/*====================================================================*
 -  Copyright (C) 2001 Leptonica.  All rights reserved.
 -
 -  Redistribution and use in source and binary forms, with or without
 -  modification, are permitted provided that the following conditions
 -  are met:
 -  1. Redistributions of source code must retain the above copyright
 -     notice, this list of conditions and the following disclaimer.
 -  2. Redistributions in binary form must reproduce the above
 -     copyright notice, this list of conditions and the following
 -     disclaimer in the documentation and/or other materials
 -     provided with the distribution.
 -
 -  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 -  ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 -  LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
 -  A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL ANY
 -  CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
 -  EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
 -  PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
 -  PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
 -  OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
 -  NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
 -  SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *====================================================================*/


/*!
 * \file maze.c
 * <pre>
 *
 *      This is a game with a pedagogical slant.  A maze is represented
 *      by a binary image.  The ON pixels (fg) are walls.  The goal is
 *      to navigate on OFF pixels (bg), using Manhattan steps
 *      (N, S, E, W), between arbitrary start and end positions.
 *      The problem is thus to find the shortest route between two points
 *      in a binary image that are 4-connected in the bg.  This is done
 *      with a breadth-first search, implemented with a queue.
 *      We also use a queue of pointers to generate the maze (image).
 *
 *          PIX             *generateBinaryMaze()
 *          static MAZEEL   *mazeelCreate()
 *
 *          PIX             *pixSearchBinaryMaze()
 *          static l_int32   localSearchForBackground()
 *
 *      Generalizing a maze to a grayscale image, the search is
 *      now for the "shortest" or least cost path, for some given
 *      cost function.
 *
 *          PIX             *pixSearchGrayMaze()
 * </pre>
 */

#ifdef HAVE_CONFIG_H
#include <config_auto.h>
#endif  /* HAVE_CONFIG_H */

#include <string.h>
#ifdef _WIN32
#include <stdlib.h>
#include <time.h>
#endif  /* _WIN32 */
#include "allheaders.h"

static const l_int32  MinMazeWidth = 50;
static const l_int32  MinMazeHeight = 50;

static const l_float32  DefaultWallProbability = 0.65f;
static const l_float32  DefaultAnisotropyRatio = 0.25f;

enum {  /* direction from parent to newly created element */
    START_LOC = 0,
    DIR_NORTH = 1,
    DIR_SOUTH = 2,
    DIR_WEST = 3,
    DIR_EAST = 4
};

struct MazeElement {
    l_float32  distance;
    l_int32    x;
    l_int32    y;
    l_uint32   val;  /* value of maze pixel at this location */
    l_int32    dir;  /* direction from parent to child */
};
typedef struct MazeElement  MAZEEL;


static MAZEEL *mazeelCreate(l_int32  x, l_int32  y, l_int32  dir);
static l_int32 localSearchForBackground(PIX  *pix, l_int32  *px,
                                        l_int32  *py, l_int32  maxrad);

#ifndef  NO_CONSOLE_IO
#define  DEBUG_PATH    0
#define  DEBUG_MAZE    0
#endif  /* ~NO_CONSOLE_IO */

/*---------------------------------------------------------------------*
 *             Binary maze generation as cellular automaton            *
 *---------------------------------------------------------------------*/
/*!
 * \brief   generateBinaryMaze()
 *
 * \param[in]    w, h  size of maze
 * \param[in]    xi, yi  initial location
 * \param[in]    wallps probability that a pixel to the side is ON
 * \param[in]    ranis ratio of prob that pixel in forward direction
 *                     is a wall to the probability that pixel in
 *                     side directions is a wall
 * \return  pix, or NULL on error
 *
 * <pre>
 * Notes:
 *      (1) We have two input probability factors that determine the
 *          density of walls and average length of straight passages.
 *          When ranis < 1.0, you are more likely to generate a wall
 *          to the side than going forward.  Enter 0.0 for either if
 *          you want to use the default values.
 *      (2) This is a type of percolation problem, and exhibits
 *          different phases for different parameters wallps and ranis.
 *          For larger values of these parameters, regions in the maze
 *          are not explored because the maze generator walls them
 *          off and cannot get through.  The boundary between the
 *          two phases in this two-dimensional parameter space goes
 *          near these values:
 *                wallps       ranis
 *                0.35         1.00
 *                0.40         0.85
 *                0.45         0.70
 *                0.50         0.50
 *                0.55         0.40
 *                0.60         0.30
 *                0.65         0.25
 *                0.70         0.19
 *                0.75         0.15
 *                0.80         0.11
 *      (3) Because there is a considerable amount of overhead in calling
 *          pixGetPixel() and pixSetPixel(), this function can be sped
 *          up with little effort using raster line pointers and the
 *          GET_DATA* and SET_DATA* macros.
 * </pre>
 */
PIX *
generateBinaryMaze(l_int32  w,
                   l_int32  h,
                   l_int32  xi,
                   l_int32  yi,
                   l_float32  wallps,
                   l_float32  ranis)
{
l_int32    x, y, dir;
l_uint32   val;
l_float32  frand, wallpf, testp;
MAZEEL    *el, *elp;
PIX       *pixd;  /* the destination maze */
PIX       *pixm;  /* for bookkeeping, to indicate pixels already visited */
L_QUEUE   *lq;

    /* On Windows, seeding is apparently necessary to get decent mazes.
     * Windows rand() returns a value up to 2^15 - 1, whereas unix
     * rand() returns a value up to 2^31 - 1.  Therefore the generated
     * mazes will differ on the two platforms. */
#ifdef _WIN32
    srand(28*333);
#endif /* _WIN32 */

    if (w < MinMazeWidth)
        w = MinMazeWidth;
    if (h < MinMazeHeight)
        h = MinMazeHeight;
    if (xi <= 0 || xi >= w)
        xi = w / 6;
    if (yi <= 0 || yi >= h)
        yi = h / 5;
    if (wallps < 0.05 || wallps > 0.95)
        wallps = DefaultWallProbability;
    if (ranis < 0.05 || ranis > 1.0)
        ranis = DefaultAnisotropyRatio;
    wallpf = wallps * ranis;

#if  DEBUG_MAZE
    lept_stderr("(w, h) = (%d, %d), (xi, yi) = (%d, %d)\n", w, h, xi, yi);
    lept_stderr("Using: prob(wall) = %7.4f, anisotropy factor = %7.4f\n",
                wallps, ranis);
#endif  /* DEBUG_MAZE */

        /* These are initialized to OFF */
    pixd = pixCreate(w, h, 1);
    pixm = pixCreate(w, h, 1);

    lq = lqueueCreate(0);

        /* Prime the queue with the first pixel; it is OFF */
    el = mazeelCreate(xi, yi, START_LOC);
    pixSetPixel(pixm, xi, yi, 1);  /* mark visited */
    lqueueAdd(lq, el);

        /* While we're at it ... */
    while (lqueueGetCount(lq) > 0) {
        elp = (MAZEEL *)lqueueRemove(lq);
        x = elp->x;
        y = elp->y;
        dir = elp->dir;
        if (x > 0) {  /* check west */
            pixGetPixel(pixm, x - 1, y, &val);
            if (val == 0) {  /* not yet visited */
                pixSetPixel(pixm, x - 1, y, 1);  /* mark visited */
                frand = (l_float32)rand() / (l_float32)RAND_MAX;
                testp = wallps;
                if (dir == DIR_WEST)
                    testp = wallpf;
                if (frand <= testp) {  /* make it a wall */
                    pixSetPixel(pixd, x - 1, y, 1);
                } else {  /* not a wall */
                    el = mazeelCreate(x - 1, y, DIR_WEST);
                    lqueueAdd(lq, el);
                }
            }
        }
        if (y > 0) {  /* check north */
            pixGetPixel(pixm, x, y - 1, &val);
            if (val == 0) {  /* not yet visited */
                pixSetPixel(pixm, x, y - 1, 1);  /* mark visited */
                frand = (l_float32)rand() / (l_float32)RAND_MAX;
                testp = wallps;
                if (dir == DIR_NORTH)
                    testp = wallpf;
                if (frand <= testp) {  /* make it a wall */
                    pixSetPixel(pixd, x, y - 1, 1);
                } else {  /* not a wall */
                    el = mazeelCreate(x, y - 1, DIR_NORTH);
                    lqueueAdd(lq, el);
                }
            }
        }
        if (x < w - 1) {  /* check east */
            pixGetPixel(pixm, x + 1, y, &val);
            if (val == 0) {  /* not yet visited */
                pixSetPixel(pixm, x + 1, y, 1);  /* mark visited */
                frand = (l_float32)rand() / (l_float32)RAND_MAX;
                testp = wallps;
                if (dir == DIR_EAST)
                    testp = wallpf;
                if (frand <= testp) {  /* make it a wall */
                    pixSetPixel(pixd, x + 1, y, 1);
                } else {  /* not a wall */
                    el = mazeelCreate(x + 1, y, DIR_EAST);
                    lqueueAdd(lq, el);
                }
            }
        }
        if (y < h - 1) {  /* check south */
            pixGetPixel(pixm, x, y + 1, &val);
            if (val == 0) {  /* not yet visited */
                pixSetPixel(pixm, x, y + 1, 1);  /* mark visited */
                frand = (l_float32)rand() / (l_float32)RAND_MAX;
                testp = wallps;
                if (dir == DIR_SOUTH)
                    testp = wallpf;
                if (frand <= testp) {  /* make it a wall */
                    pixSetPixel(pixd, x, y + 1, 1);
                } else {  /* not a wall */
                    el = mazeelCreate(x, y + 1, DIR_SOUTH);
                    lqueueAdd(lq, el);
                }
            }
        }
        LEPT_FREE(elp);
    }

    lqueueDestroy(&lq, TRUE);
    pixDestroy(&pixm);
    return pixd;
}


static MAZEEL *
mazeelCreate(l_int32  x,
             l_int32  y,
             l_int32  dir)
{
MAZEEL *el;

    el = (MAZEEL *)LEPT_CALLOC(1, sizeof(MAZEEL));
    el->x = x;
    el->y = y;
    el->dir = dir;
    return el;
}


/*---------------------------------------------------------------------*
 *                           Binary maze search                        *
 *---------------------------------------------------------------------*/
/*!
 * \brief   pixSearchBinaryMaze()
 *
 * \param[in]    pixs 1 bpp, maze
 * \param[in]    xi, yi  beginning point; use same initial point
 *                       that was used to generate the maze
 * \param[in]    xf, yf  end point, or close to it
 * \param[out]   ppixd [optional] maze with path illustrated, or
 *                     if no path possible, the part of the maze
 *                     that was searched
 * \return  pta shortest path, or NULL if either no path
 *              exists or on error
 *
 * <pre>
 * Notes:
 *      (1) Because of the overhead in calling pixGetPixel() and
 *          pixSetPixel(), we have used raster line pointers and the
 *          GET_DATA* and SET_DATA* macros for many of the pix accesses.
 *      (2) Commentary:
 *            The goal is to find the shortest path between beginning and
 *          end points, without going through walls, and there are many
 *          ways to solve this problem.
 *            We use a queue to implement a breadth-first search.  Two auxiliary
 *          "image" data structures can be used: one to mark the visited
 *          pixels and one to give the direction to the parent for each
 *          visited pixel.  The first structure is used to avoid putting
 *          pixels on the queue more than once, and the second is used
 *          for retracing back to the origin, like the breadcrumbs in
 *          Hansel and Gretel.  Each pixel taken off the queue is destroyed
 *          after it is used to locate the allowed neighbors.  In fact,
 *          only one distance image is required, if you initialize it
 *          to some value that signifies "not yet visited."  (We use
 *          a binary image for marking visited pixels because it is clearer.)
 *          This method for a simple search of a binary maze is implemented in
 *          pixSearchBinaryMaze().
 *            An alternative method would store the (manhattan) distance
 *          from the start point with each pixel on the queue.  The children
 *          of each pixel get a distance one larger than the parent.  These
 *          values can be stored in an auxiliary distance map image
 *          that is constructed simultaneously with the search.  Once the
 *          end point is reached, the distance map is used to backtrack
 *          along a minimum path.  There may be several equal length
 *          minimum paths, any one of which can be chosen this way.
 * </pre>
 */
PTA *
pixSearchBinaryMaze(PIX     *pixs,
                    l_int32  xi,
                    l_int32  yi,
                    l_int32  xf,
                    l_int32  yf,
                    PIX    **ppixd)
{
l_int32    i, j, x, y, w, h, d, found;
l_uint32   val, rpixel, gpixel, bpixel;
void     **lines1, **linem1, **linep8, **lined32;
MAZEEL    *el, *elp;
PIX       *pixd;  /* the shortest path written on the maze image */
PIX       *pixm;  /* for bookkeeping, to indicate pixels already visited */
PIX       *pixp;  /* for bookkeeping, to indicate direction to parent */
L_QUEUE   *lq;
PTA       *pta;

    if (ppixd) *ppixd = NULL;
    if (!pixs)
        return (PTA *)ERROR_PTR("pixs not defined", __func__, NULL);
    pixGetDimensions(pixs, &w, &h, &d);
    if (d != 1)
        return (PTA *)ERROR_PTR("pixs not 1 bpp", __func__, NULL);
    if (xi <= 0 || xi >= w)
        return (PTA *)ERROR_PTR("xi not valid", __func__, NULL);
    if (yi <= 0 || yi >= h)
        return (PTA *)ERROR_PTR("yi not valid", __func__, NULL);
    pixGetPixel(pixs, xi, yi, &val);
    if (val != 0)
        return (PTA *)ERROR_PTR("(xi,yi) not bg pixel", __func__, NULL);
    pixd = NULL;
    pta = NULL;

        /* Find a bg pixel near input point (xf, yf) */
    localSearchForBackground(pixs, &xf, &yf, 5);

#if  DEBUG_MAZE
    lept_stderr("(xi, yi) = (%d, %d), (xf, yf) = (%d, %d)\n",
                xi, yi, xf, yf);
#endif  /* DEBUG_MAZE */

    pixm = pixCreate(w, h, 1);  /* initialized to OFF */
    pixp = pixCreate(w, h, 8);  /* direction to parent stored as enum val */
    lines1 = pixGetLinePtrs(pixs, NULL);
    linem1 = pixGetLinePtrs(pixm, NULL);
    linep8 = pixGetLinePtrs(pixp, NULL);

    lq = lqueueCreate(0);

        /* Prime the queue with the first pixel; it is OFF */
    el = mazeelCreate(xi, yi, 0);  /* don't need direction here */
    pixSetPixel(pixm, xi, yi, 1);  /* mark visited */
    lqueueAdd(lq, el);

        /* Fill up the pix storing directions to parents,
         * stopping when we hit the point (xf, yf)  */
    found = FALSE;
    while (lqueueGetCount(lq) > 0) {
        elp = (MAZEEL *)lqueueRemove(lq);
        x = elp->x;
        y = elp->y;
        if (x == xf && y == yf) {
            found = TRUE;
            LEPT_FREE(elp);
            break;
        }

        if (x > 0) {  /* check to west */
            val = GET_DATA_BIT(linem1[y], x - 1);
            if (val == 0) {  /* not yet visited */
                SET_DATA_BIT(linem1[y], x - 1);  /* mark visited */
                val = GET_DATA_BIT(lines1[y], x - 1);
                if (val == 0) {  /* bg, not a wall */
                    SET_DATA_BYTE(linep8[y], x - 1, DIR_EAST);  /* parent E */
                    el = mazeelCreate(x - 1, y, 0);
                    lqueueAdd(lq, el);
                }
            }
        }
        if (y > 0) {  /* check north */
            val = GET_DATA_BIT(linem1[y - 1], x);
            if (val == 0) {  /* not yet visited */
                SET_DATA_BIT(linem1[y - 1], x);  /* mark visited */
                val = GET_DATA_BIT(lines1[y - 1], x);
                if (val == 0) {  /* bg, not a wall */
                    SET_DATA_BYTE(linep8[y - 1], x, DIR_SOUTH);  /* parent S */
                    el = mazeelCreate(x, y - 1, 0);
                    lqueueAdd(lq, el);
                }
            }
        }
        if (x < w - 1) {  /* check east */
            val = GET_DATA_BIT(linem1[y], x + 1);
            if (val == 0) {  /* not yet visited */
                SET_DATA_BIT(linem1[y], x + 1);  /* mark visited */
                val = GET_DATA_BIT(lines1[y], x + 1);
                if (val == 0) {  /* bg, not a wall */
                    SET_DATA_BYTE(linep8[y], x + 1, DIR_WEST);  /* parent W */
                    el = mazeelCreate(x + 1, y, 0);
                    lqueueAdd(lq, el);
                }
            }
        }
        if (y < h - 1) {  /* check south */
            val = GET_DATA_BIT(linem1[y + 1], x);
            if (val == 0) {  /* not yet visited */
                SET_DATA_BIT(linem1[y + 1], x);  /* mark visited */
                val = GET_DATA_BIT(lines1[y + 1], x);
                if (val == 0) {  /* bg, not a wall */
                    SET_DATA_BYTE(linep8[y + 1], x, DIR_NORTH);  /* parent N */
                    el = mazeelCreate(x, y + 1, 0);
                    lqueueAdd(lq, el);
                }
            }
        }
        LEPT_FREE(elp);
    }

    lqueueDestroy(&lq, TRUE);
    pixDestroy(&pixm);
    LEPT_FREE(linem1);

    if (ppixd) {
        pixd = pixUnpackBinary(pixs, 32, 1);
        *ppixd = pixd;
    }
    composeRGBPixel(255, 0, 0, &rpixel);  /* start point */
    composeRGBPixel(0, 255, 0, &gpixel);
    composeRGBPixel(0, 0, 255, &bpixel);  /* end point */

    if (found) {
        L_INFO(" Path found\n", __func__);
        pta = ptaCreate(0);
        x = xf;
        y = yf;
        while (1) {
            ptaAddPt(pta, x, y);
            if (x == xi && y == yi)
                break;
            if (pixd)  /* write 'gpixel' onto the path */
                pixSetPixel(pixd, x, y, gpixel);
            pixGetPixel(pixp, x, y, &val);
            if (val == DIR_NORTH)
                y--;
            else if (val == DIR_SOUTH)
                y++;
            else if (val == DIR_EAST)
                x++;
            else if (val == DIR_WEST)
                x--;
        }
    } else {
        L_INFO(" No path found\n", __func__);
        if (pixd) {  /* paint all visited locations */
            lined32 = pixGetLinePtrs(pixd, NULL);
            for (i = 0; i < h; i++) {
                for (j = 0; j < w; j++) {
                    if (GET_DATA_BYTE(linep8[i], j) != 0)
                        SET_DATA_FOUR_BYTES(lined32[i], j, gpixel);
                }
            }
            LEPT_FREE(lined32);
        }
    }
    if (pixd) {
        pixSetPixel(pixd, xi, yi, rpixel);
        pixSetPixel(pixd, xf, yf, bpixel);
    }

    pixDestroy(&pixp);
    LEPT_FREE(lines1);
    LEPT_FREE(linep8);
    return pta;
}


/*!
 * \brief   localSearchForBackground()
 *
 * \param[in]    pix
 * \param[out]   px, py starting position for search; return found position
 * \param[in]    maxrad max distance to search from starting location
 * \return  0 if bg pixel found; 1 if not found
 */
static l_int32
localSearchForBackground(PIX  *pix,
                         l_int32  *px,
                         l_int32  *py,
                         l_int32  maxrad)
{
l_int32   x, y, w, h, r, i, j;
l_uint32  val;

    x = *px;
    y = *py;
    pixGetPixel(pix, x, y, &val);
    if (val == 0) return 0;

        /* For each value of r, restrict the search to the boundary
         * pixels in a square centered on (x,y), clipping to the
         * image boundaries if necessary.  */
    pixGetDimensions(pix, &w, &h, NULL);
    for (r = 1; r < maxrad; r++) {
        for (i = -r; i <= r; i++) {
            if (y + i < 0 || y + i >= h)
                continue;
            for (j = -r; j <= r; j++) {
                if (x + j < 0 || x + j >= w)
                    continue;
                if (L_ABS(i) != r && L_ABS(j) != r)  /* not on "r ring" */
                    continue;
                pixGetPixel(pix, x + j, y + i, &val);
                if (val == 0) {
                    *px = x + j;
                    *py = y + i;
                    return 0;
                }
            }
        }
    }
    return 1;
}



/*---------------------------------------------------------------------*
 *                            Gray maze search                         *
 *---------------------------------------------------------------------*/
/*!
 * \brief   pixSearchGrayMaze()
 *
 * \param[in]    pixs     1 bpp maze; w and h must be >= 50
 * \param[in]    xi, yi   beginning point; use same initial point
 *                        that was used to generate the maze
 * \param[in]    xf, yf   end point, or close to it
 * \param[out]   ppixd    [optional] maze with path illustrated, or
 *                        if no path possible, the part of the maze
 *                        that was searched
 * \return  pta   shortest path, or NULL if either no path exists or on error
 *
 * <pre>
 *  Commentary:
 *      Consider first a slight generalization of the binary maze
 *      search problem.  Suppose that you can go through walls,
 *      but the cost is higher say, an increment of 3 to go into
 *      a wall pixel rather than 1?  You're still trying to find
 *      the shortest path.  One way to do this is with an ordered
 *      queue, and a simple way to visualize an ordered queue is as
 *      a set of stacks, each stack being marked with the distance
 *      of each pixel in the stack from the start.  We place the
 *      start pixel in stack 0, pop it, and process its 4 children.
 *      Each pixel is given a distance that is incremented from that
 *      of its parent 0 in this case, depending on if it is a wall
 *      pixel or not.  That value may be recorded on a distance map,
 *      according to the algorithm below.  For children of the first
 *      pixel, those not on a wall go in stack 1, and wall
 *      children go in stack 3.  Stack 0 being emptied, the process
 *      then continues with pixels being popped from stack 1.
 *      Here is the algorithm for each child pixel.  The pixel's
 *      distance value, were it to be placed on a stack, is compared
 *      with the value for it that is on the distance map.  There
 *      are three possible cases:
 *         1 If the pixel has not yet been registered, it is pushed
 *             on its stack and the distance is written to the map.
 *         2 If it has previously been registered with a higher distance,
 *             the distance on the map is relaxed to that of the
 *             current pixel, which is then placed on its stack.
 *         3 If it has previously been registered with an equal
 *             or lower value, the pixel is discarded.
 *      The pixels are popped and processed successively from
 *      stack 1, and when stack 1 is empty, popping starts on stack 2.
 *      This continues until the destination pixel is popped off
 *      a stack.   The minimum path is then derived from the distance map,
 *      going back from the end point as before.  This is just Dijkstra's
 *      algorithm for a directed graph; here, the underlying graph
 *      consisting of the pixels and four edges connecting each pixel
 *      to its 4-neighbor is a special case of a directed graph, where
 *      each edge is bi-directional.  The implementation of this generalized
 *      maze search is left as an exercise to the reader.
 *
 *      Let's generalize a bit further.  Suppose the "maze" is just
 *      a grayscale image -- think of it as an elevation map.  The cost
 *      of moving on this surface depends on the height, or the gradient,
 *      or whatever you want.  All that is required is that the cost
 *      is specified and non-negative on each link between adjacent
 *      pixels.  Now the problem becomes: find the least cost path
 *      moving on this surface between two specified end points.
 *      For example, if the cost across an edge between two pixels
 *      depends on the "gradient", you can use:
 *           cost = 1 + L_ABSdeltaV
 *      where deltaV is the difference in value between two adjacent
 *      pixels.  If the costs are all integers, we can still use an array
 *      of stacks to avoid ordering the queue e.g., by using a heap sort.
 *      This is a neat problem, because you don't even have to build a
 *      maze -- you can can use it on any grayscale image!
 *
 *      Rather than using an array of stacks, a more practical
 *      approach is to implement with a priority queue, which is
 *      a queue that is sorted so that the elements with the largest
 *      or smallest key values always come off first.  The
 *      priority queue is efficiently implemented as a heap, and
 *      this is how we do it.  Suppose you run the algorithm
 *      using a priority queue, doing the bookkeeping with an
 *      auxiliary image data structure that saves the distance of
 *      each pixel put on the queue as before, according to the method
 *      described above.  We implement it as a 2-way choice by
 *      initializing the distance array to a large value and putting
 *      a pixel on the queue if its distance is less than the value
 *      found on the array.  When you finally pop the end pixel from
 *      the queue, you're done, and you can trace the path backward,
 *      either always going downhill or using an auxiliary image to
 *      give you the direction to go at each step.  This is implemented
 *      here in searchGrayMaze.
 *
 *      Do we really have to use a sorted queue?  Can we solve this
 *      generalized maze with an unsorted queue of pixels?  Or even
 *      an unsorted stack, doing a depth-first search (DFS)?
 *      Consider a different algorithm for this generalized maze, where
 *      we travel again breadth first, but this time use a single,
 *      unsorted queue.  An auxiliary image is used as before to
 *      store the distances and to determine if pixels get pushed
 *      on the stack or dropped.  As before, we must allow pixels
 *      to be revisited, with relaxation of the distance if a shorter
 *      path arrives later.  As a result, we will in general have
 *      multiple instances of the same pixel on the stack with different
 *      distances.  However, because the queue is not ordered, some of
 *      these pixels will be popped when another instance with a lower
 *      distance is still on the stack.  Here, we're just popping them
 *      in the order they go on, rather than setting up a priority
 *      based on minimum distance.  Thus, unlike the priority queue,
 *      when a pixel is popped we have to check the distance map to
 *      see if a pixel with a lower distance has been put on the queue,
 *      and, if so, we discard the pixel we just popped.  So the
 *      "while" loop looks like this:
 *        ~ pop a pixel from the queue
 *        ~ check its distance against the distance stored in the
 *          distance map; if larger, discard
 *        ~ otherwise, for each of its neighbors:
 *            ~ compute its distance from the start pixel
 *            ~ compare this distance with that on the distance map:
 *                ~ if the distance map value higher, relax the distance
 *                  and push the pixel on the queue
 *                ~ if the distance map value is lower, discard the pixel
 *
 *      How does this loop terminate?  Before, with an ordered queue,
 *      it terminates when you pop the end pixel.  But with an unordered
 *      queue or stack, the first time you hit the end pixel, the
 *      distance is not guaranteed to be correct, because the pixels
 *      along the shortest path may not have yet been visited and relaxed.
 *      Because the shortest path can theoretically go anywhere,
 *      we must keep going.  How do we know when to stop?   Dijkstra
 *      uses an ordered queue to systematically remove nodes from
 *      further consideration.  Each time a pixel is popped, we're
 *      done with it; it's "finalized" in the Dijkstra sense because
 *      we know the shortest path to it.  However, with an unordered
 *      queue, the brute force answer is: stop when the queue
 *      or stack is empty, because then every pixel in the image
 *      has been assigned its minimum "distance" from the start pixel.
 *
 *      This is similar to the situation when you use a stack for the
 *      simpler uniform-step problem: with breadth-first search BFS
 *      the pixels on the queue are automatically ordered, so you are
 *      done when you locate the end pixel as a neighbor of a popped pixel;
 *      whereas depth-first search DFS, using a stack, requires,
 *      in general, a search of every accessible pixel.  Further, if
 *      a pixel is revisited with a smaller distance, that distance is
 *      recorded and the pixel is put on the stack again.
 *
 *      But surely, you ask, can't we stop sooner?  What if the
 *      start and end pixels are very close to each other?
 *      OK, suppose they are, and you have very high walls and a
 *      long snaking level path that is actually the minimum cost.
 *      That long path can wind back and forth across the entire
 *      maze many times before ending up at the end point, which
 *      could be just over a wall from the start.  With the unordered
 *      queue, you very quickly get a high distance for the end
 *      pixel, which will be relaxed to the minimum distance only
 *      after all the pixels of the path have been visited and placed
 *      on the queue, multiple times for many of them.  So that's the
 *      price for not ordering the queue!
 * </pre>
 */
PTA *
pixSearchGrayMaze(PIX     *pixs,
                  l_int32  xi,
                  l_int32  yi,
                  l_int32  xf,
                  l_int32  yf,
                  PIX    **ppixd)
{
l_int32   x, y, w, h, d;
l_uint32  val, valr, vals, rpixel, gpixel, bpixel;
void    **lines8, **liner32, **linep8;
l_int32   cost, dist, distparent, sival, sivals;
MAZEEL   *el, *elp;
PIX      *pixd;  /* optionally plot the path on this RGB version of pixs */
PIX      *pixr;  /* for bookkeeping, to indicate the minimum distance */
                 /* to pixels already visited */
PIX      *pixp;  /* for bookkeeping, to indicate direction to parent */
L_HEAP   *lh;
PTA      *pta;

    if (ppixd) *ppixd = NULL;
    if (!pixs)
        return (PTA *)ERROR_PTR("pixs not defined", __func__, NULL);
    pixGetDimensions(pixs, &w, &h, &d);
    if (w < 50 || h < 50)
        return (PTA *)ERROR_PTR("too small: w and h not >= 50", __func__, NULL);
    if (d != 8)
        return (PTA *)ERROR_PTR("pixs not 8 bpp", __func__, NULL);
    if (xi <= 0 || xi >= w)
        return (PTA *)ERROR_PTR("xi not valid", __func__, NULL);
    if (yi <= 0 || yi >= h)
        return (PTA *)ERROR_PTR("yi not valid", __func__, NULL);
    pixd = NULL;
    pta = NULL;

        /* Allocate stuff */
    pixr = pixCreate(w, h, 32);
    pixSetAll(pixr);  /* initialize to max value */
    pixp = pixCreate(w, h, 8);  /* direction to parent stored as enum val */
    lines8 = pixGetLinePtrs(pixs, NULL);
    linep8 = pixGetLinePtrs(pixp, NULL);
    liner32 = pixGetLinePtrs(pixr, NULL);
    lh = lheapCreate(0, L_SORT_INCREASING);  /* always remove closest pixels */

        /* Prime the heap with the first pixel */
    pixGetPixel(pixs, xi, yi, &val);
    el = mazeelCreate(xi, yi, 0);  /* don't need direction here */
    el->distance = 0;
    pixGetPixel(pixs, xi, yi, &val);
    el->val = val;
    pixSetPixel(pixr, xi, yi, 0);  /* distance is 0 */
    lheapAdd(lh, el);

        /* Breadth-first search with priority queue (implemented by
           a heap), labeling direction to parents in pixp and minimum
           distance to visited pixels in pixr.  Stop when we pull
           the destination point (xf, yf) off the queue. */
    while (lheapGetCount(lh) > 0) {
        elp = (MAZEEL *)lheapRemove(lh);
        if (!elp) {
            L_ERROR("heap broken!!\n", __func__);
            goto cleanup_stuff;
        }
        x = elp->x;
        y = elp->y;
        if (x == xf && y == yf) {  /* exit condition */
            LEPT_FREE(elp);
            break;
        }
        distparent = (l_int32)elp->distance;
        val = elp->val;
        sival = val;

        if (x > 0) {  /* check to west */
            vals = GET_DATA_BYTE(lines8[y], x - 1);
            valr = GET_DATA_FOUR_BYTES(liner32[y], x - 1);
            sivals = (l_int32)vals;
            cost = 1 + L_ABS(sivals - sival);  /* cost to move to this pixel */
            dist = distparent + cost;
            if (dist < valr) {  /* shortest path so far to this pixel */
                SET_DATA_FOUR_BYTES(liner32[y], x - 1, dist);  /* new dist */
                SET_DATA_BYTE(linep8[y], x - 1, DIR_EAST);  /* parent to E */
                el = mazeelCreate(x - 1, y, 0);
                el->val = vals;
                el->distance = dist;
                lheapAdd(lh, el);
            }
        }
        if (y > 0) {  /* check north */
            vals = GET_DATA_BYTE(lines8[y - 1], x);
            valr = GET_DATA_FOUR_BYTES(liner32[y - 1], x);
            sivals = (l_int32)vals;
            cost = 1 + L_ABS(sivals - sival);  /* cost to move to this pixel */
            dist = distparent + cost;
            if (dist < valr) {  /* shortest path so far to this pixel */
                SET_DATA_FOUR_BYTES(liner32[y - 1], x, dist);  /* new dist */
                SET_DATA_BYTE(linep8[y - 1], x, DIR_SOUTH);  /* parent to S */
                el = mazeelCreate(x, y - 1, 0);
                el->val = vals;
                el->distance = dist;
                lheapAdd(lh, el);
            }
        }
        if (x < w - 1) {  /* check east */
            vals = GET_DATA_BYTE(lines8[y], x + 1);
            valr = GET_DATA_FOUR_BYTES(liner32[y], x + 1);
            sivals = (l_int32)vals;
            cost = 1 + L_ABS(sivals - sival);  /* cost to move to this pixel */
            dist = distparent + cost;
            if (dist < valr) {  /* shortest path so far to this pixel */
                SET_DATA_FOUR_BYTES(liner32[y], x + 1, dist);  /* new dist */
                SET_DATA_BYTE(linep8[y], x + 1, DIR_WEST);  /* parent to W */
                el = mazeelCreate(x + 1, y, 0);
                el->val = vals;
                el->distance = dist;
                lheapAdd(lh, el);
            }
        }
        if (y < h - 1) {  /* check south */
            vals = GET_DATA_BYTE(lines8[y + 1], x);
            valr = GET_DATA_FOUR_BYTES(liner32[y + 1], x);
            sivals = (l_int32)vals;
            cost = 1 + L_ABS(sivals - sival);  /* cost to move to this pixel */
            dist = distparent + cost;
            if (dist < valr) {  /* shortest path so far to this pixel */
                SET_DATA_FOUR_BYTES(liner32[y + 1], x, dist);  /* new dist */
                SET_DATA_BYTE(linep8[y + 1], x, DIR_NORTH);  /* parent to N */
                el = mazeelCreate(x, y + 1, 0);
                el->val = vals;
                el->distance = dist;
                lheapAdd(lh, el);
            }
        }
        LEPT_FREE(elp);
    }

    lheapDestroy(&lh, TRUE);

    if (ppixd) {
        pixd = pixConvert8To32(pixs);
        *ppixd = pixd;
    }
    composeRGBPixel(255, 0, 0, &rpixel);  /* start point */
    composeRGBPixel(0, 255, 0, &gpixel);
    composeRGBPixel(0, 0, 255, &bpixel);  /* end point */

    x = xf;
    y = yf;
    pta = ptaCreate(0);
    while (1) {  /* write path onto pixd */
        ptaAddPt(pta, x, y);
        if (x == xi && y == yi)
            break;
        if (pixd)
            pixSetPixel(pixd, x, y, gpixel);
        pixGetPixel(pixp, x, y, &val);
        if (val == DIR_NORTH)
            y--;
        else if (val == DIR_SOUTH)
            y++;
        else if (val == DIR_EAST)
            x++;
        else if (val == DIR_WEST)
            x--;
        pixGetPixel(pixr, x, y, &val);

#if  DEBUG_PATH
        lept_stderr("(x,y) = (%d, %d); dist = %d\n", x, y, val);
#endif  /* DEBUG_PATH */

    }
    if (pixd) {
        pixSetPixel(pixd, xi, yi, rpixel);
        pixSetPixel(pixd, xf, yf, bpixel);
    }

cleanup_stuff:
    lheapDestroy(&lh, TRUE);
    pixDestroy(&pixp);
    pixDestroy(&pixr);
    LEPT_FREE(lines8);
    LEPT_FREE(linep8);
    LEPT_FREE(liner32);
    return pta;
}