))))
         (define dest (map-node M (+ dx x) (+ dy y)))
         (list (map-edge n dx dy dest))]
        [else
         empty])))))

@(chunk

 <a-star>
 (define (A* graph@ initial node-cost)
   (define-values/invoke-unit graph@ (import) (export graph^))
   (define count 0)
   <a-star-setup>

   (begin0
     (let/ec esc
       <a-star-loop>
       #f)

     (printf "visited ~a nodes\n" count))))

@(chunk

 <a-star-setup>
 <a-star-setup-closed>
 <a-star-setup-open>)

@(chunk

 <a-star-setup-closed>
 (define node->best-path (make-hash))
 (define node->best-path-cost (make-hash))
 (hash-set! node->best-path      initial empty)
 (hash-set! node->best-path-cost initial 0))

@(chunk

 <a-star-setup-open>
 (define (node-total-estimate-cost n)
   (+ (node-cost n) (hash-ref node->best-path-cost n)))
 (define (node-cmp x y)
   (<= (node-total-estimate-cost x)
       (node-total-estimate-cost y)))
 (define open-set (make-heap node-cmp))
 (heap-add! open-set initial))

@(chunk

 <a-star-loop>
 (for ([x (in-heap/consume! open-set)])
   (set! count (add1 count))
   <a-star-loop-body>))

@(chunk

 <a-star-loop-stop?>
 (define h-x (node-cost x))
 (define path-x (hash-ref node->best-path x))

 (when (zero? h-x)
   (esc (reverse path-x))))

@(chunk

 <a-star-loop-body>
 <a-star-loop-stop?>

 (define g-x (hash-ref node->best-path-cost x))
 (for ([x->y (in-list (node-edges x))])
   (define y (edge-dest x->y))
   <a-star-loop-per-neighbor>))

@(chunk

 <a-star-loop-per-neighbor>
 (define new-g-y (+ g-x (edge-cost x->y)))
 (define old-g-y
   (hash-ref node->best-path-cost y +inf.0))
 (when (< new-g-y old-g-y)
   (hash-set! node->best-path-cost y new-g-y)
   (hash-set! node->best-path y (cons x->y path-x))
   (heap-add! open-set y)))

@(chunk

 <map-display>
 (define map-scale 15)
 (define (type-color ty)
   (match ty
     [0 "yellow"]
     [1 "green"]
     [2 "red"]))
 (define (cell-square ty)
   (square map-scale "solid" (type-color ty)))
 (define (row-image M row)
   (apply beside
          (for/list ([col (in-range (matrix-num-cols M))])
            (cell-square (matrix-ref M row col)))))
 (define (map-image M)
   (apply above
          (for/list ([row (in-range (matrix-num-rows M))])
            (row-image M row)))))

@(chunk

 <path-display-line>
 (define (edge-image-on e i)
   (match-define (map-edge (map-node _ sx sy) _ _ (map-node _ dx dy)) e)
   (add-line i
             (* (+ sy 0.5) map-scale) (* (+ sx 0.5) map-scale)
             (* (+ dy 0.5) map-scale) (* (+ dx 0.5) map-scale)
             "black")))

@(chunk

 <path-display>
 (define (path-image M path)
   (foldr edge-image-on (map-image M) path)))

@(chunk

 <map-graph>
 (define-unit map@
   (import) (export graph^)

   (define node? map-node?)
   (define edge? map-edge?)
   (define edge-src map-edge-src)
   (define edge-dest map-edge-dest)

   <map-graph-cost>
   <map-graph-edges>))

@(chunk

 <map-node-cost>
 (define ((make-node-cost GX GY) n)
   (match-define (map-node M x y) n)
   ;; Jay's
   #;(+ (abs (- x GX))
        (abs (- y GY)))
   ;; RC -- diagonal movement
   (max (abs (- x GX))
        (abs (- y GY)))))

@(chunk

 <map-example>
 (define N 8)
 (define random-M
   (make-map N))
 (define random-path
   (time
    (A* map@
        (map-node random-M 0 0)
        (make-node-cost (sub1 N) (sub1 N))))))

@(chunk

 <*>
 (require rackunit
          math/matrix
          racket/unit
          racket/match
          racket/list
          data/heap
          2htdp/image
          racket/runtime-path)

 <graph-sig>

 <map-generation>
 <map-graph-rep>
 <map-graph>

 <a-star>

 <map-node-cost>
 <map-example>
 (printf "path is ~a long\n" (length random-path))
 (printf "path is: ~a\n" (map (match-lambda
                                [(map-edge src dx dy dest)
                                 (cons dx dy)])
                              random-path))

 <map-display>
 <path-display-line>
 <path-display>

 (path-image random-M random-path))</lang>

Output:

visited 35 nodes
cpu time: 94 real time: 97 gc time: 15
path is 11 long
path is: ((1 . 1) (1 . 1) (1 . -1) (1 . 0) (1 . 0) (1 . 1) (1 . 1) (0 . 1) (-1 . 1) (1 . 1) (0 . 1))
.

A diagram is also output, but you'll need to run this in DrRacket to see it.

Raku

Translation of: Sidef

<lang perl6># 20200427 Raku programming solution

class AStarGraph {

  has @.barriers =
     <2 4>,<2 5>,<2 6>,<3 6>,<4 6>,<5 6>,<5 5>,<5 4>,<5 3>,<5 2>,<4 2>,<3 2>;

  method heuristic(\start, \goal) {
     my (\D1,\D2) = 1, 1;
     my (\dx,\dy) = ( start.list »-« goal.list )».abs;
     return  (D1 * (dx + dy)) + (D2 - 2*D1) * min dx, dy
  }

  method get_vertex_neighbours(\pos) {
     gather {
        for <1 0>,<-1 0>,<0 1>,<0 -1>,<1 1>,<-1 1>,<1 -1>,<-1 -1> -> \d {
           my (\x2,\y2) = pos.list »+« d.list;
           (x2 < 0 || x2 > 7 || y2 < 0 || y2 > 7) && next;
           take x2, y2;
        }
     }
  }

  method move_cost(\a,\b) { (b ~~ any self.barriers) ?? 100 !! 1 }

}

sub AStarSearch(\start, \end, \graph) {

  my (%G,%F);

  %G{start.Str} = 0;
  %F{start.Str} = graph.heuristic(start, end);

  my @closedVertices = [];
  my @openVertices = [].push(start);
  my %cameFrom;

  while (@openVertices.Bool) {
     my $current = Nil; my $currentFscore = Inf;

     for @openVertices -> \pos {
        if (%F{pos.Str} < $currentFscore) {
           $currentFscore = %F{pos.Str};
           $current = pos
        }
     }

     if $current ~~ end {
        my @path = [].push($current);
        while %cameFrom{$current.Str}:exists {
           $current = %cameFrom{$current.Str};
           @path.push($current)
        }
        return @path.reverse, %F{end.Str}
     }

     @openVertices .=  grep: * !eqv $current;
     @closedVertices.push($current);

     for (graph.get_vertex_neighbours($current)) -> \neighbour {
        next if neighbour ~~ any @closedVertices;
        my \candidateG = %G{$current.Str}+graph.move_cost($current,neighbour);

        if !(neighbour ~~ any @openVertices) {
           @openVertices.push(neighbour)
        } elsif (candidateG ≥ %G{neighbour.Str}) {
           next
        }

        %cameFrom{neighbour.Str} = $current;
        %G{neighbour.Str} = candidateG;
        my \H = graph.heuristic(neighbour, end);
        %F{neighbour.Str} = %G{neighbour.Str} + H;
     }
  }
  die "A* failed to find a solution"

}

my \graph = AStarGraph.new; my (\route, \cost) = AStarSearch(<0 0>, <7 7>, graph);

my \w = my \h = 10;

my @grid = [ ['.' xx w ] xx h ]; for ^h -> \y { @grid[y;0] = "█"; @grid[y;*-1] = "█" } for ^w -> \x { @grid[0;x] = "█"; @grid[*-1;x] = "█" }

for (graph.barriers) -> \d { @grid[d[0]+1][d[1]+1] = "█" } for @(route) -> \d { @grid[d[0]+1][d[1]+1] = "x" }

.join.say for @grid ;

say "Path cost : ", cost, " and route : ", route;</lang>

Output:

██████████
█x.......█
█.x......█
█..x.███.█
█.x█...█.█
█.x█...█.█
█.x█████.█
█..xxxxx.█
█.......x█
██████████

Path cost : 11 and route : ((0 0) (1 1) (2 2) (3 1) (4 1) (5 1) (6 2) (6 3) (6 4) (6 5) (6 6) (7 7))

REXX

<lang rexx>/*REXX program solves the A* search problem for a (general) NxN grid. */ parse arg N sCol sRow . /*obtain optional arguments from the CL*/ if N== | N=="," then N=8 /*No grid size specified? Use default.*/ if sCol== | sCol=="," then sCol=1 /*No starting column given? " " */ if sRow== | sRow=="," then sRow=1 /* " " row " " " */ beg= '─0─' /*mark the start of the journey in grid*/ o.=.; p.=0 /*list of optimum start journey starts.*/ times=0 /*cntr/pos for number of optimizations.*/

             Pc = ' 1  1  0  0   1 -1 -1 -1 '   /*the possible column moves for a path.*/
             Pr = ' 1  0  1 -1  -1  0  1 -1 '   /* "      "     row     "    "  "   "  */

Pcm=words(Pc) /* [↑] optimized for moving right&down*/ $.=1e6; OK=0; min$=$. /*# possible directions; cost; solution*/ @Aa= " A* search algorithm on" /*a handy─dandy literal for the SAYs. */ flasher= '@. $. min$ N o. p. Pc. Pcm Pr. sCol sRow times' /*a literal list for EXPOSE.*/ call path 0 /*find a possible solution for the grid*/ @NxN= 'a ' N"x"N ' grid' /*a literal used for a SAY statement.*/ if OK then say 'A solution for the' @Aa @NxN "with a score of " @.N.N':'

      else say 'No'   @Aa   "solution for"     @NxN'.'

call show 1 /*invoke subroutine to display the grid*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ @: parse arg x,y,aChar; if arg()==3 then @.x.y=aChar; return @.x.y @p: parse arg x,y; if datatype(@.x.y, 'W') then return @.x.y<m-1; return 0 /*──────────────────────────────────────────────────────────────────────────────────────*/ barr: $=2.4 2.5 2.6 3.6 4.6 5.6 5.5 5.4 5.3 5.2 4.2 3.2 /*locations of barriers on grid*/

        do b=1  for words($);    _=word($, b);   parse var _ c '.' r;  call @ c+1,r+1,"█"
        end   /*b*/;             return

/*──────────────────────────────────────────────────────────────────────────────────────*/ move: procedure expose (flasher); parse arg m,col,row /*obtain move,col,row.*/

        do t=1  for Pcm;         nc=col + Pc.t;   nr=row + Pr.t /*a new path position. */
        if @.nc.nr==.  then do;  if opti()  then iterate        /*Costlier path?  Next.*/
                                 @.nc.nr=m;       p.1.m=nc nr   /*Empty?  A legal path.*/
                                 p.pcm.m=nr nc-1                /*used for a fast path.*/
                                 if nc==N  then if nr==N  then return 1   /*last move? */
                                 if move(m + 1,  nc, nr)  then return 1   /*  "    "   */
                                 @.nc.nr=.                      /*undo the above move. */
                            end                                 /*try a different move.*/
        end   /*t*/                                             /* [↑]  all moves tried*/
     return 0                                                   /*path isn't possible. */

/*──────────────────────────────────────────────────────────────────────────────────────*/ opti: ncm=nc-1; nrm=nr-1; if @p(ncm, nrm) then return 1

                                 if @p(ncm, nr )  then return 1
                                 if @p(nc,  nrm)  then return 1
     ncp=nc+1;   nrp=nr+1;       if @p(ncp, nr )  then return 1
                                 if @p(ncp, nrm)  then return 1
                                 if @p(nc,  nrp)  then return 1
                                 if @p(ncm, nrp)  then return 1
                                 if @p(ncp, nrp)  then return 1;         return 0

/*──────────────────────────────────────────────────────────────────────────────────────*/ path: parse arg z; t=times /*initial move can only be one of eight*/

       do #=1  for Pcm;          @.=            /*optimize for each degree of movement.*/
       if z\==0  then  if #\==z  then iterate   /*This a particular low─cost request ? */
            do c=1  for  N;    do r=1  for N;   @.c.r=.;   end  /*r*/
            end   /*c*/
       iCol=sCol;  iRow=sRow;  @.sCol.sRow= beg /*all path's initial starting  position*/
       call barr                                /*place the barriers on the grid.      */
       Pco=subword(Pc Pc, #, Pcm);  Pro=subword(Pr Pr, #, Pcm)
       parse var  Pco   Pc.1 Pc.2 Pc.3 Pc.4 Pc.5 Pc.6 Pc.7 Pc.8  /*possible directions.*/
       parse var  Pro   Pr.1 Pr.2 Pr.3 Pr.4 Pr.5 Pr.6 Pr.7 Pr.8  /*    "         "     */
            do o=1  for times;  parse var o.o c r;    @.c.r=o;     iRow=r;     iCol=c
            end   /*o*/
       fp=move(1+times, iCol, iRow);      sol=@N.N\==. & fp
       if sol  then do;    $.#=@.N.N            /*Found a solution?  Remember the cost.*/
                    OK=1;  min$=min(min$, $.#)
                    end
       end   /*#*/
     wp=1e7; wg=0;  do g=1  for Pcm; if $.g<wp & $.g>0 & t\=2  then do; wg=g; wp=$.g; end
                    end   /*g*/                 /* [↑]  find minimum non-zero path cost*/
     if wg==0  then wg=8                        /*Not found?  Then use last cost found.*/
     times=times + 1                            /*bump # times a marker has been placed*/
     o.times= p.wg.times                        /*remember this move location for PATH.*/
     if times<4  then call path 0               /*only do memoization for first 3 moves*/
     return

/*──────────────────────────────────────────────────────────────────────────────────────*/ show: ind=left(, 9 * (n<18) ); say /*the indentation of the displayed grid*/

     _=substr(copies("┼───", N),2);  say ind translate('┌'_"┐", '┬', "┼")   /*grid top.*/
                                                /* [↓]  build a display for the grid.  */
      do   c=1  for N;          if c\==1 & arg(1)  then say  ind  '├'_"┤";     L=@.
        do r=1  for N; ?=@.c.r; if c ==N & r==N & ?\==.  then ?='end'; L=L"│"center(?, 3)
        end   /*r*/                             /*done with   rank   of the grid.      */
      say ind translate(L'│', , .)              /*display a     "     "  "    "        */
      end     /*c*/                             /*a 19x19 grid can be shown 80 columns.*/
    say ind translate('└'_"┘",'┴',"┼");  return /*display the very bottom of the grid. */</lang>

output   when using the default input:

A solution for the  A*  search algorithm on a  8x8  grid with a score of  11:

          ┌───┬───┬───┬───┬───┬───┬───┬───┐
          │─0─│   │   │   │   │   │   │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │ 1 │   │   │   │   │   │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │ 2 │   │ █ │ █ │ █ │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │ 3 │ █ │   │   │   │ █ │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │ 4 │ █ │   │   │   │ █ │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │ 5 │ █ │ █ │ █ │ █ │ █ │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │ 6 │   │   │   │   │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │   │ 7 │ 8 │ 9 │10 │end│
          └───┴───┴───┴───┴───┴───┴───┴───┘

SequenceL

<lang sequencel> import <Utilities/Set.sl>; import <Utilities/Math.sl>; import <Utilities/Sequence.sl>;

Point ::= (x : int, y : int);

State ::= (open : Point(1), closed : Point(1), cameFrom : Point(2), estimate : int(2), actual : int(2));

allNeighbors := [(x : -1, y : -1), (x : 1, y : -1), (x : -1, y : 1), (x : 1, y : 1), (x : 0, y : -1), (x : -1, y : 0), (x : 0, y : 1), (x : 1, y : 0)];

defaultBarriers := [(x : 3, y : 5),(x : 3, y : 6),(x : 3, y : 7),(x : 4, y : 7), (x : 5, y : 7),(x : 6, y : 7),(x : 6, y : 6),(x : 6, y : 5),(x : 6, y : 4), (x : 6, y : 3),(x : 5, y : 3),(x : 4, y : 3)];

defaultWidth := 8; defaultHeight := 8;

main(args(2)) := aStar(defaultWidth, defaultHeight, defaultBarriers, (x : 1, y : 1), (x : defaultWidth, y : defaultHeight));

aStar(width, height, barriers(1), start, end) := let newEstimate[i,j] := heuristic(start, end) when i = start.x and j = start.y else 0 foreach i within 1...width, j within 1 ... height; newActual[i,j] := 0 foreach i within 1...width, j within 1...height; newCameFrom[i,j] := (x : 0, y : 0) foreach i within 1...width, j within 1...height;

searchResults := search((open : [start], closed : [], estimate : newEstimate, actual : newActual, cameFrom : newCameFrom), barriers, end); shortestPath := path(searchResults.cameFrom, start, end) ++ [end]; in "No Path Found" when size(searchResults.open) = 0 else "Path: " ++ toString(shortestPath) ++ "\nCost:" ++ toString(searchResults.actual[end.x, end.y]) ++ "\nMap:\n" ++ join(appendNT(drawMap(barriers,shortestPath,width, height),"\n"));

path(cameFrom(2), start, current) := let next := cameFrom[current.x, current.y]; in [] when current = start else path(cameFrom, start, next) ++ [next];

drawMap(barriers(1), path(1), width, height)[i,j] := '#' when elementOf((x:i, y:j), barriers) else 'X' when elementOf((x:i, y:j), path) else '.' foreach i within 1 ... width, j within 1 ... height;

search(state, barriers(1), end) := let nLocation := smallestEstimate(state.open, state.estimate, 2, 1, state.estimate[state.open[1].x, state.open[1].y]); n := state.open[nLocation]; neighbors := createNeighbors(n, allNeighbors, size(state.actual), size(state.actual[1])); startState := (open : state.open[1...nLocation-1] ++ state.open[nLocation+1 ... size(state.open)], closed : state.closed ++ [n], cameFrom : state.cameFrom, estimate : state.estimate, actual : state.actual); newState := findOpenNeighbors(n, startState, barriers, end, neighbors); in state when size(state.open) = 0 else state when n = end else search(newState, barriers, end);

smallestEstimate(open(1), estimate(2), index, minIndex, minEstimate) := let newEstimate := estimate[open[index].x, open[index].y]; in minIndex when index > size(open) else smallestEstimate(open, estimate, index + 1, minIndex, minEstimate) when newEstimate > minEstimate else smallestEstimate(open, estimate, index + 1, index, newEstimate);

findOpenNeighbors(n, state, barriers(1), end, neighbors(1)) := let neighbor := head(neighbors); cost := 1 + n.cost; candidate := state.actual[n.x, n.y] + calculateCost(barriers, n, neighbor); in state when size(neighbors) = 0 else findOpenNeighbors(n, state, barriers, end, tail(neighbors)) when elementOf(neighbor, state.closed) else findOpenNeighbors(n, state, barriers, end, tail(neighbors)) when elementOf(neighbor, state.open) and candidate >= state.actual[neighbor.x, neighbor.y] else findOpenNeighbors(n, (open : state.open ++ [neighbor], closed : state.closed, cameFrom : setMap(state.cameFrom, neighbor, n), estimate : setMap(state.estimate, neighbor, candidate + heuristic(neighbor, end)), actual : setMap(state.actual, neighbor, candidate)), barriers, end, tail(neighbors));

createNeighbors(n, p, w, h) := let x := n.x + p.x; y := n.y + p.y; in (x : x, y : y) when x >= 1 and x <= w and y >= 1 and y <= h;

calculateCost(barriers(1), start, end) := 100 when elementOf(end, barriers) else 1;

heuristic(start, end) := let dx := abs(start.x - end.x); dy := abs(start.y - end.y); in (dx + dy) - min(dx, dy);

setMap(map(2), point, value)[i,j] := value when point.x = i and point.y = j else map[i,j] foreach i within 1 ... size(map), j within 1 ... size(map[1]); </lang>

Output  

Path: [(x:1,y:1),(x:2,y:2),(x:3,y:3),(x:4,y:2),(x:5,y:2),(x:6,y:2),(x:7,y:3),(x:7,y:4),(x:7,y:5),(x:7,y:6),(x:7,y:7),(x:8,y:8)]
Cost:11
Map:
X.......
.X......
..X.###.
.X#...#.
.X#...#.
.X#####.
..XXXXX.
.......X

Sidef

Translation of: Python

<lang ruby>class AStarGraph {

   has barriers = [
       [2,4],[2,5],[2,6],[3,6],[4,6],[5,6],[5,5],[5,4],[5,3],[5,2],[4,2],[3,2]
   ]

   method heuristic(start, goal) {
       var (D1 = 1, D2 = 1)
       var dx = abs(start[0] - goal[0])
       var dy = abs(start[1] - goal[1])
       (D1 * (dx + dy)) + ((D2 - 2*D1) * Math.min(dx, dy))
   }

   method get_vertex_neighbours(pos) {
       gather {
           for dx, dy in [[1,0],[-1,0],[0,1],[0,-1],[1,1],[-1,1],[1,-1],[-1,-1]] {
               var x2 = (pos[0] + dx)
               var y2 = (pos[1] + dy)
               (x2<0 || x2>7 || y2<0 || y2>7) && next
               take([x2, y2])
           }
       }
   }

   method move_cost(_a, b) {
       barriers.contains(b) ? 100 : 1
   }

}

func AStarSearch(start, end, graph) {

   var G = Hash()
   var F = Hash()

   G{start} = 0
   F{start} = graph.heuristic(start, end)

   var closedVertices = []
   var openVertices = [start]
   var cameFrom = Hash()

   while (openVertices) {

       var current = nil
       var currentFscore = Inf

       for pos in openVertices {
           if (F{pos} < currentFscore) {
               currentFscore = F{pos}
               current = pos
           }
       }

       if (current == end) {
           var path = [current]
           while (cameFrom.contains(current)) {
               current = cameFrom{current}
               path << current
           }
           path.flip!
           return (path, F{end})
       }

       openVertices.remove(current)
       closedVertices.append(current)

       for neighbour in (graph.get_vertex_neighbours(current)) {
           if (closedVertices.contains(neighbour)) {
               next
           }
           var candidateG = (G{current} + graph.move_cost(current, neighbour))

           if (!openVertices.contains(neighbour)) {
               openVertices.append(neighbour)
           }
           elsif (candidateG >= G{neighbour}) {
               next
           }

           cameFrom{neighbour} = current
           G{neighbour} = candidateG
           var H = graph.heuristic(neighbour, end)
           F{neighbour} = (G{neighbour} + H)
       }
   }

   die "A* failed to find a solution"

}

var graph = AStarGraph() var (route, cost) = AStarSearch([0,0], [7,7], graph)

var w = 10 var h = 10

var grid = h.of { w.of { "." } } for y in (^h) { grid[y][0] = "█"; grid[y][-1] = "█" } for x in (^w) { grid[0][x] = "█"; grid[-1][x] = "█" }

for x,y in (graph.barriers) { grid[x+1][y+1] = "█" } for x,y in (route) { grid[x+1][y+1] = "x" }

grid.each { .join.say }

say "Path cost #{cost}: #{route}"</lang>

Output:

██████████
█x.......█
█.x......█
█..x.███.█
█.x█...█.█
█.x█...█.█
█.x█████.█
█..xxxxx.█
█.......x█
██████████
Path cost 11: [[0, 0], [1, 1], [2, 2], [3, 1], [4, 1], [5, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6], [7, 7]]

UNIX Shell

Works with: Bourne Again SHell

<lang bash>

1. !/bin/bash

1. This option will make the script exit when there is an error

set -o errexit

1. This option will make the script exit when it tries to use an unset variable

set -o nounset

declare -A grid declare -A cell_type=(

 ["empty"]=0 ["barrier"]=1
 ["start"]=2 ["end"]=3
 ["path"]=4
 ["right"]=5     ["left"]=6
 ["up"]=7        ["down"]=8
 ["left_up"]=9   ["left_down"]=10
 ["right_up"]=11 ["right_down"]=12
 )

grid_size=(10 10)

generate_rosetta_grid(){

 grid_size=(8 8)
 start=(0 0)
 end=(7 7)
 for (( i = 0; i < grid_size[0]; i++ )); do
   for (( j = 0; j < grid_size[1]; j++ )); do
     grid[$i,$j]=${cell_type[empty]}
   done
 done
 barriers=( "2,4" "2,5" "2,6" "3,6" "4,6" "5,6" "5,5" "5,4" "5,3" "5,2" "4,2" "3,2")
 for barrier in ${barriers[*]};do
   grid["$barrier"]=${cell_type[barrier]}
 done
 grid[${start[0]},${start[1]}]=${cell_type[start]}
 grid[${end[0]},${end[1]}]=${cell_type[end]}

}

abs(){

 # Number asbolute value.
 # Params:
 # ------
 # $1 -> number
 # Return:
 # number abs
 if $1 -gt 0 ;
 then
   echo "$1"
 else
   echo "$((-$1))"
 fi

}

print_table(){

 # Print table using unicode symbols.
 # Symbols:
 # " " -> empty cell
 # ◼ -> barrier
 # ◉ -> start position
 # ✪ -> goal
 # arrows -> path from start to goal
 printf ' '
 # Print letters at top.
 for ((i=0;i< grid_size[1];i++)) do
     printf "%s" $i
 done
 echo
 for ((i=0;i < grid_size[0];i++)) do
     # Print numbers.
     printf "%s" $i
     for ((j=0;j < grid_size[1];j++)) do
         cell=${grid[$i,$j]}
         if [[ $cell  -eq ${cell_type[empty]} ]];
         then
           # If cell is empty prints space
           printf " "
         elif [[ $cell -eq ${cell_type[barrier]} ]]; then
           # If cell is a barrier
           printf "■"
         elif [[ $cell -eq ${cell_type[start]} ]]; then
           # Print start and end position
           printf "◉"
         elif [[ $cell -eq ${cell_type[end]} ]]; then
           # Print end position
           printf "✪"
         elif [[ $cell -eq ${cell_type[path]} ]]; then
           # Print path
           printf "*"
         elif [[ $cell -eq ${cell_type[up]} ]]; then
           # Print path
           printf "↑"
         elif [[ $cell -eq ${cell_type[down]} ]]; then
           # Print path
           printf "↓"
         elif [[ $cell -eq ${cell_type[right]} ]]; then
           # Print path
           printf "→"
         elif [[ $cell -eq ${cell_type[left]} ]]; then
           # Print path
           printf "←"
         elif [[ $cell -eq ${cell_type[right_up]} ]]; then
           # Print path
           printf "↗"
         elif [[ $cell -eq ${cell_type[right_down]} ]]; then
           # Print path
           printf "↙"
         elif [[ $cell -eq ${cell_type[left_up]} ]]; then
           # Print path
           printf "↖"
         elif [[ $cell -eq ${cell_type[left_down]} ]]; then
           # Print path
           printf "↘"
         fi
     done
     echo
 done

}

get_neighbours(){

 # Calculates all point's neighbours
 # Params:
 # ------
 # $1 -> "x,y" formatted point position
 # Return:
 # ------
 # array of available positions
 # Skips nonexistent indices.

 neighbours=()
 for i in {-1..1},{-1..1}; do
   if [[ ( ${i%,*} -eq 0 ) && ( ${i#*,} -eq 0 ) ]]; then
     continue
   fi
   dx=${i%,*}
   dy=${i#*,}
   x=$((${1%,*}+dx))
   y=$((${1#*,}+dy))
   if $x -lt 0  || [[ $x -ge ${grid_size[0]} ]];
   then
     continue
   fi
   if | $y -ge ${grid_size[1]} ;
   then
     continue
   fi
   neighbours+=("$x,$y")
 done
 echo "${neighbours[*]}"

}

move_cost(){

 # Calculates how much will it cost
 # to travel to point b.
 # return 100 if b is barrier
 #
 # Params:
 # ------
 # $1 -> a
 # $2 -> b
 # Return:
 # ------
 # movement cost.

 barrier=${cell_type[barrier]}
 if [[ ${grid[${2%,*},${2#*,}]} -eq barrier ]];
 then
   echo 100
 else
   echo 1
 fi

}

print_raw(){

 # Print raw grid values.

 for ((i=0;i < grid_size[0];i++)) do
     for ((j=0;j < grid_size[1];j++)) do
       printf "%s" "${grid[$i,$j]}"
     done
     echo
 done

}

minimum(){

 # Minimum between two numbers
 # Params:
 # ------
 # $1 -> a
 # $2 -> b
 # Return:
 # ------
 # less value

 if $1 -lt $2 ;
 then
   echo "$1"
 else
   echo "$2"
 fi

}

heuristic_cost(){

 # Chebyshev distance heuristic score
 # if we can move one square either
 # adjacent or diagonal

 d=1
 d2=1
 dx=$(abs $((${1#*,} - ${2#*,})))
 dy=$(abs $((${1%,*} - ${2%,*})))
 echo "$(((d*(dx + dy))+(d2 - 2 * d)*$(minimum dx dy)))"

}

contains(){

 for el in "${2[@]}"; do
   echo "$el"
 done

}

contains_value() {

   # Check if element exists in array
   # Params:
   # ------
   # $1 -> array
   # $2 -> element to find.
   # Returns:
   # 1 if element exists in array
   # 0 otherwise.

   local array="$1[@]"
   arr=("${!array}")
   local seeking=$2
   local in=0
   for element in ${arr[*]}; do
       if [ "$element" = "$seeking" ]; then
           in=1
           break
       fi
   done
   echo "$in"

}

reverse_array(){

 # Reverse given array.
 # Params:
 # ------
 # $1 -> array
 # Return:
 # ------
 # reversed array.
 local array="$1[@]"
 arr=("${!array}")
 result=()
 for (( idx=${#arr[@]}-1 ; idx>=0 ; idx-- )) ; do
   result+=("${arr[$idx]}")
 done
 echo "${result[@]}"

}

find_path(){

 declare -A fScore
 declare -A gScore
 declare -A cameFrom
 declare -a openVertices
 declare -a closedVertices
 for (( i = 0; i < grid_size[0]; i++ )); do
   for (( j = 0; j < grid_size[1]; j++ )); do
     gScore[$i,$j]=$((1<<62))
     fScore[$i,$j]=$((1<<62))
   done
 done
 gScore["${start[0]},${start[1]}"]=0
 fScore["${start[0]},${start[1]}"]=$(heuristic_cost "${start[0]},${start[1]}" "${end[0]},${end[1]}")
 openVertices+=("${start[0]},${start[1]}")

 while [[ -n "${openVertices[*]}" ]]; do

   current=-1
   currentFscore=0
   for pos in ${openVertices[*]}; do
     if $current -eq -1  ||
        [[ ${fScore["$pos"]} -lt $currentFscore ]]; then
       currentFscore=${fScore["$pos"]}
       current=$pos
     fi
   done
   if [[ "$current" = "${end[0]},${end[1]}" ]]; then
     path=( "$current" )
     while [ ${cameFrom["$current"]+_} ]; do
       current=${cameFrom["$current"]}
       path+=("$current")
     done
     reverse_array path
     return 0
   fi
   openVertices=( "$( echo "${openVertices[@]/$current}" | xargs )" )
   closedVertices+=( "$current" )
   neighbours=( "$(get_neighbours "$current")" )

   for neighbour in ${neighbours[*]}; do
     if $(contains_value closedVertices "$neighbour") -eq 1 ; then
       continue
     fi
     mCost="$(move_cost "$current" "$neighbour")"
     candidateG=$(( ${gScore["$current"]}+mCost ))
     if $candidateG -gt 100 ; then
       continue
     fi
     if $(contains_value openVertices "$neighbour") -eq 0 ; then
       openVertices+=("$neighbour")
     elif [[ $candidateG -gt ${gScore[$neighbour]} ]]; then
       continue
     fi
     cameFrom["$neighbour"]="$current"
     gScore["$neighbour"]=$candidateG
     heuristic_score=$(heuristic_cost "$neighbour" "${end[0]},${end[1]}")
     fScore["$neighbour"]=$(( candidateG+heuristic_score ))
   done
 done

}

map_to_arrows(){

 local array="$1[@]"
 arr=("${!array}")
 last="${start[0]},${start[1]}"
 for el in ${arr[*]}; do
   if   [[ $((${el#*,}-${last#*,})) -eq -1 ]] &&
        [[ $((${el%,*}-${last%,*})) -eq -1 ]]; then
     grid["$last"]=${cell_type[left_up]}
   elif [[ $((${el#*,}-${last#*,})) -eq -1 ]] &&
        [[ $((${el%,*}-${last%,*})) -eq 1 ]]; then
          grid["$last"]=${cell_type[right_down]}
   elif [[ $((${el#*,}-${last#*,})) -eq 1 ]] &&
        [[ $((${el%,*}-${last%,*})) -eq -1 ]]; then
     grid["$last"]=${cell_type[right_up]}
   elif [[ $((${el#*,}-${last#*,})) -eq 1 ]] &&
        [[ $((${el%,*}-${last%,*})) -eq 1 ]]; then
     grid["$last"]=${cell_type[left_down]}
   elif [[ $((${el#*,}-${last#*,})) -eq -1 ]];then
     grid["$last"]=${cell_type[left]}
   elif [[ $((${el%,*}-${last%,*})) -eq -1 ]];then
     grid["$last"]=${cell_type[up]}
   elif [[ $((${el#*,}-${last#*,})) -eq 1 ]];then
     grid["$last"]=${cell_type[right]}
   elif [[ $((${el%,*}-${last%,*})) -eq 1 ]];then
     grid["$last"]=${cell_type[down]}
   else
     grid["$last"]=${cell_type[path]}
   fi
   last=$el
 done
 grid[${start[0]},${start[1]}]=${cell_type[start]}
 grid[${end[0]},${end[1]}]=${cell_type[end]}

}

main(){

 generate_rosetta_grid
 path=( "$(find_path)" )
 pstr="$(echo "${path[*]}" | xargs | sed "s/space:/ → /g")"
 echo path: "$pstr"
 if -z $pstr ; then
   echo "No path found."
 else
   map_to_arrows path
   print_table
 fi

}

main "$@"

</lang>

Output:

path: 0,0 → 1,0 → 2,0 → 3,0 → 4,0 → 5,1 → 6,2 → 7,3 → 7,4 → 7,5 → 7,6 → 7,7
 01234567
0◉
1↓
2↓   ■■■
3↓ ■   ■
4↘ ■   ■
5 ↘■■■■■
6  ↘
7   →→→→✪

Wren

Translation of: Sidef

<lang ecmascript>var Equals = Fn.new { |p1, p2| p1[0] == p2[0] && p1[1] == p2[1] }

var Contains = Fn.new { |pairs, p|

   for (pair in pairs) {
       if (Equals.call(p, pair)) return true
   }
   return false

}

var Remove = Fn.new { |pairs, p|

   for (pair in pairs) {
       if (Equals.call(p, pair)) {
           pairs.remove(pair)
           return
       }
   }

}

class AStarGraph {

   construct new() {
       _barriers = [[2,4], [2,5], [2,6], [3,6], [4,6], [5,6], [5,5], [5,4], [5,3], [5,2], [4,2], [3,2]]
   }

   barriers { _barriers }

   heuristic(start, goal) {
       var D1 = 1
       var D2 = 1
       var dx = (start[0] - goal[0]).abs
       var dy = (start[1] - goal[1]).abs
       return D1 * (dx + dy) + (D2 - 2*D1) * dx.min(dy)
   }

   getVertexNeighbors(pos) {
       var n = []
       for (d in [[1,0], [-1,0], [0,1], [0,-1], [1,1], [-1,1], [1,-1], [-1,-1]]) {
           var x2 = pos[0] + d[0]
           var y2 = pos[1] + d[1]
           if (x2 < 0 || x2 > 7 || y2 < 0 || y2 > 7) continue
           n.add([x2, y2])
       }
       return n
   }

   moveCost(b) { Contains.call(_barriers, b) ? 100 : 1 }

}

var AStarSearch = Fn.new { |start, end, graph|

   var G = {start.toString: 0}
   var F = {start.toString: graph.heuristic(start, end)}
   var closedVertices = []
   var openVertices = [start]
   var cameFrom  = {}
   while (openVertices.count > 0) {
       var current = null
       var currentFscore = 1 / 0
       for (pos in openVertices) {
           var v
           if ((v = F[pos.toString]) && v && v < currentFscore) {
               currentFscore = v
               current = pos
           }
       }
       if (Equals.call(current, end)) {
           var path = [current]
           while (cameFrom.containsKey(current.toString)) {
               current = cameFrom[current.toString]
               path.add(current)
           }
           path = path[-1..0]
           return [path, F[end.toString]]
       }
       Remove.call(openVertices, current)
       closedVertices.add(current)
       for (neighbor in graph.getVertexNeighbors(current)) {
           if (Contains.call(closedVertices, neighbor)) continue
           var candidateG = G[current.toString] + graph.moveCost(neighbor)
           if (!Contains.call(openVertices, neighbor)) {
               openVertices.add(neighbor)
           } else if (candidateG >= G[neighbor.toString]) continue
           cameFrom[neighbor.toString] = current
           G[neighbor.toString] = candidateG
           var H = graph.heuristic(neighbor, end)
           F[neighbor.toString] = G[neighbor.toString] + H
       }
   }
   Fiber.abort("A* failed to find a solution")

}

var graph = AStarGraph.new() var rc = AStarSearch.call([0,0], [7,7], graph) var route = rc[0] var cost = rc[1] var w = 10 var h = 10 var grid = List.filled(h, null) for (i in 0...h) grid[i] = List.filled(w, ".") for (y in 0...h) {

   grid[y][0]  = "█"
   grid[y][-1] = "█"

} for (x in 0...w) {

   grid[0][x]  = "█"
   grid[-1][x] = "█"

} for (p in graph.barriers) {

   var x = p[0]
   var y = p[1]
   grid[x+1][y+1] = "█"

} for (p in route) {

   var x = p[0]
   var y = p[1]
   grid[x+1][y+1] = "x"

} for (line in grid) System.print(line.join()) System.print("\npath cost %(cost): %(route)")</lang>

Output:

██████████
█x.......█
█.x......█
█..x.███.█
█.x█...█.█
█.x█...█.█
█.x█████.█
█..xxxxx.█
█.......x█
██████████

path cost 11: [[0, 0], [1, 1], [2, 2], [3, 1], [4, 1], [5, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6], [7, 7]]

zkl

Translation of: Python

<lang zkl> // we use strings as hash keys: (x,y)-->"x,y", keys are a single pair fcn toKey(xy){ xy.concat(",") }

fcn AStarSearch(start,end,graph){

  G:=Dictionary(); # Actual movement cost to each position from the start position
  F:=Dictionary(); # Estimated movement cost of start to end going via this position
     #Initialize starting values
  kstart:=toKey(start);
  G[kstart]=0;
  F[kstart]=graph.heuristic(start,end);
  closedVertices,openVertices,cameFrom := List(),List(start),Dictionary();

  while(openVertices){
     # Get the vertex in the open list with the lowest F score
     current,currentFscore := Void, Void;
     foreach pos in (openVertices){
        kpos:=toKey(pos);
        if(current==Void or F[kpos]<currentFscore)

currentFscore,current = F[kpos],pos;

# Check if we have reached the goal if(current==end){ # Yes! Retrace our route backward path,kcurrent := List(current),toKey(current); while(current = cameFrom.find(kcurrent)){ path.append(current); kcurrent=toKey(current); } return(path.reverse(),F[toKey(end)]) # Done! }

# Mark the current vertex as closed openVertices.remove(current); if(not closedVertices.holds(current)) closedVertices.append(current);

# Update scores for vertices near the current position foreach neighbor in (graph.get_vertex_neighbors(current)){ if(closedVertices.holds(neighbor)) continue; # We have already processed this node exhaustively kneighbor:=toKey(neighbor); candidateG:=G[toKey(current)] + graph.move_cost(current, neighbor);

if(not openVertices.holds(neighbor)) openVertices.append(neighbor); # Discovered a new vertex else if(candidateG>=G[kneighbor]) continue; # This G score is worse than previously found

# Adopt this G score cameFrom[kneighbor]=current; G[kneighbor]=candidateG; F[kneighbor]=G[kneighbor] + graph.heuristic(neighbor,end); }

     }
  } // while
  throw(Exception.AssertionError("A* failed to find a solution"));

}</lang> <lang zkl>class [static] AStarGraph{ # Define a class board like grid with barriers

  var [const] barriers =
     T(        T(3,2),T(4,2),T(5,2),   // T is RO List

T(5,3), T(2,4), T(5,4), T(2,5), T(5,5), T(2,6),T(3,6),T(4,6),T(5,6) );

  fcn heuristic(start,goal){  // (x,y),(x,y)
  # Use Chebyshev distance heuristic if we can move one square either
  # adjacent or diagonal
     D,D2,dx,dy := 1,1, (start[0] - goal[0]).abs(), (start[1] - goal[1]).abs();
     D*(dx + dy) + (D2 - 2*D)*dx.min(dy);
  }
  fcn get_vertex_neighbors([(x,y)]){      # Move like a chess king
     var moves=Walker.cproduct([-1..1],[-1..1]).walk();  // 8 moves + (0,0)
     moves.pump(List,'wrap([(dx,dy)]){

x2,y2 := x + dx, y + dy; if((dx==dy==0) or x2 < 0 or x2 > 7 or y2 < 0 or y2 > 7) Void.Skip; else T(x2,y2);

     })
  }
  fcn move_cost(a,b){  // ( (x,y),(x,y) )
     if(barriers.holds(b))

return(100); # Extremely high cost to enter barrier squares

     1 # Normal movement cost
  }

}</lang> <lang zkl>graph:=AStarGraph; route,cost := AStarSearch(T(0,0), T(7,7), graph); println("Route: ", route.apply(fcn(xy){ String("(",toKey(xy),")") }).concat(",")); println("Cost: ", cost);

  // graph the solution:

grid:=(10).pump(List,List.createLong(10," ").copy); foreach x,y in (graph.barriers){ grid[x][y]="#" } foreach x,y in (route){ grid[x][y]="+" } grid[0][0] = "S"; grid[7][7] = "E"; foreach line in (grid){ println(line.concat()) }</lang>

Output:

Route: (0,0),(1,1),(2,2),(3,1),(4,0),(5,1),(6,2),(7,3),(7,4),(7,5),(7,6),(7,7)
Cost: 11
S
 +
  + ###
 +#   #
+ #   #
 +#####
  +
   ++++E