`

【转】A*寻路 -- 弗洛伊德(Floyd)算法

 
阅读更多

http://chaimzane.iteye.com/blog/1629062

 

源码下载

 

弗洛伊德(Floyd)算法过程:
1、用D[v][w]记录每一对顶点的最短距离。
2、依次扫描每一个点,并以其为基点再遍历所有每一对顶点D[][]的值,看看是否可用过该基点让这对顶点间的距离更小。
算法理解:
最短距离有三种情况:
1、两点的直达距离最短。(如下图<v,x>)
2、两点间只通过一个中间点而距离最短。(图<v,u>)
3、两点间用通过两各以上的顶点而距离最短。(图<v,w>)

对于第一种情况:在初始化的时候就已经找出来了且以后也不会更改到。
对于第二种情况:弗洛伊德算法的基本操作就是对于每一对顶点,遍历所有其它顶点,看看可否通过这一个顶点让这对顶点距离更短,也就是遍历了图中所有的三角形(算法中对同一个三角形扫描了九次,原则上只用扫描三次即可,但要加入判断,效率更低)。
对 于第三种情况:如下图的五边形,可先找一点(比如x,使<v,u>=2),就变成了四边形问题,再找一点(比如y,使<u,w>=2),可变成三角形问题了(v,u,w),也就变成第二种情况了,由此对于n边形也可以一步步转化成四边形三角形问题。(这里面不用担心哪个点要 先找哪个点要后找,因为找了任一个点都可以使其变成(n-1)边形的问题)。

 

 


 

使用前

 

使用后

 

其本思路:

  1. 使用A*得出基本路径
  2. 删除路径中方向相同的节点 比如 [0,1],[0,2],[0,3],[1,2] 可表现为 [0,1][0,3][1,2]
  3. 把余下的节点做为转角,代入flody算法进行计算,最后得出最简洁的方法。

在用flody计算两两转角是否连通时,需要获得一直线上经过的格子。可参考:http://25swf.blogbus.com/logs/82350359.html

flody算法:参考 http://www.itweb2.com/article/system/317.htm 
A*参考:http://eidiot.net/2007/04/17/a-star-pathfinding/

 

package {
    /**
     * ...
     * @author sliz http://game-develop.net/blog/
     */
    import flash.display.Bitmap;
    import flash.display.BitmapData;
    import flash.display.Sprite;
    import flash.display.StageAlign;
    import flash.display.StageScaleMode;
    import flash.events.Event;
    import flash.events.MouseEvent;
    import flash.geom.Point;
    import flash.geom.Rectangle;
    import flash.text.TextField;
    import flash.utils.getTimer;
    import sliz.miniui.Button;
    import sliz.miniui.Checkbox;
    import sliz.miniui.Label;
    import sliz.miniui.LabelInput;
    import sliz.miniui.layouts.BoxLayout;
    import sliz.miniui.Window;
 
    public class Game2 extends Sprite {
        private var _cellSize:int = 5;
        private var _grid:Grid;
        private var _player:Sprite;
        private var _index:int;
        private var _path:Array;
 
        private var tf:Label;
        private var astar:AStar;
 
        private var path:Sprite = new Sprite();
        private var image:Bitmap = new Bitmap(new BitmapData(1, 1));
        private var imageWrapper:Sprite = new Sprite();
 
        public function Game2(){
            stage.align = StageAlign.TOP_LEFT;
            stage.scaleMode = StageScaleMode.NO_SCALE;
            addChild(imageWrapper);
            imageWrapper.addChild(image);
            makePlayer();
 
            var w:Window = new Window(this, 20, 20, "tool");
            numCols = new LabelInput("numCols ", "numCols");
            numCols.setValue("50");
            w.add(numCols);
            numRows = new LabelInput("numRows ", "numRows");
            w.add(numRows);
            numRows.setValue("50");
            cellSize = new LabelInput("cellSize", "cellSize");
            cellSize.setValue("10");
            w.add(cellSize);
            density = new LabelInput("density ", "density");
            density.setValue("0.1");
            w.add(density);
            isEight = new Checkbox("是否8方向");
            isEight.setToggle(true);
            w.add(isEight);
            tf = new Label("info");
            w.add(tf);
            w.add(new sliz.miniui.Link("author sliz"));
            w.add(new sliz.miniui.Link("source", "http://code.google.com/p/actionscriptiui/"));
            var btn:Button = new Button("新建", 0, 0, null, newMap);
            w.add(btn, null, 0.8);
            w.setLayout(new BoxLayout(w, 1, 5));
            w.doLayout();
            imageWrapper.addEventListener(MouseEvent.CLICK, onGridClick);
            addEventListener(Event.ENTER_FRAME, onEnterFrame);
            imageWrapper.addChild(path);
            makeGrid();
        }
 
        private function newMap(e:Event):void {
            makeGrid();
        }
 
        private function changeMode(e:Event):void {
        /*if (_grid.getType()==1) {
           _grid.calculateLinks(0);
           (e.currentTarget as Button).text = "  四方向  ";
           }else {
           _grid.calculateLinks(1);
           (e.currentTarget as Button).text = "  八方向  ";
         }*/
        }
 
        private function makePlayer():void {
            _player = new Sprite();
            _player.graphics.beginFill(0xff00ff);
            _player.graphics.drawCircle(0, 0, 2);
            _player.graphics.endFill();
            imageWrapper.addChild(_player);
        }
 
        private function makeGrid():void {
            var rows:int = int(numRows.getValue());
            var cols:int = int(numCols.getValue());
            _cellSize = int(cellSize.getValue());
            _grid = new Grid(cols, rows);
            for (var i:int = 0; i < rows * cols * Number(density.getValue()); i++){
                _grid.setWalkable(Math.floor(Math.random() * cols), Math.floor(Math.random() * rows), false);
            }
            _grid.setWalkable(0, 0, true);
            _grid.setWalkable(cols / 2, rows / 2, false);
            if (isEight.getToggle())
                _grid.calculateLinks();
            else
                _grid.calculateLinks(1);
            astar = new AStar(_grid);
            drawGrid();
            isClick = false;
            _player.x = 0;
            _player.y = 0;
            path.graphics.clear();
        }
 
 
        private function drawGrid():void {
            image.bitmapData = new BitmapData(_grid.numCols * _cellSize, _grid.numRows * _cellSize, false, 0xffffff);
            for (var i:int = 0; i < _grid.numCols; i++){
                for (var j:int = 0; j < _grid.numRows; j++){
                    var node:Node = _grid.getNode(i, j);
                    if (!node.walkable){
                        image.bitmapData.fillRect(new Rectangle(i * _cellSize, j * _cellSize, _cellSize, _cellSize), getColor(node));
                    }
                }
            }
        }
 
        private function getColor(node:Node):uint {
            if (!node.walkable)
                return 0;
            if (node == _grid.startNode)
                return 0xcccccc;
            if (node == _grid.endNode)
                return 0xcccccc;
            return 0xffffff;
        }
 
        private function onGridClick(event:MouseEvent):void {
            var xpos:int = Math.floor(mouseX / _cellSize);
            var ypos:int = Math.floor(mouseY / _cellSize);
            xpos = Math.min(xpos, _grid.numCols - 1);
            ypos = Math.min(ypos, _grid.numRows - 1);
            _grid.setEndNode(xpos, ypos);
 
            xpos = Math.floor(_player.x / _cellSize);
            ypos = Math.floor(_player.y / _cellSize);
            _grid.setStartNode(xpos, ypos);
            findPath();
            //path.graphics.clear();
            //path.graphics.lineStyle(0, 0xff0000,0.5);
            //path.graphics.moveTo(_player.x, _player.y);
        }
 
        private function findPath():void {
            var time:int = getTimer();
            if (astar.findPath()){
                _index = 0;
                isClick = true;
 
                astar.floyd();
                _path = astar.floydPath;
                time = getTimer() - time;
                tf.text = time + "ms   length:" + astar.path.length;
                trace(astar.floydPath);
                path.graphics.clear();
                for (var i:int = 0; i < astar.floydPath.length; i++){
                    var p:Node = astar.floydPath[i];
                    path.graphics.lineStyle(0, 0xff0000);
                    path.graphics.drawCircle((p.x + 0.5) * _cellSize, (p.y + 0.5) * _cellSize, 2);
 
                    path.graphics.lineStyle(0, 0xff0000, 0.5);
                    path.graphics.moveTo(_player.x, _player.y);
                }
            } else {
                time = getTimer() - time;
                tf.text = time + "ms 找不到";
            }
        }
 
        private var isClick:Boolean = false;
        private var numCols:LabelInput;
        private var numRows:LabelInput;
        private var cellSize:LabelInput;
        private var density:LabelInput;
        private var isEight:Checkbox;
 
        private function onEnterFrame(event:Event):void {
            if (!isClick){
                return;
            }
            var targetX:Number = _path[_index].x * _cellSize + _cellSize / 2;
            var targetY:Number = _path[_index].y * _cellSize + _cellSize / 2;
            var dx:Number = targetX - _player.x;
            var dy:Number = targetY - _player.y;
            var dist:Number = Math.sqrt(dx * dx + dy * dy);
            if (dist < 1){
                _index++;
                if (_index >= _path.length){
                    isClick = false;
                }
            } else {
                _player.x += dx * .5;
                _player.y += dy * .5;
                path.graphics.lineTo(_player.x, _player.y);
            }
        }
    }
}
 
import flash.geom.Point;
 
class AStar {
    //private var _open:Array;
    private var _open:BinaryHeap;
    private var _grid:Grid;
    private var _endNode:Node;
    private var _startNode:Node;
    private var _path:Array;
    private var _floydPath:Array;
    public var heuristic:Function;
    private var _straightCost:Number = 1.0;
    private var _diagCost:Number = Math.SQRT2;
    private var nowversion:int = 1;
 
    public function AStar(grid:Grid){
        this._grid = grid;
        heuristic = euclidian2;
 
    }
 
    private function justMin(x:Object, y:Object):Boolean {
        return x.f < y.f;
    }
 
    public function findPath():Boolean {
        _endNode = _grid.endNode;
        nowversion++;
        _startNode = _grid.startNode;
        //_open = [];
        _open = new BinaryHeap(justMin);
        _startNode.g = 0;
        return search();
    }
 
    public function floyd():void {
        if (path == null)
            return;
        _floydPath = path.concat();
        var len:int = _floydPath.length;
        if (len > 2){
            var vector:Node = new Node(0, 0);
            var tempVector:Node = new Node(0, 0);
            floydVector(vector, _floydPath[len - 1], _floydPath[len - 2]);
            for (var i:int = _floydPath.length - 3; i >= 0; i--){
                floydVector(tempVector, _floydPath[i + 1], _floydPath[i]);
                if (vector.x == tempVector.x && vector.y == tempVector.y){
                    _floydPath.splice(i + 1, 1);
                } else {
                    vector.x = tempVector.x;
                    vector.y = tempVector.y;
                }
            }
        }
        len = _floydPath.length;
        for (i = len - 1; i >= 0; i--){
            for (var j:int = 0; j <= i - 2; j++){
                if (floydCrossAble(_floydPath[i], _floydPath[j])){
                    for (var k:int = i - 1; k > j; k--){
                        _floydPath.splice(k, 1);
                    }
                    i = j;
                    len = _floydPath.length;
                    break;
                }
            }
        }
    }
 
    private function floydCrossAble(n1:Node, n2:Node):Boolean {
        var ps:Array = bresenhamNodes(new Point(n1.x, n1.y), new Point(n2.x, n2.y));
        for (var i:int = ps.length - 2; i > 0; i--){
            if (!_grid.getNode(ps[i].x, ps[i].y).walkable){
                return false;
            }
        }
        return true;
    }
 
    private function bresenhamNodes(p1:Point, p2:Point):Array {
        var steep:Boolean = Math.abs(p2.y - p1.y) > Math.abs(p2.x - p1.x);
        if (steep){
            var temp:int = p1.x;
            p1.x = p1.y;
            p1.y = temp;
            temp = p2.x;
            p2.x = p2.y;
            p2.y = temp;
        }
        var stepX:int = p2.x > p1.x ? 1 : (p2.x < p1.x ? -1 : 0);
        var stepY:int = p2.y > p1.y ? 1 : (p2.y < p1.y ? -1 : 0);
        var deltay:Number = (p2.y - p1.y) / Math.abs(p2.x - p1.x);
        var ret:Array = [];
        var nowX:Number = p1.x + stepX;
        var nowY:Number = p1.y + deltay;
        if (steep){
            ret.push(new Point(p1.y, p1.x));
        } else {
            ret.push(new Point(p1.x, p1.y));
        }
        while (nowX != p2.x){
            var fy:int = Math.floor(nowY)
            var cy:int = Math.ceil(nowY);
            if (steep){
                ret.push(new Point(fy, nowX));
            } else {
                ret.push(new Point(nowX, fy));
            }
            if (fy != cy){
                if (steep){
                    ret.push(new Point(cy, nowX));
                } else {
                    ret.push(new Point(nowX, cy));
                }
            }
            nowX += stepX;
            nowY += deltay;
        }
        if (steep){
            ret.push(new Point(p2.y, p2.x));
        } else {
            ret.push(new Point(p2.x, p2.y));
        }
        return ret;
    }
 
    private function floydVector(target:Node, n1:Node, n2:Node):void {
        target.x = n1.x - n2.x;
        target.y = n1.y - n2.y;
    }
 
    public function search():Boolean {
        var node:Node = _startNode;
        node.version = nowversion;
        while (node != _endNode){
            var len:int = node.links.length;
            for (var i:int = 0; i < len; i++){
                var test:Node = node.links[i].node;
                var cost:Number = node.links[i].cost;
                var g:Number = node.g + cost;
                var h:Number = heuristic(test);
                var f:Number = g + h;
                if (test.version == nowversion){
                    if (test.f > f){
                        test.f = f;
                        test.g = g;
                        test.h = h;
                        test.parent = node;
                    }
                } else {
                    test.f = f;
                    test.g = g;
                    test.h = h;
                    test.parent = node;
                    _open.ins(test);
                    test.version = nowversion;
                }
 
            }
            if (_open.a.length == 1){
                return false;
            }
            node = _open.pop() as Node;
        }
        buildPath();
        return true;
    }
 
    private function buildPath():void {
        _path = [];
        var node:Node = _endNode;
        _path.push(node);
        while (node != _startNode){
            node = node.parent;
            _path.unshift(node);
        }
    }
 
    public function get path():Array {
        return _path;
    }
 
    public function get floydPath():Array {
        return _floydPath;
    }
 
    public function manhattan(node:Node):Number {
        return Math.abs(node.x - _endNode.x) + Math.abs(node.y - _endNode.y);
    }
 
    public function manhattan2(node:Node):Number {
        var dx:Number = Math.abs(node.x - _endNode.x);
        var dy:Number = Math.abs(node.y - _endNode.y);
        return dx + dy + Math.abs(dx - dy) / 1000;
    }
 
    public function euclidian(node:Node):Number {
        var dx:Number = node.x - _endNode.x;
        var dy:Number = node.y - _endNode.y;
        return Math.sqrt(dx * dx + dy * dy);
    }
 
    private var TwoOneTwoZero:Number = 2 * Math.cos(Math.PI / 3);
 
    public function chineseCheckersEuclidian2(node:Node):Number {
        var y:int = node.y / TwoOneTwoZero;
        var x:int = node.x + node.y / 2;
        var dx:Number = x - _endNode.x - _endNode.y / 2;
        var dy:Number = y - _endNode.y / TwoOneTwoZero;
        return sqrt(dx * dx + dy * dy);
    }
 
    private function sqrt(x:Number):Number {
        return Math.sqrt(x);
    }
 
    public function euclidian2(node:Node):Number {
        var dx:Number = node.x - _endNode.x;
        var dy:Number = node.y - _endNode.y;
        return dx * dx + dy * dy;
    }
 
    public function diagonal(node:Node):Number {
        var dx:Number = Math.abs(node.x - _endNode.x);
        var dy:Number = Math.abs(node.y - _endNode.y);
        var diag:Number = Math.min(dx, dy);
        var straight:Number = dx + dy;
        return _diagCost * diag + _straightCost * (straight - 2 * diag);
    }
}
 
 
class BinaryHeap {
    public var a:Array = [];
    public var justMinFun:Function = function(x:Object, y:Object):Boolean {
        return x < y;
    };
 
    public function BinaryHeap(justMinFun:Function = null){
        a.push(-1);
        if (justMinFun != null)
            this.justMinFun = justMinFun;
    }
 
    public function ins(value:Object):void {
        var p:int = a.length;
        a[p] = value;
        var pp:int = p >> 1;
        while (p > 1 && justMinFun(a[p], a[pp])){
            var temp:Object = a[p];
            a[p] = a[pp];
            a[pp] = temp;
            p = pp;
            pp = p >> 1;
        }
    }
 
    public function pop():Object {
        var min:Object = a[1];
        a[1] = a[a.length - 1];
        a.pop();
        var p:int = 1;
        var l:int = a.length;
        var sp1:int = p << 1;
        var sp2:int = sp1 + 1;
        while (sp1 < l){
            if (sp2 < l){
                var minp:int = justMinFun(a[sp2], a[sp1]) ? sp2 : sp1;
            } else {
                minp = sp1;
            }
            if (justMinFun(a[minp], a[p])){
                var temp:Object = a[p];
                a[p] = a[minp];
                a[minp] = temp;
                p = minp;
                sp1 = p << 1;
                sp2 = sp1 + 1;
            } else {
                break;
            }
        }
        return min;
    }
}
 
class Grid {
 
    private var _startNode:Node;
    private var _endNode:Node;
    private var _nodes:Array;
    private var _numCols:int;
    private var _numRows:int;
 
    private var type:int;
 
    private var _straightCost:Number = 1.0;
    private var _diagCost:Number = Math.SQRT2;
 
    public function Grid(numCols:int, numRows:int){
        _numCols = numCols;
        _numRows = numRows;
        _nodes = new Array();
 
        for (var i:int = 0; i < _numCols; i++){
            _nodes[i] = new Array();
            for (var j:int = 0; j < _numRows; j++){
                _nodes[i][j] = new Node(i, j);
            }
        }
    }
 
    /**
     *
     * @param   type    0四方向 1八方向 2跳棋
     */
    public function calculateLinks(type:int = 0):void {
        this.type = type;
        for (var i:int = 0; i < _numCols; i++){
            for (var j:int = 0; j < _numRows; j++){
                initNodeLink(_nodes[i][j], type);
            }
        }
    }
 
    public function getType():int {
        return type;
    }
 
    /**
     *
     * @param   node
     * @param   type    0八方向 1四方向 2跳棋
     */
    private function initNodeLink(node:Node, type:int):void {
        var startX:int = Math.max(0, node.x - 1);
        var endX:int = Math.min(numCols - 1, node.x + 1);
        var startY:int = Math.max(0, node.y - 1);
        var endY:int = Math.min(numRows - 1, node.y + 1);
        node.links = [];
        for (var i:int = startX; i <= endX; i++){
            for (var j:int = startY; j <= endY; j++){
                var test:Node = getNode(i, j);
                if (test == node || !test.walkable){
                    continue;
                }
                if (type != 2 && i != node.x && j != node.y){
                    var test2:Node = getNode(node.x, j);
                    if (!test2.walkable){
                        continue;
                    }
                    test2 = getNode(i, node.y);
                    if (!test2.walkable){
                        continue;
                    }
                }
                var cost:Number = _straightCost;
                if (!((node.x == test.x) || (node.y == test.y))){
                    if (type == 1){
                        continue;
                    }
                    if (type == 2 && (node.x - test.x) * (node.y - test.y) == 1){
                        continue;
                    }
                    if (type == 2){
                        cost = _straightCost;
                    } else {
                        cost = _diagCost;
                    }
                }
                node.links.push(new Link(test, cost));
            }
        }
    }
 
    public function getNode(x:int, y:int):Node {
        return _nodes[x][y];
    }
 
    public function setEndNode(x:int, y:int):void {
        _endNode = _nodes[x][y];
    }
 
    public function setStartNode(x:int, y:int):void {
        _startNode = _nodes[x][y];
    }
 
    public function setWalkable(x:int, y:int, value:Boolean):void {
        _nodes[x][y].walkable = value;
    }
 
    public function get endNode():Node {
        return _endNode;
    }
 
    public function get numCols():int {
        return _numCols;
    }
 
    public function get numRows():int {
        return _numRows;
    }
 
    public function get startNode():Node {
        return _startNode;
    }
 
}
 
class Link {
    public var node:Node;
    public var cost:Number;
 
    public function Link(node:Node, cost:Number){
        this.node = node;
        this.cost = cost;
    }
 
}
 
class Node {
    public var x:int;
    public var y:int;
    public var f:Number;
    public var g:Number;
    public var h:Number;
    public var walkable:Boolean = true;
    public var parent:Node;
    //public var costMultiplier:Number = 1.0;
    public var version:int = 1;
    public var links:Array;
 
    //public var index:int;
    public function Node(x:int, y:int){
        this.x = x;
        this.y = y;
    }
 
    public function toString():String {
        return "x:" + x + " y:" + y;
    }
}

 

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