A*寻路 -- 弗洛伊德（Floyd）算法

来源：互联网发布：阿卡索外教怎么样知乎编辑：程序博客网时间：2024/05/17 01:39

转：http://www.itweb2.com/article/system/317.htm

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：

使用A*得出基本路径
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]
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/

Java代码

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;
}
}