缘起

最近阅读<<我的第一本算法书>>(【日】石田保辉;宫崎修一)
本系列笔记拟采用golang练习之

A*(A-Star)算法

A*(A-Star)算法也是一种在图中求解最短路径问题的算法,
由狄克斯特拉算法发展而来。
A*算法不仅会考虑从起点到候补顶点的距离,
还会考虑从当前所在顶点到终点的估算距离。
距离估算值越接近当前顶点到终点的实际值,
A*算法的搜索效率也就越高.
当距离估算值小于实际距离时,
是一定可以得到正确答案的.
A*算法在游戏编程中经常被用于计算敌人追赶玩家时的行动路线等.
摘自 <<我的第一本算法书>> 【日】石田保辉;宫崎修一

场景

如下图, 某游戏中, 地图是网格状的, 我方在S点, 敌人在G点, 空白区域是湖泊/树林等不可到达区域:

现在需要追击敌人, 因此需要计算S点到G点的最短行进路线.
A*算法是以狄克斯特拉算法为基础, 区别是在计算候选节点的权重时, 需要同时考虑测量权重和估算权重.
此场景中, 使用S点到G点坐标的直线距离作为估算权重.
估算权重的作用就像牵引风筝的绳子, 使得每次选取的候选节点, 尽量是靠往终点方向.

流程

1. 给定若干顶点, 以及顶点间的若干条边, 寻找从指定起点srcNode到指定终点dstNode的最小权重路径
2. 设定srcNode的权重为0, 其他顶点的权重为无穷大
3. 计算所有节点到dstNode节点的估算距离, 以x,y坐标的直线距离作为估算值
4. 节点.总权重 = 节点.测量权重 + 节点.估算权重
5. 将srcNode节点送入候选堆
6. for 候选堆不为空:
   1. 从候选堆pop顶点node, node是总权重最小的候选节点
   2. 如果node.id == dstNode.id, 循环结束
   3. 遍历从node出发的所有边, 将边的终点to的测量权重, 更新为min(to.测量权重, node.测量权重+边.权重)
   4. 如果to.测量权重 > node.测量权重+边.权重, 说明更新有效
   5. 如果更新有效, 判断to是否在堆中, 如果是, 则上浮以维护堆秩序, 否则, 将to节点push入候选堆
7. 判断dstNode的测量权重是否被更新(!=无穷大), 如果是则说明存在最短路径
8. 反向查找最短路径:
   1. 设定当前节点current = 终点
   2. push节点current进路径队列
   3. 遍历终点为current的边, 查找符合条件的node:边的起点.测量权重 = current.测量权重-边.权重
   4. push节点node进路径队列
   5. 循环1-4, 直到current == srcNode, 查找完成

设计

• INode: 顶点接口, 支持xy坐标和估算权重
• ILine: 边接口
• IPathFinder: 最短路径查找算法接口
• IComparator: 顶点比较接口
• IHeap: 顶点堆接口
• tNode: 顶点, 实现INode
• tLine: 边, 实现ILine
• tNodeWeightComparator: 基于权重的顶点比较器, 实现IComparator接口
• tArrayHeap: 堆的实现
• tAStarPathFinder: A*算法的实现, 使用xy坐标的直线距离作为估算权重

单元测试

a_star_finder_test.go

package graph
import (
 "fmt"
 "strings"
 "testing"
)
import astar "learning/gooop/graph/a_star"
func Test_AStarFinder(t *testing.T) {
 fnAssertTrue := func(b bool, msg string) {
 if !b {
 t.Fatal(msg)
 }
 }
 // 设定顶点
 nodes := []astar.INode {
 astar.NewNode("11", 1, 1),
 astar.NewNode("21", 2, 1),
 astar.NewNode("31", 3, 1),
 astar.NewNode("12", 1, 2),
 astar.NewNode("32", 3, 2),
 astar.NewNode("13", 1, 3),
 astar.NewNode("33", 3, 3),
 astar.NewNode("43", 4, 3),
 astar.NewNode("53", 5, 3),
 astar.NewNode("63", 6, 3),
 astar.NewNode("73", 7, 3),
 astar.NewNode("14", 1, 4),
 astar.NewNode("34", 3, 4),
 astar.NewNode("74", 7, 4),
 astar.NewNode("15", 1, 5),
 astar.NewNode("35", 3, 5),
 astar.NewNode("55", 5, 5),
 astar.NewNode("65", 6, 5),
 astar.NewNode("75", 7, 5),
 astar.NewNode("16", 1, 6),
 astar.NewNode("36", 3, 6),
 astar.NewNode("56", 5, 6),
 astar.NewNode("17", 1, 7),
 astar.NewNode("27", 2, 7),
 astar.NewNode("37", 3, 7),
 astar.NewNode("47", 4, 7),
 astar.NewNode("57", 5, 7),
 astar.NewNode("67", 6, 7),
 astar.NewNode("77", 7, 7),
 }
 // 为相邻点创建边
 var lines []astar.ILine
 mapNodes := make(map[string]astar.INode, len(nodes))
 for _,it := range nodes {
 k := fmt.Sprintf("%v,%v", it.GetX(), it.GetY())
 mapNodes[k] = it
 }
 for _,it := range nodes {
 if up,ok := mapNodes[fmt.Sprintf("%v,%v", it.GetX(), it.GetY() - 1)];ok {
 lines = append(lines, astar.NewLine(it.ID(), up.ID(), 1), astar.NewLine(up.ID(), it.ID(), 1))
 }
 if down,ok := mapNodes[fmt.Sprintf("%v,%v", it.GetX(), it.GetY() + 1)];ok {
 lines = append(lines, astar.NewLine(it.ID(), down.ID(), 1), astar.NewLine(down.ID(), it.ID(), 1))
 }
 if left,ok := mapNodes[fmt.Sprintf("%v,%v", it.GetX()-1, it.GetY())];ok {
 lines = append(lines, astar.NewLine(it.ID(), left.ID(), 1), astar.NewLine(left.ID(), it.ID(), 1))
 }
 if right,ok := mapNodes[fmt.Sprintf("%v,%v", it.GetX()+1, it.GetY())];ok {
 lines = append(lines, astar.NewLine(it.ID(), right.ID(), 1), astar.NewLine(right.ID(), it.ID(), 1))
 }
 }
 // a*算法 查找最短路径
 ok,path := astar.AStarPathFinder.FindPath(nodes, lines, "33", "77")
 if !ok {
 t.Fatal("failed to find min path")
 }
 fnPathToString := func(nodes []astar.INode) string {
 items := make([]string, len(nodes))
 for i,it := range nodes {
 items[i] = fmt.Sprintf("%s", it)
 }
 return strings.Join(items, " ")
 }
 pathString := fnPathToString(path)
 t.Log(pathString)
 fnAssertTrue(pathString == "33(0+6) 34(1+5) 35(2+4) 36(3+4) 37(4+4) 47(5+3) 57(6+2) 67(7+1) 77(8+0)", "incorrect path")
}

测试输出

$ go test -v a_star_finder_test.go 
=== RUN Test_AStarFinder
 a_star_finder_test.go:96: 33(0+6) 34(1+5) 35(2+4) 36(3+4) 37(4+4) 47(5+3) 57(6+2) 67(7+1) 77(8+0)
--- PASS: Test_AStarFinder (0.00s)
PASS
ok command-line-arguments 0.002s

INode.go

顶点接口, 支持xy坐标和估算权重

package a_star
type INode interface {
 ID() string
 GetX() int
 GetY() int
 SetX(int)
 SetY(int)
 SetEstimatedWeight(int)
 SetMeasuredWeight(int)
 GetMeasuredWeight() int
 GetTotalWeight() int
}
const MaxWeight = int(0x7fffffff_00000000)

ILine.go

边接口

package a_star
type ILine interface {
 From() string
 To() string
 Weight() int
}

IPathFinder.go

最短路径查找算法接口

package a_star
type IPathFinder interface {
 FindPath(nodes []INode, lines []ILine, from string, to string) (bool,[]INode)
}

IComparator.go

顶点比较接口

package a_star
type IComparator interface {
 Less(a interface{}, b interface{}) bool
}

IHeap.go

顶点堆接口

package a_star
type IHeap interface {
 Size() int
 IsEmpty() bool
 IsNotEmpty() bool
 Push(node interface{})
 Pop() (bool, interface{})
 IndexOf(node interface{}) int
 ShiftUp(i int)
}

tNode.go

顶点, 实现INode

package a_star
import "fmt"
type tNode struct {
 id string
 x int
 y int
 measuredWeight int
 estimatedWeight int
}
func NewNode(id string, x int, y int) INode {
 return &tNode{
 id,x, y, MaxWeight,0,
 }
}
func (me *tNode) ID() string {
 return me.id
}
func (me *tNode) GetX() int {
 return me.x
}
func (me *tNode) GetY() int {
 return me.y
}
func (me *tNode) SetX(x int) {
 me.x = x
}
func (me *tNode) SetY(y int) {
 me.y = y
}
func (me *tNode) SetEstimatedWeight(w int) {
 me.estimatedWeight = w
}
func (me *tNode) SetMeasuredWeight(w int) {
 me.measuredWeight = w
}
func (me *tNode) GetMeasuredWeight() int {
 return me.measuredWeight
}
func (me *tNode) GetTotalWeight() int {
 return me.estimatedWeight + me.measuredWeight
}
func (me *tNode) String() string {
 return fmt.Sprintf("%s(%v+%v)", me.id, me.measuredWeight, me.estimatedWeight)
}

tLine.go

边, 实现ILine

package a_star
type tLine struct {
 from string
 to string
 weight int
}
func NewLine(from string, to string, weight int) ILine {
 return &tLine{
 from,to,weight,
 }
}
func (me *tLine) From() string {
 return me.from
}
func (me *tLine) To() string {
 return me.to
}
func (me *tLine) Weight() int {
 return me.weight
}

tNodeWeightComparator.go

基于权重的顶点比较器, 实现IComparator接口

package a_star
import "errors"
type tNodeWeightComparator struct {
}
func newNodeWeightComparator() IComparator {
 return &tNodeWeightComparator{
 }
}
func (me *tNodeWeightComparator) Less(a interface{}, b interface{}) bool {
 if a == nil || b == nil {
 panic(gNullArgumentError)
 }
 n1 := a.(INode)
 n2 := b.(INode)
 return n1.GetTotalWeight() <= n2.GetTotalWeight()
}
var gNullArgumentError = errors.New("null argument error")

tArrayHeap.go

堆的实现

package a_star
import (
 "errors"
 "fmt"
 "strings"
)
type tArrayHeap struct {
 comparator IComparator
 items []interface{}
 size int
 version int64
}
func newArrayHeap(comparator IComparator) IHeap {
 return &tArrayHeap{
 comparator: comparator,
 items: make([]interface{}, 0),
 size: 0,
 version: 0,
 }
}
func (me *tArrayHeap) Size() int {
 return me.size
}
func (me *tArrayHeap) IsEmpty() bool {
 return me.size <= 0
}
func (me *tArrayHeap) IsNotEmpty() bool {
 return !me.IsEmpty()
}
func (me *tArrayHeap) Push(value interface{}) {
 me.version++
 me.ensureSize(me.size + 1)
 me.items[me.size] = value
 me.size++
 me.ShiftUp(me.size - 1)
 me.version++
}
func (me *tArrayHeap) ensureSize(size int) {
 for ;len(me.items) < size; {
 me.items = append(me.items, nil)
 }
}
func (me *tArrayHeap) parentOf(i int) int {
 return (i - 1) / 2
}
func (me *tArrayHeap) leftChildOf(i int) int {
 return i*2 + 1
}
func (me *tArrayHeap) rightChildOf(i int) int {
 return me.leftChildOf(i) + 1
}
func (me *tArrayHeap) last() (i int, v interface{}) {
 if me.IsEmpty() {
 return -1, nil
 }
 i = me.size - 1
 v = me.items[i]
 return i,v
}
func (me *tArrayHeap) IndexOf(node interface{}) int {
 n := -1
 for i,it := range me.items {
 if it == node {
 n = i
 break
 }
 }
 return n
}
func (me *tArrayHeap) ShiftUp(i int) {
 if i <= 0 {
 return
 }
 v := me.items[i]
 pi := me.parentOf(i)
 pv := me.items[pi]
 if me.comparator.Less(v, pv) {
 me.items[pi], me.items[i] = v, pv
 me.ShiftUp(pi)
 }
}
func (me *tArrayHeap) Pop() (bool, interface{}) {
 if me.IsEmpty() {
 return false, nil
 }
 me.version++
 top := me.items[0]
 li, lv := me.last()
 me.items[0] = nil
 me.size--
 if me.IsEmpty() {
 return true, top
 }
 me.items[0] = lv
 me.items[li] = nil
 me.shiftDown(0)
 me.version++
 return true, top
}
func (me *tArrayHeap) shiftDown(i int) {
 pv := me.items[i]
 ok, ci, cv := me.minChildOf(i)
 if ok && me.comparator.Less(cv, pv) {
 me.items[i], me.items[ci] = cv, pv
 me.shiftDown(ci)
 }
}
func (me *tArrayHeap) minChildOf(p int) (ok bool, i int, v interface{}) {
 li := me.leftChildOf(p)
 if li >= me.size {
 return false, 0, nil
 }
 lv := me.items[li]
 ri := me.rightChildOf(p)
 if ri >= me.size {
 return true, li, lv
 }
 rv := me.items[ri]
 if me.comparator.Less(lv, rv) {
 return true, li, lv
 } else {
 return true, ri, rv
 }
}
func (me *tArrayHeap) String() string {
 level := 0
 lines := make([]string, 0)
 lines = append(lines, "")
 for {
 n := 1<<level
 min := n - 1
 max := n + min - 1
 if min >= me.size {
 break
 }
 line := make([]string, 0)
 for i := min;i <= max;i++ {
 if i >= me.size {
 break
 }
 line = append(line, fmt.Sprintf("%4d", me.items[i]))
 }
 lines = append(lines, strings.Join(line, ","))
 level++
 }
 return strings.Join(lines, "\n")
}
var gNoMoreElementsError = errors.New("no more elements")

tAStarPathFinder.go

A*算法的实现, 使用xy坐标的直线距离作为估算权重

package a_star
import "math"
type tAStarPathFinder struct {
}
func newAStarPathFinder() IPathFinder {
 return &tAStarPathFinder{}
}
func (me *tAStarPathFinder) FindPath(nodes []INode, lines []ILine, srcID string, dstID string) (bool,[]INode) {
 // 节点索引
 mapNodes := make(map[string]INode, 0)
 for _,it := range nodes {
 mapNodes[it.ID()] = it
 }
 srcNode, ok := mapNodes[srcID]
 if !ok {
 return false, nil
 }
 dstNode,ok := mapNodes[dstID]
 if !ok {
 return false, nil
 }
 // 边的索引
 mapFromLines := make(map[string][]ILine, 0)
 mapToLines := make(map[string][]ILine, 0)
 for _, it := range lines {
 if v,ok := mapFromLines[it.From()];ok {
 mapFromLines[it.From()] = append(v, it)
 } else {
 mapFromLines[it.From()] = []ILine{ it }
 }
 if v,ok := mapToLines[it.To()];ok {
 mapToLines[it.To()] = append(v, it)
 } else {
 mapToLines[it.To()] = []ILine{ it }
 }
 }
 for _,it := range nodes {
 // 设置src节点的weight为0, 其他节点的weight为MaxWeight
 if it.ID() == srcID {
 it.SetMeasuredWeight(0)
 } else {
 it.SetMeasuredWeight(MaxWeight)
 }
 // 计算每个节点到dst节点的估算距离
 if it.ID() == dstID {
 it.SetEstimatedWeight(0)
 } else {
 it.SetEstimatedWeight(me.distance(it.GetX(), it.GetY(), dstNode.GetY(), dstNode.GetY()))
 }
 }
 // 将起点push到堆
 heap := newArrayHeap(newNodeWeightComparator())
 heap.Push(srcNode)
 // 遍历候选节点
 for heap.IsNotEmpty() {
 _, top := heap.Pop()
 from := top.(INode)
 if from.ID() == dstID {
 break
 }
 links, ok := mapFromLines[from.ID()]
 if ok {
 for _,line := range links {
 if to,ok := mapNodes[line.To()];ok {
 if me.updateMeasuredWeight(from, to, line) {
 n := heap.IndexOf(to)
 if n >= 0 {
 heap.ShiftUp(n)
 } else {
 heap.Push(to)
 }
 }
 }
 }
 }
 }
 // 逆向查找最短路径
 if dstNode.GetMeasuredWeight() >= MaxWeight {
 return false, nil
 }
 path := []INode{ dstNode }
 current := dstNode
 maxRound := len(lines)
 for ;current != srcNode && maxRound > 0;maxRound-- {
 linkedLines, _ := mapToLines[current.ID()]
 for _,line := range linkedLines {
 from, _ := mapNodes[line.From()]
 if from.GetMeasuredWeight() == current.GetMeasuredWeight() - line.Weight() {
 current = from
 path = append(path, from)
 }
 }
 }
 if current != srcNode {
 return false, nil
 }
 me.reverse(path)
 return true, path
}
func (me *tAStarPathFinder) distance(x0, y0, x1, y1 int) int {
 dx := x0 - x1
 dy := y0 - y1
 return int(math.Round(math.Sqrt(float64(dx * dx + dy * dy))))
}
func (me *tAStarPathFinder) reverse(nodes []INode) {
 for i,j := 0, len(nodes)-1;i < j;i,j=i+1,j-1 {
 nodes[i], nodes[j] = nodes[j], nodes[i]
 }
}
func (me *tAStarPathFinder) updateMeasuredWeight(from INode, to INode, line ILine) bool {
 w := me.min(from.GetMeasuredWeight() + line.Weight(), to.GetMeasuredWeight())
 if to.GetMeasuredWeight() > w {
 to.SetMeasuredWeight(w)
 return true
 }
 return false
}
func (me *tAStarPathFinder) min(a, b int) int {
 if a <= b {
 return a
 }
 return b
}
var AStarPathFinder = newAStarPathFinder()

(end)

本文来自:简书

感谢作者:老罗话编程

查看原文:手撸golang 基本数据结构与算法 图的最短路径 A*(A-Star)算法

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