# 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.

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).

## 11l

Translation of: Python
```F AStarSearch(start, end, barriers)
F heuristic(start, goal)
V D = 1
V D2 = 1
V dx = abs(start[0] - goal[0])
V dy = abs(start[1] - goal[1])
R D * (dx + dy) + (D2 - 2 * D) * min(dx, dy)

F get_vertex_neighbours(pos)
[(Int, Int)] n
L(dx, dy) [(1, 0), (-1, 0), (0, 1), (0, -1), (1, 1), (-1, 1), (1, -1), (-1, -1)]
V x2 = pos[0] + dx
V y2 = pos[1] + dy
I x2 < 0 | x2 > 7 | y2 < 0 | y2 > 7
L.continue
n.append((x2, y2))
R n

F move_cost(a, b)
L(barrier) @barriers
I b C barrier
R 100
R 1

[(Int, Int) = Int] G
[(Int, Int) = Int] f

G[start] = 0
f[start] = heuristic(start, end)

Set[(Int, Int)] closedVertices
V openVertices = Set([start])
[(Int, Int) = (Int, Int)] cameFrom

L openVertices.len > 0
(Int, Int)? current
V currentFscore = 0
L(pos) openVertices
I current == N | f[pos] < currentFscore
currentFscore = f[pos]
current = pos

I current == end
V path = [current]
L current C cameFrom
current = cameFrom[current]
path.append(current)
path.reverse()
R (path, f[end])

openVertices.remove(current)

L(neighbour) get_vertex_neighbours(current)
I neighbour C closedVertices
L.continue
V candidateG = G[current] + move_cost(current, neighbour)

I neighbour !C openVertices
E I candidateG >= G[neighbour]
L.continue

cameFrom[neighbour] = current
G[neighbour] = candidateG
V H = heuristic(neighbour, end)
f[neighbour] = G[neighbour] + H

X.throw RuntimeError(‘A* failed to find a solution’)

V (result, cost) = AStarSearch((0, 0), (7, 7), [[(2, 4), (2, 5), (2, 6), (3, 6), (4, 6), (5, 6), (5, 5), (5, 4), (5, 3), (5, 2), (4, 2), (3, 2)]])
print(‘route ’result)
print(‘cost ’cost)```
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
```

## C

```#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include <float.h>
/* and not not_eq */
#include <iso646.h>
/* add -lm to command line to compile with this header */
#include <math.h>

#define map_size_rows 10
#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;

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');
}

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;
}
```
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#

```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

// 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)
{
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)
{
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
}
}

// 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)
{
}
}

// 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();
}
}
}
```
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++

```#include <list>
#include <algorithm>
#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;
}
```
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

```;; * Using external libraries with quicklisp

;; * 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 " "))))
(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)))
```
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

```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;
}
```
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

```'###############################
'###   A* search algorithm   ###
'###############################

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

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)
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)
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
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
end if
end```
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

```// 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]
}
```
```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)
}
```
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
```

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

```{-# 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)
```

The simple graph combinators:

```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) ->
| dx <- [-1..1], dy <- [-1..1]
, not (dx == 0 && dy == 0)
, 0 <= x+dx && x+dx <= a
, 0 <= y+dy && y+dy <= b]

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
```

Finally, the search algorithm, as given in Wikipedia.

```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

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
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
```

Example

```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)
```
```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```

## J

Implementation:

```barrier=: 2 4,2 5,2 6,3 6,4 6,5 6,5 5,5 4,5 3,5 2,4 2,:3 2
price=: _,.~_,~100 barrier} 8 8\$1
dirs=: 0 0-.~>,{,~<i:1
start=: 0 0
end=: 7 7

next=: {{
frontier=. ~.locs=. ,/dests=. (\$price)|"1 ({:"2 y)+"1/dirs
paths=. ,/y,"2 1/"2 dests
costs=. ,x+(<"1 dests){price
deals=. (1+locs <.//. costs) <. (<"1 frontier) { values
keep=. costs < (frontier i. locs) { deals
(keep#costs);keep#paths
}}

Asrch=: {{
values=: (\$price)\$_
best=: (\$price)\$a:
paths=: ,:,:start
costs=: ,0
while. #paths do.
dests=. <"1 {:"2 paths
values=: costs dests} values
best=: (<"2 paths) dests} best
'costs paths'=.costs next paths
end.
((<end){values) ; (<end){best
}}
```

```   Asrch''
┌──┬───┐
│11│0 0│
│  │1 1│
│  │2 2│
│  │3 1│
│  │4 1│
│  │5 1│
│  │6 2│
│  │7 3│
│  │7 4│
│  │7 5│
│  │7 6│
│  │7 7│
└──┴───┘
'A B'=: Asrch''
'x' (<"1 B)} '. #'{~(i.~~.@,) price
x.......
.x......
..x.###.
.x#...#.
.x#...#.
.x#####.
..x.....
...xxxxx
```

Note that this is based on a literal reading of the task, where we are paying a cost to move into a new square -- here, we are not paying for the cost of the start square, because we never move into that square. If we paid to move into the start square, the final cost would have to include that price.

## Java

```package astar;

import java.util.List;
import java.util.ArrayList;
import java.util.Collections;
import java.util.PriorityQueue;
import java.util.Comparator;
import java.util.Queue;

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;
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
}
while (this.now.x != this.xstart || this.now.y != this.ystart) {
this.now = this.now.parent;
}
return this.path;
}
/*
**This function is the step of expanding nodes
**
**
*/
public void expandAStar(int[][] maze, int xstart, int ystart, boolean diag){
if(maze[stateNode.getR()][stateNode.getC()] == 2){
if(isNodeILegal(stateNode, stateNode.expandDirection())){
}
}
/*
** Calculate distance for goal three methods shown.
**
**
*/
public void AStarSearch(){
this.start.setCostToGoal(this.start.calculateCost(this.goal));
this.start.setPathCost(0);
this.start.setAStartCost(this.start.getPathCost() + this.start.getCostToGoal());
Mazecoord intialNode = this.start;
Mazecoord stateNode = intialNode;
//explored<Queue> is empty
while (true){
if(frontier.isEmpty()){
System.out.println("fail");
System.out.println(explored.size());
System.exit(-1);
}
}
/*
** Second method.
**
**
*/
/**
* calculate the cost from current node to goal.
* @param goal : goal
* @return : cost from current node to goal. use Manhattan distance.
*/
public int calculateCost(Mazecoord goal){
int rState = this.getR();
int rGoal = goal.getR();
int diffR = rState - rGoal;
int diffC = this.getC() - goal.getC();
if(diffR * diffC > 0) {     // same sign
return Math.abs(diffR) + Math.abs(diffC);
} else {
return Math.max(Math.abs(diffR), Math.abs(diffC));
}
}

public Coord getFather(){
return this.father;
}

public void setFather(Mazecoord node){
this.father = node;
}

public int getAStartCost() {
return AStartCost;
}

public void setAStartCost(int aStartCost) {
AStartCost = aStartCost;
}

public int getCostToGoal() {
return costToGoal;
}

public void setCostToGoal(int costToGoal) {
this.costToGoal = costToGoal;
}
/*
** Third method.
**
**
*/
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"
}
}
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
}
*/
}
}
}
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();
}
}
}
}
```
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

```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;
}
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;
}
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 ) + "<br />[";
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();
}
```
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

```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));
```
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.

```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
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
```
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

```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)
}
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)

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)) {
}
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")
}
```
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

```-- 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
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
```
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.

```# 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)]

#---------------------------------------------------------------------------------------------------

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

#---------------------------------------------------------------------------------------------------

func d(current, neighbor: Cell): int =
## Return the cost for the move from "current" to "neighbor".
if neighbor in Barriers: 100 else: 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))

#---------------------------------------------------------------------------------------------------

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.reverse()

#---------------------------------------------------------------------------------------------------

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:
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)

#---------------------------------------------------------------------------------------------------

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'

#---------------------------------------------------------------------------------------------------

proc printSolution(path: seq[Cell]; cost: int) =
## Print the solution.
var pathLine = "Path: "
let start = pathLine.len
for cell in path:
echo pathLine
echo(&"Cost: {cost}\n\n")
drawBoard(path)

#---------------------------------------------------------------------------------------------------

let (path, cost) = search((0, 0), (7, 7))
if cost == 0:
echo "No solution found"
else:
printSolution(path, cost)
```
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.

```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)

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 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 =
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 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 =
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
| 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 ()
```
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

```; 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))))))))
```
Output:
```(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)
```
```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)
```

## Perl

```#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/A*_search_algorithm
use warnings;
use List::AllUtils qw( nsort_by );

sub distance
{
my (\$r1, \$c1, \$r2, \$c2) = split /[, ]/, "@_";
sqrt( (\$r1-\$r2)**2 + (\$c1-\$c2)**2 );
}

my \$start = '0,0';
my \$finish = '7,7';
my %barrier = map {\$_, 100}
split ' ', '2,4 2,5 2,6 3,6 4,6 5,6 5,5 5,4 5,3 5,2 4,2 3,2';
my %values = ( \$start, 0 );
my @new = [ \$start, 0 ];
my %from;
my \$mid;
while( ! exists \$values{\$finish} and @new )
{
my \$pick = (shift @new)->[0];
for my \$n ( nsort_by { distance(\$_, \$finish) } # heuristic
grep !/-|8/ && ! exists \$values{\$_},
glob \$pick =~ s/\d+/{@{[\$&-1]},\$&,@{[\$&+1]}}/gr
)
{
\$from{\$n} = \$pick;
\$values{\$n} = \$values{\$pick} + ( \$barrier{\$n} // 1 );
my \$new = [ \$n, my \$dist = \$values{\$n} ];
my \$low = 0; # binary insertion into @new (the priority queue)
my \$high = @new;
\$new[\$mid = \$low + \$high >> 1][1] <= \$dist
? (\$low = \$mid + 1) : (\$high = \$mid) while \$low < \$high;
splice @new, \$low, 0, \$new; # insert in order
}
}

my \$grid = "s.......\n" . ('.' x 8 . "\n") x 7;
substr \$grid, /,/ * \$` * 9 + \$', 1, 'b' for keys %barrier;
my @path = my \$pos = \$finish; # the walkback to get path
while( \$pos ne \$start )
{
substr \$grid, \$pos =~ /,/ ? \$` * 9 + \$' : die, 1, 'x';
unshift @path, \$pos = \$from{\$pos};
}
print "\$grid\nvalue \$values{\$finish}  path @path\n";
```
Output:
```s.......
.x......
..x.bbb.
.xb...b.
.xb...b.
.xbbbbb.
..x.....
...xxxxx

value 11  path 0,0 1,1 2,2 3,1 4,1 5,1 6,2 7,3 7,4 7,5 7,6 7,7
```

### Extra Credit

```#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/A*_search_algorithm
use warnings; # extra credit
use List::AllUtils qw( sum );

my \$start = <<END;
087
654
321
END
my \$finish = <<END;
123
456
780
END

my @tiles = \$finish =~ /[1-9a-z]/g;
my \$width = index \$start, "\n";
my \$gap = qr/.{\$width}/s;
my \$mod = \$width + 1;
my %rc = map {\$_, int(\$_ / \$mod) . ',' . (\$_ % \$mod)} 0 .. length(\$start) - 2;
my %finishrc = map { \$_, [ split /,/, \$rc{index \$finish, \$_} ] } @tiles;
my %found = ( \$start, 1 );
my @new = [ \$start, heuristic(\$start) ]; # a priority queue
my %from;
my \$mid;
while( ! exists \$found{\$finish} and @new )
{
my \$pick = (shift @new)->[0];
for my \$n ( grep ! exists \$found{\$_},
\$pick =~ s/0(\w)/\${1}0/r,         # up
\$pick =~ s/(\w)0/0\$1/r,           # down
\$pick =~ s/0(\$gap)(\w)/\$2\${1}0/r, # left
\$pick =~ s/(\w)(\$gap)0/0\$2\$1/r,   # right
)
{
\$found{\$n} = \$from{\$n} = \$pick;
my \$new = [ \$n, my \$dist = heuristic( \$n ) ];
my \$low = 0; # binary insertion into @new (the priority queue)
my \$high = @new;
\$new[\$mid = \$low + \$high >> 1][1] <= \$dist
? (\$low = \$mid + 1) : (\$high = \$mid) while \$low < \$high;
splice @new, \$low, 0, \$new; # insert in order
}
}

#use Data::Dump 'dd'; dd \%found;
my \$count = keys %found;
exists \$found{\$finish} or die "no solution found with \$count\n";
my @path = my \$pos = \$finish; # the walkback to get path
unshift @path, \$pos = \$from{\$pos} while \$pos ne \$start;
my \$steps = @path - 1;
my \$states = keys %found;
#print "\$_\n" for @path;
my (undef, \$w) = split ' ', qx(stty size);
while( @path )
{
my @section = splice @path, 0, int( \$w / (\$mod + 1) );
while( \$section[0] )
{
s/(.*)\n/ print "\$1  "; ''/e for @section;
print "\n";
}
print "\n";
}
print "steps: \$steps  states: \$states\n";

sub heuristic
{
my \$from = shift;
sum map
{
my (\$r1, \$c1) = split /,/, \$rc{index \$from, \$_};
my (\$r2, \$c2) = @{ \$finishrc{\$_} };
abs(\$r1 - \$r2) + abs(\$c1 - \$c2)
} @tiles;
}
```
Output:
```087  807  870  874  874  874  874  874  074  704  740  741  741  741  741  741  041
654  654  654  650  651  651  651  051  851  851  851  850  852  852  852  052  752
321  321  321  321  320  302  032  632  632  632  632  632  630  603  063  863  863

401  410  412  412  412  412  412  012  102  120  123  123
752  752  750  753  753  753  053  453  453  453  450  456
863  863  863  860  806  086  786  786  786  786  786  780

steps: 28  states: 53
```

k

## 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
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
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()
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()
setd(grid,true,seen)
atom t1 = time()+1
bool found = false
integer count = 0, mc
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
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

```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)
\$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"
```
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)
```

## Picat

```% Picat's tabling system uses an algorithm like Dijkstra's to find an optimal solution.
% Picat's planner supports A* search with heuristics.
% See the program for the 15-puzzle at https://rosettacode.org/wiki/15_puzzle_solver#Picat
%
main =>
Maze = new_array(8,8),
Obs = [(2,4), (2,5), (2,6), (3,6), (4,6), (5,6), (5,5), (5,4), (5,3), (5,2), (4,2), (3,2)],
foreach ((R0,C0) in Obs)
Maze[R0+1,C0+1] = 100
end,
foreach (R in 1..8, C in 1..8)
(var(Maze[R,C]) -> Maze[R,C] = 1; true)
end,
search((1,1),(8,8),Maze,Cost,Path),
writeln(cost=Cost),
println([(R0,C0) : (R1,C1) in Path, R0 = R1-1, C0 = C1-1]).

table (+,+,+,min,-)
search(G,G,_Maze,Cost,Path) => Cost = 0, Path = [G].
search(S@(R,C),G,Maze,Cost,Path) =>
neibs(R,C,Neibs),
member(S1,Neibs),
S1 = (R1,C1),
search(S1,G,Maze,Cost1,Path1),
Cost = Cost1+Maze[R1,C1],
Path = [S|Path1].

neibs(R,C,Neibs) =>
Neibs = [(R1,C1) : Dr in [-1,0,1], Dc in [-1,0,1], R1 = R+Dr, C1 = C+Dc,
R1 >= 1, R1 <= 8, C1 >= 1, C1 <= 8, (R,C) != (R1,C1)].```
Output:
```cost = 11
[(0,0),(1,0),(2,0),(3,0),(4,0),(5,1),(6,2),(6,3),(6,4),(6,5),(6,6),(7,7)]
```

## 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
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)

#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:
elif candidateG >= G[neighbour]:
continue #This G score is worse than previously found

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()
```
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 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))

@(chunk
<a-star-loop>
(for ([x (in-heap/consume! open-set)])
<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))

@(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)
(* (+ 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))
```
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
```# 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;
```
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

```/*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*/
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. */
```
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

```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
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]);```
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
```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}"
```
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
```#!/bin/bash

# This option will make the script exit when there is an error
set -o errexit
# 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 -lt 0 || \$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

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#*,}-```