手撸golang 基本数据结构与算法 图的最短路径 贝尔曼-福特算法

Posted 2021-09-28 ioly

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

贝尔曼-福特算法

贝尔曼-福特（Bellman-Ford）算法是一种在图中求解最短路径问题的算法。
最短路径问题就是在加权图指定了起点和终点的前提下，
寻找从起点到终点的路径中权重总和最小的那条路径。

摘自 <<我的第一本算法书>> 【日】石田保辉；宫崎修一

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from到指定终点to的最小权重路径
2. 设定from的权重为0, 其他顶点的权重为无穷大
3. 将from节点送入候选队列

4. for 候选队列不为空:

   1. 从候选队列出队一个顶点node
   2. 遍历从node出发的所有边, 将边的终点权重, 更新为min(终点权重, node.权重+边.权重)
   3. 如果终点权重 > node.权重+边.权重, 说明更新有效, 则将终点push到候选队列
5. 判断终点的权重是否被更新(!=无穷大), 如果是则说明存在最短路径

6. 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.权重 = current.权重-边.权重
   4. push节点node进路径队列
   5. 循环1-4, 直到current == from, 查找完成

目标

• 实现并验证贝尔曼-福特算法

设计

• INode: 顶点接口
• ILine: 边接口
• IPathFinder: 最短路径查找算法接口
• iNodeQueue: 顶点队列接口, FIFO队列
• tNode: 顶点, 实现INode
• tLine: 边, 实现ILine
• tFIFOQueue: FIFO节点队列的实现
• tBellmanFoldFinder: 贝尔曼-福特算法的实现

单元测试

bellman_fold_test.go

package graph

import (
    "fmt"
    bf "learning/gooop/graph/bellman_fold"
    "strings"
    "testing"
)

func Test_BellmanFold(t *testing.T) {
    fnAssertTrue := func(b bool, msg string) {
        if !b {
            t.Fatal(msg)
        }
    }

    nodes := []bf.INode{
        bf.NewNode("a"),
        bf.NewNode("b"),
        bf.NewNode("c"),
        bf.NewNode("d"),
        bf.NewNode("e"),
        bf.NewNode("f"),
        bf.NewNode("g"),
    }

    lines := []bf.ILine {
        bf.NewLine("a", "b", 9),
        bf.NewLine("a", "c", 2),

        bf.NewLine("b", "c", 6),
        bf.NewLine("b", "d", 3),
        bf.NewLine("b", "e", 1),

        bf.NewLine("c", "d", 2),
        bf.NewLine("c", "f", 9),

        bf.NewLine("d", "e", 5),
        bf.NewLine("d", "f", 6),

        bf.NewLine("e", "f", 3),
        bf.NewLine("e", "g", 7),

        bf.NewLine("f", "g", 4),
    }

    for _,it := range lines[:] {
        lines = append(lines, bf.NewLine(it.To(), it.From(), it.Weight()))
    }

    ok,path := bf.BellmanFoldFinder.FindPath(nodes, lines, "a", "g")
    if !ok {
        t.Fatal("failed to find min path")
    }
    fnPathToString := func(nodes []bf.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)
    fnAssertTrue(pathString == "a(0) c(2) d(4) f(10) g(14)", "incorrect path")
}

测试输出

$ go test -v bellman_fold_test.go 
=== RUN   Test_BellmanFold
    bellman_fold_test.go:63: a(0) c(2) d(4) f(10) g(14)
--- PASS: Test_BellmanFold (0.00s)
PASS
ok      command-line-arguments  0.002s

INode.go

顶点接口

package bellman_fold

type INode interface {
    ID() string
    GetWeight() int
    SetWeight(int)
}

const MaxWeight = int(0x7fffffff_ffffffff)

ILine.go

边接口

package bellman_fold

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

IPathFinder.go

最短路径查找算法接口

package bellman_fold

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

iNodeQueue.go

顶点队列接口, FIFO队列

package bellman_fold

type iNodeQueue interface {
    Clear()
    Size() int
    Empty() bool
    Push(node INode)
    Poll() (bool, INode)
}

tNode.go

顶点, 实现INode

package bellman_fold

import "fmt"

type tNode struct {
    id string
    weight int
}

func NewNode(id string) INode {
    return &tNode{
        id,MaxWeight,
    }
}

func (me *tNode) ID() string {
    return me.id
}

func (me *tNode) GetWeight() int {
    return me.weight
}

func (me *tNode) SetWeight(w int) {
    me.weight = w
}

func (me *tNode) String() string {
    return fmt.Sprintf("%s(%v)", me.id, me.weight)
}

tLine.go

边, 实现ILine

package bellman_fold

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
}

tFIFOQueue.go

FIFO节点队列的实现

package bellman_fold

type tFIFOQueue struct {
    nodes []INode
    capacity int
    rindex int
    windex int
}

func newFIFOQueue() iNodeQueue {
    it := &tFIFOQueue{}
    it.Clear()
    return it
}

func (me *tFIFOQueue) Clear() {
    me.nodes = make([]INode, 0)
    me.capacity = 0
    me.rindex = -1
    me.windex = -1
}

func (me *tFIFOQueue) Size() int {
    return me.windex - me.rindex
}

func (me *tFIFOQueue) Empty() bool {
    return me.Size() <= 0
}

func (me *tFIFOQueue) Push(node INode) {
    me.ensureSpace(1)
    me.windex++
    me.nodes[me.windex] = node
}

func (me *tFIFOQueue) ensureSpace(size int) {
    for me.capacity < me.windex + size + 1 {
        me.nodes = append(me.nodes, nil)
        me.capacity++
    }
}

func (me *tFIFOQueue) Poll() (bool, INode) {
    if me.Empty() {
        return false, nil
    }

    me.rindex++
    it := me.nodes[me.rindex]
    me.nodes[me.rindex] = nil

    if me.rindex > me.capacity / 2 {
        size := me.Size()
        offset := me.rindex + 1
        for i := 0;i < size;i++ {
            me.nodes[i], me.nodes[i + offset] = me.nodes[i + offset], nil
        }

        me.rindex -= offset
        me.windex -= offset
    }

    return true, it
}

tBellmanFoldFinder.go

贝尔曼-福特算法的实现

package bellman_fold

type tBellmanFoldFinder struct {
}


func newBellmanFoldFinder() IPathFinder {
    return &tBellmanFoldFinder{
    }
}

func (me *tBellmanFoldFinder) FindPath(nodes []INode, lines []ILine, fromID string, toID string) (bool,[]INode) {
    // 节点索引
    mapNodes := make(map[string]INode, 0)
    for _,it := range nodes {
        mapNodes[it.ID()] = it
    }

    fromNode, ok := mapNodes[fromID]
    if !ok {
        return false, nil
    }

    toNode,ok := mapNodes[toID]
    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 }
        }
    }

    // 设置from节点的weight为0, 其他节点的weight为MaxWeight
    for _,it := range nodes {
        if it.ID() == fromID {
            it.SetWeight(0)
        } else {
            it.SetWeight(MaxWeight)
        }
    }

    // 循环更新所有节点的权重 直到不再变化
    fromNode.SetWeight(0)
    queue := newFIFOQueue()
    queue.Push(fromNode)
    for !queue.Empty() {
        ok,from := queue.Poll()
        if !ok {
            panic("unexpected !ok")
        }

        affectedLines, ok := mapFromLines[from.ID()]
        if ok {
            for _,line := range affectedLines {
                if to,ok := mapNodes[line.To()];ok {
                    if me.updateWeight(from, to, line) {
                        queue.Push(to)
                    }
                }
            }
        }
    }

    // 逆向查找最短路径
    if toNode.GetWeight() >= MaxWeight {
        return false, nil
    }

    queue.Clear()
    queue.Push(toNode)
    current := toNode
    maxRound := len(lines)
    for ;current != fromNode && maxRound > 0;maxRound-- {
        linkedLines, _ := mapToLines[current.ID()]
        for _,line := range linkedLines {
            from, _ := mapNodes[line.From()]
            if from.GetWeight() == current.GetWeight() - line.Weight() {
                current = from
                queue.Push(from)
            }
        }
    }

    if current != fromNode {
        return false, nil
    }

    // 返回
    result := make([]INode, queue.Size())
    for i := queue.Size() - 1;i >= 0;i-- {
        _,result[i] = queue.Poll()
    }
    return true, result
}

func (me *tBellmanFoldFinder) updateWeight(from INode, to INode, line ILine) bool {
    w := me.min(from.GetWeight() + line.Weight(), to.GetWeight())
    if to.GetWeight() > w {
        to.SetWeight(w)
        return true
    }

    return false
}

func (me *tBellmanFoldFinder) min(a, b int) int {
    if a <= b {
        return a
    }
    return b
}

var BellmanFoldFinder = newBellmanFoldFinder()

(end)