缘起

最近浏览 << 我的第一本算法书 >>(【日】石田保辉；宫崎修一)
本系列笔记拟采纳 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门路问题的算法。最短门路问题就是在加权图指定了终点和起点的前提下，寻找从终点到起点的门路中权重总和最小的那条门路。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, 寻找从指定终点 from 到指定起点 to 的最小权重门路
2. 设定 from 的权重为 0, 其余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 登程的所有边, 将边的起点权重, 更新为 min(起点权重, node. 权重 + 边. 权重)
   3. 如果起点权重 > node. 权重 + 边. 权重, 阐明更新无效, 则将起点 push 到候选队列
5. 判断起点的权重是否被更新 (!= 无穷大), 如果是则阐明存在最短门路

6. 反向查找最短门路:

   1. 设定以后节点 current = 起点
   2. push 节点 current 进门路队列
   3. 遍历起点为 current 的边, 查找符合条件的 node: 边的终点. 权重 = current. 权重 - 边. 权重
   4. push 节点 node 进门路队列
   5. 循环 1 -4, 直到 current == from, 查找实现

指标

• 实现并验证贝尔曼 - 福特算法

设计

• INode: 顶点接口
• ILine: 边接口
• IPathFinder: 最短门路查找算法接口
• 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")
}

测试输入

$ 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

最短门路查找算法接口

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, 其余节点的 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 直到不再变动
    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)
                    }
                }
            }
        }
    }

    // 逆向查找最短门路
    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)