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

  • Elli_ON
  • 2
  • 2021-03-02 08:59

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)



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