文心一言 VS 讯飞星火 VS chatgpt （383）-- 算法导论24.5 3题

package main

import(
"fmt"
"math"
)

// 定义图的边
typeEdgestruct{
From,Toint
Weightfloat64
}

// 计算最短路径权重的函数（使用 Floyd-Warshall 算法作为示例）
func shortestPaths(graph [][]Edge, numVertices int)[][]float64{
// 初始化距离矩阵
    dist :=make([][]float64, numVertices)
for i :=range dist {
        dist[i]=make([]float64, numVertices)
for j :=range dist[i]{
if i == j {
                dist[i][j]=0
}else{
                dist[i][j]= math.Inf(1)// 初始化为正无穷大
}
}
}

// 设置边的权重
for _, vertex :=range graph {
for _, edge :=range vertex {
            dist[edge.From][edge.To]= edge.Weight
}
}

// Floyd-Warshall 算法
for k :=0; k < numVertices; k++{
for i :=0; i < numVertices; i++{
for j :=0; j < numVertices; j++{
if dist[i][k]!= math.Inf(1)&& dist[k][j]!= math.Inf(1)&& dist[i][k]+dist[k][j]< dist[i][j]{
                    dist[i][j]= dist[i][k]+ dist[k][j]
}
}
}
}

return dist
}

// 检查三角不等式
func checkTriangleInequality(graph [][]Edge, source int, numVertices int){
    dist := shortestPaths(graph, numVertices)

for _, vertex :=range graph {
for _, edge :=range vertex {
            u, v := edge.From, edge.To
            w := edge.Weight
            deltaSU := dist[source][u]
            deltaSV := dist[source][v]

            fmt.Printf("Checking (%d, %d) with weight %f: δ(s,u) = %f, δ(s,v) = %f\n", u, v, w, deltaSU, deltaSV)
if!(deltaSV <= deltaSU+w){
                fmt.Printf("Triangle inequality violated for (%d, %d)!\n", u, v)
}else{
                fmt.Println("Triangle inequality holds.")
}
}
}
}

func main(){
// 示例图（邻接表表示）
    graph :=[][]Edge{
{{0,1,1.0},{0,2,4.0}},
{{1,2,2.0},{1,3,-math.Inf(1)}},// 包含 -∞ 的边
{{2,3,5.0}},
{},
}

    source :=0
    numVertices :=len(graph)

    checkTriangleInequality(graph, source, numVertices)
}

package main

import(
"fmt"
"math"
)

// Edge represents a directed edge with a weight
typeEdgestruct{
fromint
    to     int
    weight float64
}

// Graph represents a directed graph with weights
typeGraphstruct{
    vertices int
    edges    []Edge
}

// BellmanFord implements the Bellman-Ford algorithm to find shortest paths from source vertex
func BellmanFord(graph *Graph, source int)([]float64,error){
    distance :=make([]float64, graph.vertices)
for i :=range distance {
        distance[i]= math.Inf(1)// Initialize distances to infinity
}
    distance[source]=0// Distance to the source is zero

// Relax edges repeatedly
for i :=0; i < graph.vertices-1; i++{
for _, edge :=range graph.edges {
if distance[edge.from]!= math.Inf(1)&& distance[edge.from]+edge.weight < distance[edge.to]{
                distance[edge.to]= distance[edge.from]+ edge.weight
}
}
}

// Check for negative-weight cycles
for _, edge :=range graph.edges {
if distance[edge.from]!= math.Inf(1)&& distance[edge.from]+edge.weight < distance[edge.to]{
returnnil, fmt.Errorf("graph contains a negative-weight cycle")
}
}

return distance,nil
}

func main(){
// Example usage of Bellman-Ford algorithm
    graph :=&Graph{
        vertices:5,
        edges:[]Edge{
{from:0, to:1, weight:6},
{from:0, to:2, weight:7},
{from:1, to:2, weight:8},
{from:1, to:3, weight:5},
{from:1, to:4, weight:-4},
{from:2, to:3, weight:-3},
{from:2, to:4, weight:9},
{from:3, to:1, weight:-2},
{from:4, to:0, weight:2},
{from:4, to:3, weight:7},
},
}

    source :=0
    distances, err :=BellmanFord(graph, source)
if err !=nil{
        fmt.Println("Error:", err)
}else{
        fmt.Println("Shortest path distances from source:", distances)
}
}

package main

import(
"fmt"
"math"
)

// Weight represents the weight of an edge, which can be finite or infinite.
typeWeightfloat64

const(
InfinityWeight= math.Inf(1)
NegativeInfinityWeight= math.Inf(-1)
)

// ShortestPathWeight represents the shortest path weight from source to a vertex.
typeShortestPathWeightmap[string]Weight

// Graph represents a weighted directed graph.
typeGraphstruct{
Vertices[]string
Edgesmap[string]map[string]Weight
Sourcestring
ShortestPathsShortestPathWeight
}

// NewGraph creates a new graph with given vertices and edges.
func NewGraph(vertices []string, edges map[string]map[string]Weight, source string)*Graph{
return&Graph{
Vertices:    vertices,
Edges:       edges,
Source:      source,
ShortestPaths:make(ShortestPathWeight),
}
}

// InitializeShortestPaths initializes the shortest path weights from the source.
func (g *Graph)InitializeShortestPaths(){
for _, v :=range g.Vertices{
if v == g.Source{
            g.ShortestPaths[v]=0
}else{
            g.ShortestPaths[v]=Infinity
}
}
}

// Relax relaxes the edge (u, v) if a shorter path is found.
func (g *Graph)Relax(u, v string, weight Weight){
if g.ShortestPaths[u]+ weight < g.ShortestPaths[v]{
        g.ShortestPaths[v]= g.ShortestPaths[u]+ weight
}
}

// VerifyTriangleInequality verifies the triangle inequality for all edges.
func (g *Graph)VerifyTriangleInequality()bool{
for u, neighbors :=range g.Edges{
for v, weight :=range neighbors {
if g.ShortestPaths[u]!=NegativeInfinity&& g.ShortestPaths[v]!=Infinity{
if g.ShortestPaths[u]> g.ShortestPaths[v]+weight {
returnfalse
}
}
}
}
returntrue
}

func main(){
    vertices :=[]string{"s","u","v"}
    edges :=map[string]map[string]Weight{
"s":{"u":10},
"u":{"v":5},
"v":{},
}
    source :="s"

    graph :=NewGraph(vertices, edges, source)
    graph.InitializeShortestPaths()

// Relax edges to find shortest paths
    graph.Relax("s","u",10)
    graph.Relax("u","v",5)

    fmt.Println("Shortest Paths:", graph.ShortestPaths)
    fmt.Println("Triangle Inequality Verified:", graph.VerifyTriangleInequality())
}

package main
import(
"fmt"
"math"
)
// Edge represents a directed edge with a weight.
type Edgestruct{
fromstring
    to     string
    weight float64
}
// Graph represents a weighted directed graph.
type Graphstruct{
    edges map[string][]Edge
}
// NewGraph creates a new Graph.
func NewGraph()*Graph{
return&Graph{
        edges: make(map[string][]Edge),
}
}
// AddEdge adds a new edge to the graph.
func (g *Graph)AddEdge(from, to string, weight float64){
    g.edges[from]= append(g.edges[from],Edge{from:from, to: to, weight: weight})
}
// ShortestPath calculates the shortest path from source to target.
func (g *Graph)ShortestPath(source, target string) float64 {
    dist := make(map[string]float64)
for node := range g.edges {
        dist[node]= math.Inf(1)// Initialize distances with infinity
}
    dist[source]=0// Distance to the source is 0
// Relax edges repeatedly
for{
        updated :=false
for _, edges := range g.edges {
for _, edge := range edges {
if dist[edge.from]!= math.Inf(1)&& dist[edge.from]+edge.weight < dist[edge.to]{
                    dist[edge.to]= dist[edge.from]+ edge.weight
                    updated =true
}
}
}
if!updated {
break
}
}
return dist[target]
}
func main(){
    g :=NewGraph()
    g.AddEdge("s","u",1)
    g.AddEdge("u","v",2)
    g.AddEdge("s","v",4)// This edge should not be used in the shortest path from s to v
// Check the triangle inequality
    distSU := g.ShortestPath("s","u")
    distUV := g.ShortestPath("u","v")
    distSV := g.ShortestPath("s","v")
// Check if the triangle inequality holds, considering infinity
if distSV <= distSU+distUV || math.IsInf(distSU,1)|| math.IsInf(distUV,1){
        fmt.Println("Triangle inequality holds.")
}else{
        fmt.Println("Triangle inequality does not hold.")
}
}

请注意，上述代码中的`Graph`类型、`calculateDelta`函数和`getWeight`函数需要根据实际的图结构定义来实现。这个代码只是为了展示在处理\(∞\)和\(-∞\)情况时的大致框架。

## chatgpt：

引理24.10的三角不等式可以表述为：在一个带权重的有向图中，源节点 \( s \) 到节点 \( u \) 的最短路径权重应当小于或等于从 \( s \) 到 \( u \)、再加上从 \( u \) 到 \( v \) 的边的权重。这一定理本质上依赖于路径的加法性质，并且对于所有的节点及其之间的边有效。

要考虑最短路径权重为 \( \infty \) 和 \( -\infty \) 的情况，我们需要首先确定这些权重的意义：

1.**最短路径权重为 \( \infty \)**：表示从源节点 \( s \) 到节点 \( u \) 不可达，因此 \( \delta(s, u)= \infty \)。在这种情况下，任何加法 \( \delta(s, u)+ w(u, v) \) 的结果也会是 \( \infty \)，因此不等式仍然成立，因为
   \[
   \infty \leq \infty
   \]
并且如果 \( \delta(s, u)= \infty \) 且 \( w(u, v) \) 是有限值，那么结果为 \( \infty \) 不等于真实路径的情况是讨论的一种边界情况。

2.**最短路径权重为 \( -\infty \)**：表示存在路径可达性问题，通常会出现在有负权重环的情况下。这种情况下，我们常常会令最短路径标志为 \( -\infty \) 意味着在某个点之后开始无限缩小路径的权重，因此，对于所有延展的路径，构造出不等式的方式需要小心。

因此，在处理这些情况时，对引理24.10的表达和证明可以考虑如下逻辑：

### 改进的引理证明

对于每一条边 \( (u, v) \in E \)，我们考虑以下几种情况：

1.**假设** \( \delta(s, u)= \infty \)：
-由于 \( \delta(s, u) \) 是不可达的，那么对于任何值 \( w(u, v) \),
   \[
   \delta(s, u)+ w(u, v)= \infty + w(u, v)= \infty
   \]
不等式自然成立。

2.**假设** \( \delta(s, u)=-\infty \)：
-由于这里实际上意味着到达的路径存在负环，可形成无穷下降，因此所有后续添加 \( w(u, v) \) 的边也将导致总体路径权重趋近于负无穷，保持不等式成立。

3.**假设** \( \delta(s, u) \) 和 \( w(u, v) \) 是有限值：
-传统的三角不等式依旧成立：
   \[
   \delta(s, u)+ w(u, v) \text{结果也将是凡是} \delta(s, v) \text{的可达性无影响}
   \]

这种讨论确保了不等式在所有情况下均成立。

### Go 语言实现

以下是简单的Go实现，可以帮助判断并在图中采用三角不等式进行更新：

```go
package main

import (
    "fmt"
    "math"
)

const inf = math.Inf(1)  // 定义正无穷
const negInf = math.Inf(-1) // 定义负无穷

// Graph 存储图的结构
type Graph struct {
    vertices int
    edges    [][]float64
}

// NewGraph 创建一个新图
func NewGraph(v int) *Graph {
    g := &Graph{vertices: v}
    g.edges = make([][]float64, v)
    for i := range g.edges {
        g.edges[i] = make([]float64, v)
        for j := range g.edges[i] {
            g.edges[i][j] = inf // 初始化为无穷大
        }
    }
    return g
}

// AddEdge 添加边到图
func (g *Graph) AddEdge(u, v int, weight float64) {
    g.edges[u][v] = weight
}

// FloydWarshall 实现 Floyd-Warshall 算法
func (g *Graph) FloydWarshall() [][]float64 {
    dist := make([][]float64, g.vertices)
    for i := range dist {
        dist[i] = make([]float64, g.vertices)
        copy(dist[i], g.edges[i])
    }

    for k := 0; k < g.vertices; k++ {
        for i := 0; i < g.vertices; i++ {
            for j := 0; j < g.vertices; j++ {
                if dist[i][k] != inf && dist[k][j] != inf {
                    dist[i][j] = min(dist[i][j], dist[i][k]+dist[k][j])
                }
            }
        }
    }

    return dist
}

// min 辅助函数，返回最小值
func min(a, b float64) float64 {
    if a < b {
        return a
    }
    return b
}

func main() {
    g := NewGraph(4)
    g.AddEdge(0, 1, 3)
    g.AddEdge(1, 2, 4)
    g.AddEdge(2, 3, 2)
    g.AddEdge(0, 3, 10)
    g.AddEdge(3, 1, -6) // 负环边

    distances := g.FloydWarshall()

    for i := range distances {
        for j := range distances[i] {
            if distances[i][j] == inf {
                fmt.Print("∞ ")
            } else {
                fmt.Printf("%v ", distances[i][j])
            }
        }
        fmt.Println()
    }
}

package main

import(
"fmt"
"math"
)

// 定义最短路径权重类型
typePathWeightfloat64

// 检查三角不等式是否成立
func checkTriangleInequality(deltaSU, wUV PathWeight)bool{
return deltaSU <= deltaSU + wUV
}

func main(){
// 示例权重
    deltaSU :=PathWeight(math.Inf(1))// 表示无穷大
    wUV :=PathWeight(5)

// 检查不等式
if checkTriangleInequality(deltaSU, wUV){
        fmt.Println("三角不等式成立:", deltaSU,"<=", deltaSU+wUV)
}else{
        fmt.Println("三角不等式不成立")
}

// 示例权重
    deltaSU =PathWeight(math.Inf(-1))// 表示负无穷大
    wUV =PathWeight(5)

// 检查不等式
if checkTriangleInequality(deltaSU, wUV){
        fmt.Println("三角不等式成立:", deltaSU,"<=", deltaSU+wUV)
}else{
        fmt.Println("三角不等式不成立")
}
}

package main

import(
"fmt"
"math"
)

const(
    inf  = math.Inf(1)
    negInf = math.Inf(-1)
)

typeEdgestruct{
    to     int
    weight float64
}

func shortestPath(graph [][]Edge, source int)[]float64{
    n :=len(graph)
    dist :=make([]float64, n)
for i :=range dist {
if i == source {
            dist[i]=0
}else{
            dist[i]= inf
}
}

for i :=range graph {
for _, edge :=range graph[i]{
if dist[i]+edge.weight < dist[edge.to]{
                dist[edge.to]= dist[i]+ edge.weight
}
}
}

return dist
}

func main(){
    graph :=[][]Edge{
{{to:1, weight:1},{to:2, weight:4}},
{{to:2, weight:2},{to:3, weight:5}},
{{to:3, weight:1}},
{},
}

    source :=0
    dist := shortestPath(graph, source)

for i, d :=range dist {
if d == inf {
            fmt.Printf("Distance from %d to %d is ∞\n", source, i)
}elseif d == negInf {
            fmt.Printf("Distance from %d to %d is -∞\n", source, i)
}else{
            fmt.Printf("Distance from %d to %d is %.2f\n", source, i, d)
}
}
}