deepseek VS chatgpt （400）-- 算法导论25.2 9题

package main

import (
    "fmt"
)

//传递闭包计算函数
func transitiveClosure(graph [][]bool) [][]bool {
    n := len(graph)
    // 打印原始图
    fmt.Println("Original Graph:")
    printGraph(graph)

    // 创建一个用于存储传递闭包的图
    closure := make([][]bool, n)
    for i := range closure {
        closure[i] = make([]bool, n)
        copy(closure[i], graph[i])
    }

    // Floyd-Warshall算法
    for k := 0; k < n; k++ {
        for i := 0; i < n; i++ {
            for j := 0; j < n; j++ {
                closure[i][j] = closure[i][j] || (closure[i][k] && closure[k][j])
            }
        }
    }

    // 打印传递闭包
    fmt.Println("Transitive Closure:")
    printGraph(closure)
    return closure
}

// 打印图的函数
func printGraph(graph [][]bool) {
    for _, row := range graph {
        for _, val := range row {
            if val {
                fmt.Print("1 ")
            } else {
                fmt.Print("0 ")
            }
        }
        fmt.Println()
    }
}

func main() {
    // 示例图（有向图）
    graph := [][]bool{
        {false, true, false, false},
        {false, false, true, false},
        {false, false, false, true},
        {false, false, false, false},
    }

    // 计算传递闭包
    transitiveClosure(graph)
}

package main

import (
    "fmt"
)

func main() {
    V := 4
    edges := [][]int{
        {0, 1},
        {1, 2},
        {2, 0},
        {1, 3},
    }
    closure := transitiveClosure(V, edges)
    for u := range closure {
        for v := range closure[u] {
            fmt.Printf("%d→%d ", u, v)
        }
        fmt.Println()
    }
}

func transitiveClosure(V int, edges [][]int)map[int]map[int]bool {
    sccIDs, sccList := kosaraju(V, edges)
    dagEdges := buildDAG(sccIDs, edges)
    dagClosure := computeDAGTransitiveClosure(len(sccList), dagEdges)

    closure := make(map[int]map[int]bool)
    for i := 0; i < V; i++ {
        closure[i] = make(map[int]bool)
    }

    for aID, reachable := range dagClosure {
        for bID := range reachable {
            for _, u := range sccList[aID] {
                for _, v := range sccList[bID] {
                    closure[u][v] = true
                }
            }
        }
    }

    for _, scc := range sccList {
        for _, u := range scc {
            for _, v := range scc {
                closure[u][v] = true
            }
        }
    }

    return closure
}

func kosaraju(V int, edges [][]int) ([]int, [][]int) {
    adj := make([][]int, V)
    reverseAdj := make([][]int, V)
    for _, e := range edges {
        u, v := e[0], e[1]
        adj[u] = append(adj[u], v)
        reverseAdj[v] = append(reverseAdj[v], u)
    }

    visited := make([]bool, V)
    order := []int{}

    var dfs1 func(int)
    dfs1 = func(u int) {
        visited[u] = true
        for _, v := range adj[u] {
            if !visited[v] {
                dfs1(v)
            }
        }
        order = append(order, u)
    }

    for u := 0; u < V; u++ {
        if !visited[u] {
            dfs1(u)
        }
    }

    visited = make([]bool, V)
    sccIDs := make([]int, V)
    sccList := [][]int{}
    id := 0

    var dfs2 func(int)
    dfs2 = func(u int) {
        visited[u] = true
        sccIDs[u] = id
        iflen(sccList) <= id {
            sccList = append(sccList, []int{u})
        } else {
            sccList[id] = append(sccList[id], u)
        }
        for _, v := range reverseAdj[u] {
            if !visited[v] {
                dfs2(v)
            }
        }
    }

    for i := len(order) - 1; i >= 0; i-- {
        u := order[i]
        if !visited[u] {
            dfs2(u)
            id++
        }
    }

    return sccIDs, sccList
}

func buildDAG(sccIDs []int, edges [][]int) [][]int {
    maxID := 0
    for _, id := range sccIDs {
        if id > maxID {
            maxID = id
        }
    }
    dagEdges := make([][]int, maxID+1)
    edgeSet := make(map[[2]int]bool)

    for _, e := range edges {
        u, v := e[0], e[1]
        uSCC := sccIDs[u]
        vSCC := sccIDs[v]
        if uSCC != vSCC && !edgeSet[[2]int{uSCC, vSCC}] {
            dagEdges[uSCC] = append(dagEdges[uSCC], vSCC)
            edgeSet[[2]int{uSCC, vSCC}] = true
        }
    }

    return dagEdges
}

func computeDAGTransitiveClosure(n int, dagEdges [][]int) []map[int]bool {
    closure := make([]map[int]bool, n)
    for i := range closure {
        closure[i] = make(map[int]bool)
        closure[i][i] = true
    }

    inDegree := make([]int, n)
    for u := range dagEdges {
        for _, v := range dagEdges[u] {
            inDegree[v]++
        }
    }

    queue := []int{}
    for i := 0; i < n; i++ {
        if inDegree[i] == 0 {
            queue = append(queue, i)
        }
    }

    order := []int{}
    forlen(queue) > 0 {
        u := queue[0]
        queue = queue[1:]
        order = append(order, u)
        for _, v := range dagEdges[u] {
            inDegree[v]--
            if inDegree[v] == 0 {
                queue = append(queue, v)
            }
        }
    }

    for i := len(order) - 1; i >= 0; i-- {
        u := order[i]
        for _, v := range dagEdges[u] {
            for node := range closure[v] {
                closure[u][node] = true
            }
        }
    }

    return closure
}

package main

import (
    "container/list"
    "fmt"
)

type Graph struct {
    vertices []*list.List
}

func NewGraph(size int) *Graph {
    g := &Graph{vertices: make([]*list.List, size)}
    for i := range g.vertices {
        g.vertices[i] = list.New()
    }
    return g
}

// 添加边
func (g *Graph) AddEdge(u, v int) {
    g.vertices[u].PushBack(v)
}

// Kosaraju算法的第一步：DFS并填充栈
func dfsFirst(graph *Graph, u int, visited []bool, stack *list.List) {
    visited[u] = true
    for e := graph.vertices[u].Front(); e != nil; e = e.Next() {
        v := e.Value.(int)
        if !visited[v] {
            dfsFirst(graph, v, visited, stack)
        }
    }
    stack.PushBack(u)
}

// Kosaraju算法的第二步：在转置图上的DFS
func dfsSecond(graph *Graph, u int, visited []bool, component *list.List) {
    visited[u] = true
    component.PushBack(u)
    for e := graph.vertices[u].Front(); e != nil; e = e.Next() {
        v := e.Value.(int)
        if !visited[v] {
            dfsSecond(graph, v, visited, component)
        }
    }
}

// 获取强连通分量
func GetStronglyConnectedComponents(graph *Graph) [][]int {
    var result [][]int
    stack := list.New()
    visited := make([]bool, len(graph.vertices))

    for i := range graph.vertices {
        if !visited[i] {
            dfsFirst(graph, i, visited, stack)
        }
    }

    // 创建转置图...
    // 然后继续进行第二次DFS...

    return result
}

func main() {
    graph := NewGraph(5)
    graph.AddEdge(1, 0)
    graph.AddEdge(0, 2)
    graph.AddEdge(2, 1)
    graph.AddEdge(0, 3)
    graph.AddEdge(3, 4)

    fmt.Println(GetStronglyConnectedComponents(graph))
}

package main

import (
    "fmt"
)

// 计算强连通分量（Tarjan 算法）
func tarjan(graph [][]int) [][]int {
    var index, sccCount int
    n := len(graph)
    indices := make([]int, n)
    for i := range indices {
        indices[i] = -1
    }
    lowlinks := make([]int, n)
    onStack := make([]bool, n)
    stack := []int{}
    sccs := [][]int{}

    var strongConnect func(v int)
    strongConnect = func(v int) {
        indices[v] = index
        lowlinks[v] = index
        index++
        stack = append(stack, v)
        onStack[v] = true

        for _, w := range graph[v] {
            if indices[w] == -1 {
                strongConnect(w)
                if lowlinks[w] < lowlinks[v] {
                    lowlinks[v] = lowlinks[w]
                }
            } elseif onStack[w] {
                if indices[w] < lowlinks[v] {
                    lowlinks[v] = indices[w]
                }
            }
        }

        if lowlinks[v] == indices[v] {
            scc := []int{}
            for {
                w := stack[len(stack)-1]
                stack = stack[:len(stack)-1]
                onStack[w] = false
                scc = append(scc, w)
                if w == v {
                    break
                }
            }
            sccs = append(sccs, scc)
            sccCount++
        }
    }

    for v := 0; v < n; v++ {
        if indices[v] == -1 {
            strongConnect(v)
        }
    }

    return sccs
}

// 模拟计算有向无环图的传递闭包
func calculateDAGTransitiveClosure(graph [][]int) [][]bool {
    n := len(graph)
    closure := make([][]bool, n)
    for i := range closure {
        closure[i] = make([]bool, n)
        for j := range closure[i] {
            if i == j {
                closure[i][j] = true
            }
        }
    }

    for u := 0; u < n; u++ {
        for _, v := range graph[u] {
            closure[u][v] = true
        }
    }

    for k := 0; k < n; k++ {
        for i := 0; i < n; i++ {
            for j := 0; j < n; j++ {
                closure[i][j] = closure[i][j] || (closure[i][k] && closure[k][j])
            }
        }
    }

    return closure
}

// 计算通用有向图的传递闭包
func calculateTransitiveClosure(graph [][]int) [][]bool {
    sccs := tarjan(graph)
    n := len(graph)
    closure := make([][]bool, n)
    for i := range closure {
        closure[i] = make([]bool, n)
        for j := range closure[i] {
            if i == j {
                closure[i][j] = true
            }
        }
    }

    // 计算有向无环图的传递闭包
    dagClosure := calculateDAGTransitiveClosure(graph)

    // 恢复传递闭包到原始图
    for i := 0; i < n; i++ {
        for j := 0; j < n; j++ {
            closure[i][j] = dagClosure[i][j]
        }
    }

    return closure
}

func main() {
    graph := [][]int{
        {1},
        {2},
        {0},
        {4},
        {5},
        {3},
    }

    closure := calculateTransitiveClosure(graph)
    fmt.Println(closure)
}

package main

import (
    "fmt"
)

// Graph represents a directed graph
type Graph struct {
    vertices int
    edges    map[int]map[int]struct{}
}

// NewGraph creates a new graph with a given number of vertices
func NewGraph(vertices int) *Graph {
    return &Graph{
        vertices: vertices,
        edges:    make(map[int]map[int]struct{}),
    }
}

// AddEdge adds an edge to the graph
func (g *Graph) AddEdge(from, to int) {
    if _, exists := g.edges[from]; !exists {
        g.edges[from] = make(map[int]struct{})
    }
    g.edges[from][to] = struct{}{}
}

// TarjanSCC performs Tarjan's algorithm to find SCCs (placeholder)
func (g *Graph) TarjanSCC() [][]int {
    // Implement Tarjan's algorithm to find SCCs
    // Return a slice of slices, where each inner slice represents an SCC
    return [][]int{}
}

// TransitiveClosureDAG computes the transitive closure of a DAG (placeholder)
func TransitiveClosureDAG(dag *Graph) *Graph {
    // Implement the given function f to compute the transitive closure of a DAG
    // Return a new graph representing the transitive closure
    return dag // Placeholder return
}

// ExpandSCCs expands SCCs back to the original graph (placeholder)
func ExpandSCCs(sccGraph *Graph, sccs [][]int, closureDAG *Graph) *Graph {
    // Expand the SCCs back to the original graph structure
    // Return a new graph representing the transitive closure of the original graph
    return closureDAG // Placeholder return
}

// TransitiveClosure computes the transitive closure of a general directed graph
func TransitiveClosure(g *Graph) *Graph {
    sccs := g.TarjanSCC()
    dag := NewGraph(len(sccs))

    // Build the DAG from SCCs
    sccMap := make(map[int]int) // Maps original vertices to SCC indices
    for i, scc := range sccs {
        for _, v := range scc {
            sccMap[v] = i
        }
    }

    for u := 0; u < g.vertices; u++ {
        for v := range g.edges[u] {
            if sccMap[u] != sccMap[v] {
                dag.AddEdge(sccMap[u], sccMap[v])
            }
        }
    }

    closureDAG := TransitiveClosureDAG(dag)
    closureGraph := ExpandSCCs(dag, sccs, closureDAG)

    return closureGraph
}

func main() {
    // Example usage
    g := NewGraph(5)
    g.AddEdge(0, 1)
    g.AddEdge(1, 2)
    g.AddEdge(2, 0) // Creates a cycle
    g.AddEdge(3, 4)

    closure := TransitiveClosure(g)

    // Print the closure graph (implementation depends on the graph representation used)
    fmt.Println("Transitive Closure Graph:")
    // Placeholder for printing the graph
}

package main

import (
    "fmt"
)

// TransitiveClosure calculates the transitive closure of a DAG in O(f(|V|, |E|)) time.
func TransitiveClosure(dag map[int][]int)map[int][]int {
    // Placeholder for the actual implementation of the transitive closure calculation.
    return dag // This should be replaced with the actual logic.
}

// FindSCCs finds strongly connected components using Tarjan's algorithm.
func FindSCCs(graph map[int][]int) [][]int {
    // Placeholder for the actual implementation of Tarjan's algorithm.
    return [][]int{} // This should be replaced with the actual logic.
}

// MergeSCCResults merges the results of transitive closures of individual SCCs.
func MergeSCCResults(sccs [][]int, sccClosures []map[int][]int)map[int][]int {
    result := make(map[int][]int)
    for i, scc := range sccs {
        closure := sccClosures[i]
        for u := range closure {
            for _, v := range closure[u] {
                result[u] = append(result[u], v)
            }
        }
    }
    return result
}

// ComputeTransitiveClosure computes the transitive closure of a general directed graph.
func ComputeTransitiveClosure(graph map[int][]int)map[int][]int {
    sccs := FindSCCs(graph)
    sccClosures := make([]map[int][]int, len(sccs))
    for i, scc := range sccs {
        sccGraph := make(map[int][]int)
        for _, u := range scc {
            for _, v := range graph[u] {
                if contains(scc, v) {
                    sccGraph[u] = append(sccGraph[u], v)
                }
            }
        }
        sccClosures[i] = TransitiveClosure(sccGraph)
    }
    return MergeSCCResults(sccs, sccClosures)
}

// Helper function to check if an element is in a slice.
func contains(slice []int, item int)bool {
    for _, v := range slice {
        if v == item {
            returntrue
        }
    }
    returnfalse
}

func main() {
    // Example usage:
    graph := map[int][]int{
        0: {1},
        1: {2},
        2: {0, 3},
        3: {4},
        4: {},
    }

    transitiveClosure := ComputeTransitiveClosure(graph)
    fmt.Println("Transitive Closure:", transitiveClosure)
}

// Build DAG adjacency list 
compAdj := make(map[int]map[int]bool)
for u := 0; u < n; u++ {
    for _, v := range edges[u] {
        uComp := components[u]
        vComp := components[v]
        if uComp != vComp {
            if compAdj[uComp] == nil {
                compAdj[uComp] = make(map[int]bool)
            }
            compAdj[uComp][vComp] = true 
        }
    }
}

// Compute DAG closure using BFS 
dagClosure := make(map[int]map[int]bool)
for comp := range compToNodes {
    visited := make(map[int]bool)
    queue := []int{comp}
    visited[comp] = true 
    for len(queue) > 0 {
        current := queue[0]
        queue = queue[1:]
        for neighbor := range compAdj[current] {
            if !visited[neighbor] {
                visited[neighbor] = true 
                queue = append(queue, neighbor)
            }
        }
    }
    dagClosure[comp] = visited 
}

// Build transitive closure for original graph 
closure := make([][]bool, n)
for i := range closure {
    closure[i] = make([]bool, n)
}

// Add edges within each SCC 
for _, nodes := range compToNodes {
    for _, u := range nodes {
        for _, v := range nodes {
            if u != v {
                closure[u][v] = true 
            }
        }
    }
}

// Add edges between SCCs based on DAG closure 
for aComp, reachable := range dagClosure {
    for bComp := range reachable {
        if aComp == bComp {
            continue 
        }
        for _, u := range compToNodes[aComp] {
            for _, v := range compToNodes[bComp] {
                closure[u][v] = true 
            }
        }
    }
}

return closure 

var dfs1 func(int)
dfs1 = func(u int) {
    visited[u] = true 
    for _, v := range adj[u] {
        if !visited[v] {
            dfs1(v)
        }
    }
    order = append(order, u)
}

for u := 0; u < n; u++ {
    if !visited[u] {
        dfs1(u)
    }
}

reverseAdj := make([][]int, n)
for u := 0; u < n; u++ {
    for _, v := range adj[u] {
        reverseAdj[v] = append(reverseAdj[v], u)
    }
}

components := make([]int, n)
currentComp := 0 
visited = make([]bool, n)

var dfs2 func(int)
dfs2 = func(u int) {
    visited[u] = true 
    components[u] = currentComp 
    for _, v := range reverseAdj[u] {
        if !visited[v] {
            dfs2(v)
        }
    }
}

for i := n - 1; i >= 0; i-- {
    u := order[i]
    if !visited[u] {
        dfs2(u)
        currentComp++
    }
}

return components 

package main
import (
    "fmt"
)
// 假设的DAG传递闭包计算函数
func computeTransitiveClosureDAG(V, E int) {
    // 这里应该是使用f(|V|, |E|)算法计算DAG的传递闭包
    fmt.Println("Compute transitive closure for DAG with", V, "vertices and", E, "edges")
}
// Tarjan算法实现强连通分量查找（简化版）
func findSCCs(graph [][]int) [][]int {
    // 这里应该是Tarjan算法的实现
    // 返回值是图的强连通分量列表
    return [][]int{} // 简化处理
}
// 计算原图的传递闭包
func computeTransitiveClosure(graph [][]int) {
    SCCs := findSCCs(graph)
    V := len(graph)
    E := 0// 这里应该计算边的数量，简化处理
    // 计算DAG的传递闭包
    computeTransitiveClosureDAG(V, E)
    // 构建原图的传递闭包
    // 这里需要根据SCCs之间的可达性来更新原图的传递闭包
}
func main() {
    graph := [][]int{
        // 图的邻接表表示
    }
    computeTransitiveClosure(graph)
}

func findSCCs(graph [][]int) [][]int {
    // 使用Tarjan或Kosaraju算法实现
}

func buildCondensation(sccs [][]int, graph [][]int) ([][]int, []int) {
    // sccs是各个SCC的列表
    // 为每个节点分配所在的SCC编号
    // 然后构建缩点后的图的邻接表
}

func dagTransitiveClosure(condensedGraph [][]int) [][]bool {
    // 使用已知的DAG传递闭包算法，比如拓扑排序后动态规划
}

func generateTransitiveClosure(sccs [][]int, sccMap []int, closure [][]bool) map[int]map[int]bool {
    // 对于每个SCC内的节点，添加所有可能的边
    // 对于closure中为true的SCC对，添加所有跨边
}

package main

import (
    "fmt"
)

// 强连通分量分解（示例使用Kosaraju算法）
func findSCCs(graph [][]int) [][]int {
    n := len(graph)
    visited := make([]bool, n)
    order := make([]int, 0)
    reverseGraph := make([][]int, n)
    
    var dfs1 func(int)
    dfs1 = func(u int) {
        visited[u] = true
        for _, v := range graph[u] {
            if !visited[v] {
                dfs1(v)
            }
        }
        order = append(order, u)
    }
    
    for u := 0; u < n; u++ {
        if !visited[u] {
            dfs1(u)
        }
    }
    
    for u := 0; u < n; u++ {
        for _, v := range graph[u] {
            reverseGraph[v] = append(reverseGraph[v], u)
        }
    }
    
    visited = make([]bool, n)
    var scc []int
    var sccs [][]int
    
    var dfs2 func(int)
    dfs2 = func(u int) {
        visited[u] = true
        scc = append(scc, u)
        for _, v := range reverseGraph[u] {
            if !visited[v] {
                dfs2(v)
            }
        }
    }
    
    for i := len(order) - 1; i >= 0; i-- {
        u := order[i]
        if !visited[u] {
            scc = make([]int, 0)
            dfs2(u)
            sccs = append(sccs, scc)
        }
    }
    return sccs
}

// 构建缩点后的DAG
func buildCondensation(sccs [][]int, graph [][]int) ([][]int, []int) {
    n := len(graph)
    sccID := make([]int, n)
    for id, scc := range sccs {
        for _, u := range scc {
            sccID[u] = id
        }
    }
    
    condensed := make([][]int, len(sccs))
    added := make(map[int]bool)
    
    for u := 0; u < n; u++ {
        for _, v := range graph[u] {
            from := sccID[u]
            to := sccID[v]
            if from != to && !added[to] {
                condensed[from] = append(condensed[from], to)
                added[to] = true
            }
        }
        added = make(map[int]bool) // Reset for next u
    }
    return condensed, sccID
}

// DAG的传递闭包（示例使用Floyd-Warshall）
func dagTransitiveClosure(condensedGraph [][]int) [][]bool {
    n := len(condensedGraph)
    closure := make([][]bool, n)
    for i := range closure {
        closure[i] = make([]bool, n)
    }
    
    // Initialize direct edges
    for u := range condensedGraph {
        for _, v := range condensedGraph[u] {
            closure[u][v] = true
        }
    }
    
    // Floyd-Warshall
    for k := 0; k < n; k++ {
        for i := 0; i < n; i++ {
            for j := 0; j < n; j++ {
                closure[i][j] = closure[i][j] || (closure[i][k] && closure[k][j])
            }
        }
    }
    return closure
}

// 生成传递闭包E*
func generateTransitiveClosure(sccs [][]int, sccID []int, closure [][]bool)map[int]map[int]bool {
    eStar := make(map[int]map[int]bool)
    
    // 初始化邻接表
    for u := 0; u < len(sccID); u++ {
        eStar[u] = make(map[int]bool)
    }
    
    // 添加同一SCC内的所有边
    for _, scc := range sccs {
        for _, u := range scc {
            for _, v := range scc {
                eStar[u][v] = true
            }
        }
    }
    
    // 添加跨SCC的边
    for aID := range sccs {
        for bID := range sccs {
            if closure[aID][bID] {
                for _, u := range sccs[aID] {
                    for _, v := range sccs[bID] {
                        eStar[u][v] = true
                    }
                }
            }
        }
    }
    return eStar
}

func main() {
    // 示例输入（有向图）
    graph := [][]int{
        {1},          // 0 -> 1
        {2},          // 1 -> 2
        {0, 3},       // 2 -> 0, 2 -> 3
        {4},          // 3 -> 4
        {},           // 4
    }
    
    // Step 1: 找到SCC
    sccs := findSCCs(graph)
    fmt.Println("SCCs:", sccs)
    
    // Step 2: 构建缩点后的DAG
    condensed, sccID := buildCondensation(sccs, graph)
    fmt.Println("Condensed DAG:", condensed)
    
    // Step 3: 计算缩点图的传递闭包
    closure := dagTransitiveClosure(condensed)
    fmt.Println("Closure:")
    for _, row := range closure {
        fmt.Println(row)
    }
    
    // Step 4: 生成原图的传递闭包E*
    eStar := generateTransitiveClosure(sccs, sccID, closure)
    
    // 输出部分边（示例）
    fmt.Println("\nSample edges in E*:")
    for u := 0; u < 5; u++ {
        for v := 0; v < 5; v++ {
            if eStar[u][v] {
                fmt.Printf("%d -> %d\n", u, v)
            }
        }
    }
}

package main

import (
    "fmt"
    "github.com/yourusername/yourgraphlibrary"// 假设你有一个图库
)

// 计算强连通分量
func findSCCs(graph *yourgraphlibrary.Graph) [][]int {
    // 使用 Tarjan 或 Kosaraju 算法
    returnnil// 返回强连通分量的列表
}

// 构造压缩图
func constructDAG(sccs [][]int, graph *yourgraphlibrary.Graph) *yourgraphlibrary.Graph {
    // 将每个强连通分量缩点，得到 DAG
    returnnil// 返回 DAG
}

// 计算 DAG 的传递闭包
func computeDAGTransitiveClosure(dag *yourgraphlibrary.Graph) *yourgraphlibrary.Graph {
    // 使用题目假设的算法
    returnnil// 返回 DAG 的传递闭包
}

// 展开传递闭包
func expandTransitiveClosure(dagTransitiveClosure *yourgraphlibrary.Graph, sccs [][]int) *yourgraphlibrary.Graph {
    // 将 DAG 的传递闭包展开回原图的传递闭包
    returnnil// 返回原图的传递闭包
}

// 计算通用有向图的传递闭包
func computeTransitiveClosure(graph *yourgraphlibrary.Graph) *yourgraphlibrary.Graph {
    sccs := findSCCs(graph)
    dag := constructDAG(sccs, graph)
    dagTransitiveClosure := computeDAGTransitiveClosure(dag)
    transitiveClosure := expandTransitiveClosure(dagTransitiveClosure, sccs)
    return transitiveClosure
}

func main() {
    graph := yourgraphlibrary.NewGraph()
    // 添加顶点和边
    transitiveClosure := computeTransitiveClosure(graph)
    fmt.Println("传递闭包：", transitiveClosure)
}

package main

import (
    "fmt"
)

// 假设 f 是一个已实现的函数，计算DAG的传递闭包
// 这里我们用一个简单的Floyd-Warshall算法作为示例
func f(n int, edges [][2]int) [][2]int {
    // 初始化邻接矩阵
    adj := make([][]bool, n)
    for i := range adj {
        adj[i] = make([]bool, n)
    }
    for _, edge := range edges {
        u, v := edge[0], edge[1]
        adj[u][v] = true
    }

    // Floyd-Warshall
    for k := 0; k < n; k++ {
        for i := 0; i < n; i++ {
            if adj[i][k] {
                for j := 0; j < n; j++ {
                    if adj[k][j] {
                        adj[i][j] = true
                    }
                }
            }
        }
    }

    // 收集所有边
    transitiveClosure := [][2]int{}
    for i := 0; i < n; i++ {
        for j := 0; j < n; j++ {
            if adj[i][j] {
                transitiveClosure = append(transitiveClosure, [2]int{i, j})
            }
        }
    }
    return transitiveClosure
}

// 计算传递约简（这里简化为只保留必要的边）
func transitiveReduction(n int, edges [][2]int) [][2]int {
    // 简单实现：移除如果存在u->v且存在w使得u->w->v，则移除u->v
    reduced := make([][2]int, 0, len(edges))
    keep := make(map[[2]int]bool)
    for _, edge := range edges {
        u, v := edge[0], edge[1]
        if !existsPath(reduced, u, v) {
            reduced = append(reduced, edge)
            keep[edge] = true
        }
    }
    return reduced
}

// 检查在reduced中是否存在从u到v的路径
func existsPath(reduced [][2]int, u, v int)bool {
    visited := make([]bool, len(reduced))
    var dfs func(int)bool
    dfs = func(current int)bool {
        if current == v {
            returntrue
        }
        for _, edge := range reduced {
            if edge[0] == current && !visited[edge[1]] {
                visited[edge[1]] = true
                if dfs(edge[1]) {
                    returntrue
                }
            }
        }
        returnfalse
    }
    return dfs(u)
}

// 计算传递闭包
func computeTransitiveClosure(n int, edges [][2]int) [][2]int {
    // 步骤1：传递约简
    reducedEdges := transitiveReduction(n, edges)

    // 步骤2：在约简后的图上计算传递闭包
    reducedClosure := f(n, reducedEdges)

    // 步骤3：扩展到原图的传递闭包
    // 这里假设f已经包含了所有必要的传递信息
    // 如果需要进一步扩展，可以根据具体需求实现
    return reducedClosure
}

func main() {
    // 示例图
    n := 4
    edges := [][2]int{
        {0, 1},
        {1, 2},
        {2, 3},
        {0, 3},
    }

    closure := computeTransitiveClosure(n, edges)
    fmt.Println("传递闭包的边:")
    for _, edge := range closure {
        fmt.Printf("(%d, %d)\n", edge[0], edge[1])
    }
}