A* search algorithm

From Rosetta Code
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 for route finding that uses heuristics to quickly find an approximate solution.


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*!


See also


Related tasks



C

<lang c>

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

/* and not not_eq */

  1. include <iso646.h>

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

  1. include <math.h>
  1. define map_size_rows 10
  2. define map_size_cols 10

char map[map_size_rows][map_size_cols] = { 

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

};

/* description of graph node */ struct stop { 

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

};

int ind[map_size_rows][map_size_cols] = { 

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

};

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

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

};

int main() { 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

           if (b) {
               continue;
           }

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

           b = 1;

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

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

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

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

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

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

   return 0;

} </lang>

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

C++

<lang cpp>

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

class point { public: 

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

};

class map { public: 

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

};

class node { public: 

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

};

class aStar { public: 

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

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

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

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

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

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

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

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

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

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

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

};

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

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

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

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

} </lang>

Output:
██████████
█x.......█
█x.......█
█x...███.█
█x.█...█.█
█x.█...█.█
█.x█████.█
█..xxxx..█
█......xx█
██████████

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

Racket

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

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

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

 <a-star>
 (define (A* graph@ initial node-cost)
   (define-values/invoke-unit graph@ (import) (export graph^))
   (define count 0)
   <a-star-setup>
    
   (begin0
     (let/ec esc
       <a-star-loop>
       #f)
    
     (printf "visited ~a nodes\n" count))))

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

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

@(chunk 

 <map-graph>
 (define-unit map@
   (import) (export graph^)
    
   (define node? map-node?)
   (define edge? map-edge?)
   (define edge-src map-edge-src)
   (define edge-dest map-edge-dest)
    
   <map-graph-cost>
   <map-graph-edges>))

@(chunk 

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

@(chunk 

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

@(chunk 

 <*>
 (require rackunit
          math/matrix
          racket/unit
          racket/match
          racket/list
          data/heap
          2htdp/image
          racket/runtime-path)
    
 <graph-sig>
    
 <map-generation>
 <map-graph-rep>
 <map-graph>
    
 <a-star>
    
 <map-node-cost>
 <map-example>
 (printf "path is ~a long\n" (length random-path))
 (printf "path is: ~a\n" (map (match-lambda
                                [(map-edge src dx dy dest)
                                 (cons dx dy)])
                              random-path))
    
 <map-display>
 <path-display-line>
 <path-display>

 (path-image random-M random-path))</lang>
Output:
visited 35 nodes
cpu time: 94 real time: 97 gc time: 15
path is 11 long
path is: ((1 . 1) (1 . 1) (1 . -1) (1 . 0) (1 . 0) (1 . 1) (1 . 1) (0 . 1) (-1 . 1) (1 . 1) (0 . 1))
.

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

Python

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

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

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

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

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

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

def AStarSearch(start, end, graph):

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

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

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

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

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

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

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

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

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

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

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

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

REXX

Since the REXX language doesn't employ shortcuts, the   shorter   function could be re-coded for greater speed: <lang rexx>shorter: if @1(nc-1, nr-1) then return 1 

         if @1(nc-1, nr  )  then return 1
         if @1(nc  , nr-1)  then return 1
         if @1(nc+1, nr  )  then return 1
         if @1(nc+1, nr-1)  then return 1
         if @1(nc  , nr+1)  then return 1
         if @1(nc-1, nr+1)  then return 1
         if @1(nc+1, nr+1)  then return 1
                                 return 0</lang>

However, this grid (with the current barriers) don't pose that much computation to solve.

A significant portion of the REXX code address the issue of presenting a decent grid representation. <lang rexx>/*REXX program solves the A* search problem for a (general) NxN grid. */ parse arg N sCol sRow . /*obtain optional arguments from the CL*/ if N== | N=="," then N=8 /*No grid size specified? Use default.*/ if sCol== | sCol=="," then sCol=1 /*No starting column given? " " */ if sRow== | sRow=="," then sRow=1 /* " " row " " " */ NN=N**2; NxN='a ' N"x"N ' grid' /* row [↓] [↓] c=column*/ @.=; do c=1 for N; do r=1 for N; @.c.r=.; end /*r*/; end /*c*/ !.=0; do i=0 for 10; !.i=1; end /*i*/ /*mark as numbers.*/

$= 2.4 2.5 2.6 3.6 4.6 5.6 5.5 5.4 5.3 5.2 4.2 3.2 3.1 /*locations of barriers on grid.*/

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

beg= '-0-' /*mark beginning of the journey (path).*/ 

             Pc = ' 1  1  0  0    1 -1 -1 -1 '  /*the legal "column" moves for a path. */
             Pr = ' 1  0  1 -1   -1  0  1 -1 '  /* "    "     "row"    "    "  "    "  */
                                                /* [↑]  optimized for moving right&down*/

Pc.possM=words(Pc) /*number of possible moves for a path. */ parse var Pc Pc.1 Pc.2 Pc.3 Pc.4 Pc.5 Pc.6 Pc.7 Pc.8 /*parse the legal moves by hand.*/ parse var Pr Pr.1 Pr.2 Pr.3 Pr.4 Pr.5 Pr.6 Pr.7 Pr.8 /* " " " " " " */ @.sCol.sRow= beg /*the path's starting position. */ @Aa= " A* search algorithm on" /*a handy-dandy literal for the SAYs. */ OK=move(1, sCol, sRow) if OK & @.N.N==. then say 'No' @Aa "solution for" NxN'.' 

                 else say 'A solution for the'  @Aa    NxN   'with a score of '  @.N.N"."

ind=left(, 9 * (n<18) ); say /*the indentation of the displayed grid*/ _=substr( copies("┼───", N), 2); say ind translate('┌'_"┐", '┬', "┼") /*part of a grid*/ 

                                                /* [↓]  build a display for the grid.  */
  do   c=1  for N  by +1;     if c\==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('└'_"┘", '┴', "┼") /*display the last part of 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 @1: parse arg x,y; parse var @.x.y dig 2; if !.dig then return @.x.y<#-1; return 0 /*──────────────────────────────────────────────────────────────────────────────────────*/ move: procedure expose @. !. Pc. Pr. N NN; parse arg #,col,row /*obtain move,col,row.*/ 

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

/*──────────────────────────────────────────────────────────────────────────────────────*/ shorter: return @1(nc-1, nr ) | @1(nc-1, nr-1) | @1(nc-1, nr+1) | @1(nc+1, nr ) |, 

               @1(nc+1, nr-1)  |  @1(nc+1, nr-1)  |  @1(nc  , nr-1)  |  @1(nc  , nr+1)</lang>
output   when using the default input:
A solution for the  A*  search algorithm on a  8x8  grid with a score of  12.

          ┌───┬───┬───┬───┬───┬───┬───┬───┐
          │-0-│   │   │   │   │   │   │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │ 1 │   │   │ 4 │ 5 │ 6 │   │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │ 2 │ 3 │ █ │ █ │ █ │ 7 │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │ █ │ █ │   │   │   │ █ │ 8 │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │ █ │   │   │   │ █ │ 9 │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │ █ │ █ │ █ │ █ │ █ │10 │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │   │   │   │   │   │11 │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │   │   │   │   │   │   │   │end│
          └───┴───┴───┴───┴───┴───┴───┴───┘

Sidef

Translation of: Python

<lang ruby>class AStarGraph { 

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

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

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

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

}

func AStarSearch(start, end, graph) { 

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

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

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

   while (openVertices) {

       var current = nil
       var currentFscore = Inf

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

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

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

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

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

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

   die "A* failed to find a solution"

}

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

var w = 10 var h = 10

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

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

grid.each { .join.say }

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

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

zkl

Translation of: Python

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

fcn AStarSearch(start,end,graph){ 

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

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

currentFscore,current = F[kpos],pos;

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

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

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

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

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

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

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

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

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

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

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

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

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

     1 # Normal movement cost
  }

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

  // graph the solution:

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

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