【算法笔记自学】第 10 章 提高篇（4）——图算法专题

Polaris北极星少女2024-07-14 19:27

10.1图的定义和相关术语

cpp 复制代码 
#include <cstdio>

const int MAXN = 100;
int degree[MAXN] = {0};

int main() {
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int j = 0; j < m; j++) {
        scanf("%d%d", &u, &v);
        degree[u]++;
        degree[v]++;
    }
    for (int i = 0; i < n; i++) {
        printf("%d", degree[i]);
        if (i < n - 1) {
            printf(" ");
        }
    }
    return 0;
}
cpp 复制代码 
#include <cstdio>

const int MAXN = 100;
int inDegree[MAXN] = {0};
int outDegree[MAXN] = {0};

int main() {
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int j = 0; j < m; j++) {
        scanf("%d%d", &u, &v);
        outDegree[u]++;
        inDegree[v]++;
    }
    for (int i = 0; i < n; i++) {
        printf("%d %d\n", inDegree[i], outDegree[i]);
    }
    return 0;
}

10.2图的存储

cpp 复制代码 
#include <cstdio>
#include <cstring>

const int MAXN = 100;
int G[MAXN][MAXN];

int main() {
    memset(G, 0, sizeof(G));
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u][v] = G[v][u] = 1;
    }
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            printf("%d", G[i][j]);
            printf(j < n - 1 ? " " : "\n");
        }
    }
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <cstring>

const int MAXN = 100;
int G[MAXN][MAXN];

int main() {
    memset(G, 0, sizeof(G));
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u][v] = 1;
    }
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            printf("%d", G[i][j]);
            printf(j < n - 1 ? " " : "\n");
        }
    }
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];

int main() {
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        G[v].push_back(u);
    }
    for (int i = 0; i < n; i++) {
        printf("%d(%d)", i, (int)G[i].size());
        for (int j = 0; j < G[i].size(); j++) {
            printf(" %d", G[i][j]);
        }
        printf("\n");
    }
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <vector>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];

int main() {
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
    }
    for (int i = 0; i < n; i++) {
        printf("%d(%d)", i, (int)G[i].size());
        for (int j = 0; j < G[i].size(); j++) {
            printf(" %d", G[i][j]);
        }
        printf("\n");
    }
    return 0;
}

10.3图的遍历

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];
bool vis[MAXN];

void DFS(int u) {
    vis[u] = true;
    for (int i = 0; i < G[u].size(); i++) {
        int v = G[u][i];
        if (!vis[v]) {
            DFS(v);
        }
    }
}

int main() {
    memset(vis, false, sizeof(vis));
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        G[v].push_back(u);
    }
    int blockCount = 0;
    for (int i = 0; i < n; i++) {
        if (!vis[i]) {
            DFS(i);
            blockCount++;
        }
    }
    printf("%d", blockCount);
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <cstring>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];
bool vis[MAXN];

void DFS(int u) {
    vis[u] = true;
    for (int i = 0; i < G[u].size(); i++) {
        int v = G[u][i];
        if (!vis[v]) {
            DFS(v);
        }
    }
}

int main() {
    memset(vis, false, sizeof(vis));
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        G[v].push_back(u);
    }
    int blockCount = 0;
    for (int i = 0; i < n; i++) {
        if (!vis[i]) {
            DFS(i);
            blockCount++;
        }
    }
    printf(blockCount == 1 ? "Yes" : "No");
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <cstring>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];
int vis[MAXN];

bool isCyclic(int u) {
    vis[u] = 0;
    for (int i = 0; i < G[u].size(); i++) {
        int v = G[u][i];
        if (vis[v] == -1 && isCyclic(v)) {
            return true;
        } else if (vis[v] == 0) {
            return true;
        }
    }
    vis[u] = 1;
    return false;
}

int main() {
    memset(vis, -1, sizeof(vis));
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
    }
    int isCyclicResult = false;
    for (int i = 0; i < n; i++) {
        if (vis[i] == -1 && isCyclic(i)) {
            isCyclicResult = true;
        }
        if (isCyclicResult) {
            break;
        }
    }
    printf(isCyclicResult ? "Yes" : "No");
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <vector>
#include <algorithm>
#include <cstring>
using namespace std;

const int MAXN = 100;
int weight[MAXN];
vector<int> G[MAXN];
bool vis[MAXN];

int DFS(int u) {
    vis[u] = true;
    int weightSum = weight[u];
    for (int i = 0; i < G[u].size(); i++) {
        int v = G[u][i];
        if (!vis[v]) {
            weightSum += DFS(v);
        }
    }
    return weightSum;
}

int main() {
    memset(vis, false, sizeof(vis));
    int n, m, u, v;
    scanf("%d%d", &n, &m);
    for (int i = 0; i < n; i++) {
        scanf("%d", &weight[i]);
    }
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        G[v].push_back(u);
    }
    int maxWeightSum = 0;
    for (int i = 0; i < n; i++) {
        if (!vis[i]) {
            maxWeightSum = max(maxWeightSum, DFS(i));
        }
    }
    printf("%d", maxWeightSum);
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];
bool inQueue[MAXN] = {false};
int layers[MAXN];

void BFS(int s) {
    queue<int> q;
    q.push(s);
    inQueue[s] = true;
    int layer = 0;
    while (!q.empty()) {
        int cnt = q.size();
        for (int i = 0; i < cnt; i++) {
            int front = q.front();
            q.pop();
            layers[front] = layer;
            for (int j = 0; j < G[front].size(); j++) {
                int v = G[front][j];
                if (!inQueue[v]) {
                    q.push(v);
                    inQueue[v] = true;
                }
            }
        }
        layer++;
    }
}

int main() {
    int n, m, start, u, v;
    scanf("%d%d%d", &n, &m, &start);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        G[v].push_back(u);
    }
    BFS(start);
    for (int i = 0; i < n; i++) {
        printf("%d", layers[i]);
        if (i < n - 1) {
            printf(" ");
        }
    }
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
using namespace std;

const int MAXN = 100;
vector<int> G[MAXN];
bool inQueue[MAXN] = {false};

int BFS(int s, int maxLayer) {
    queue<int> q;
    q.push(s);
    inQueue[s] = true;
    int layer = 0;
    int vertexCount = 0;
    while (!q.empty() && layer <= maxLayer) {
        int cnt = q.size();
        vertexCount += cnt;
        for (int i = 0; i < cnt; i++) {
            int front = q.front();
            q.pop();
            for (int j = 0; j < G[front].size(); j++) {
                int v = G[front][j];
                if (!inQueue[v]) {
                    q.push(v);
                    inQueue[v] = true;
                }
            }
        }
        layer++;
    }
    return vertexCount;
}

int main() {
    int n, m, start, maxLayer, u, v;
    scanf("%d%d%d%d", &n, &m, &start, &maxLayer);
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
    }
    int vertexCount = BFS(start, maxLayer);
    printf("%d", vertexCount);
    return 0;
}

10.4最短路径

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 100;
const int INF = 1e9;

struct Edge {
    int v, dis;
    Edge(int _v, int _dis) {
        v = _v, dis = _dis;
    }
};

vector<Edge> G[MAXN];
int d[MAXN];
bool vis[MAXN];

void dijkstra(int n, int s) {
    fill(d, d + MAXN, INF);
    memset(vis, false, sizeof(vis));
    d[s] = 0;
    for (int i = 0; i < n; i++) {
        int u = -1, minDis = INF;
        for (int j = 0; j < n; j++) {
            if (!vis[j] && d[j] < minDis) {
                u = j;
                minDis = d[j];
            }
        }
        if (u == -1) {
            return;
        }
        vis[u] = true;
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int dis = G[u][j].dis;
            if (!vis[v] && d[u] + dis < d[v]) {
                d[v] = d[u] + dis;
            }
        }
    }
}

int main() {
    int n, m, s, t;
    scanf("%d%d%d%d", &n, &m, &s, &t);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        G[v].push_back(Edge(u, w));
    }
    dijkstra(n, s);
    if (d[t] == INF) {
        printf("-1");
    } else {
        printf("%d", d[t]);
    }
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 100;
const int INF = 1e9;

struct Edge {
    int v, dis;
    Edge(int _v, int _dis) {
        v = _v, dis = _dis;
    }
};

vector<Edge> G[MAXN];
int d[MAXN];
bool vis[MAXN];

void dijkstra(int n, int s) {
    fill(d, d + MAXN, INF);
    memset(vis, false, sizeof(vis));
    d[s] = 0;
    for (int i = 0; i < n; i++) {
        int u = -1, minDis = INF;
        for (int j = 0; j < n; j++) {
            if (!vis[j] && d[j] < minDis) {
                u = j;
                minDis = d[j];
            }
        }
        if (u == -1) {
            return;
        }
        vis[u] = true;
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int dis = G[u][j].dis;
            if (!vis[v] && d[u] + dis < d[v]) {
                d[v] = d[u] + dis;
            }
        }
    }
}

int main() {
    int n, m, s;
    scanf("%d%d%d", &n, &m, &s);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        G[v].push_back(Edge(u, w));
    }
    dijkstra(n, s);
    for (int i = 0; i < n; i++) {
        if (d[i] == INF) {
            printf("-1");
        } else {
            printf("%d", d[i]);
        }
        if (i < n - 1) {
            printf(" ");
        }
    }
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 100;
const int INF = 1e9;

struct Edge {
    int v, dis, cost;
    Edge(int _v, int _dis, int _cost) {
        v = _v, dis = _dis, cost = _cost;
    }
};

vector<Edge> G[MAXN];
int d[MAXN], c[MAXN];
bool vis[MAXN];

void dijkstra(int n, int s) {
    fill(d, d + MAXN, INF);
    fill(c, c + MAXN, INF);
    memset(vis, false, sizeof(vis));
    d[s] = 0;
    c[s] = 0;
    for (int i = 0; i < n; i++) {
        int u = -1, minDis = INF;
        for (int j = 0; j < n; j++) {
            if (!vis[j] && d[j] < minDis) {
                u = j;
                minDis = d[j];
            }
        }
        if (u == -1) {
            return;
        }
        vis[u] = true;
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int dis = G[u][j].dis;
            int cost = G[u][j].cost;
            if (!vis[v]) {
                if (d[u] + dis < d[v]) {
                    d[v] = d[u] + dis;
                    c[v] = c[u] + cost;
                } else if (d[u] + dis == d[v] && c[u] + cost < c[v]) {
                    c[v] = c[u] + cost;
                }
            }
        }
    }
}

int main() {
    int n, m, s, t;
    scanf("%d%d%d%d", &n, &m, &s, &t);
    int u, v, dis, cost;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d%d", &u, &v, &dis, &cost);
        G[u].push_back(Edge(v, dis, cost));
        G[v].push_back(Edge(u, dis, cost));
    }
    dijkstra(n, s);
    printf("%d %d", d[t], c[t]);
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 100;
const int INF = 1e9;

struct Edge {
    int v, dis;
    Edge(int _v, int _dis) {
        v = _v, dis = _dis;
    }
};

vector<Edge> G[MAXN];
int d[MAXN], pathCount[MAXN];
bool vis[MAXN];

void dijkstra(int n, int s) {
    fill(d, d + MAXN, INF);
    memset(pathCount, 0, sizeof(pathCount));
    memset(vis, false, sizeof(vis));
    d[s] = 0;
    pathCount[s] = 1;
    for (int i = 0; i < n; i++) {
        int u = -1, minDis = INF;
        for (int j = 0; j < n; j++) {
            if (!vis[j] && d[j] < minDis) {
                u = j;
                minDis = d[j];
            }
        }
        if (u == -1) {
            return;
        }
        vis[u] = true;
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int dis = G[u][j].dis;
            if (!vis[v]) {
                if (d[u] + dis < d[v]) {
                    d[v] = d[u] + dis;
                    pathCount[v] = pathCount[u];
                } else if (d[u] + dis == d[v]) {
                    pathCount[v] += pathCount[u];
                }
            }
        }
    }
}

int main() {
    int n, m, s, t;
    scanf("%d%d%d%d", &n, &m, &s, &t);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        G[v].push_back(Edge(u, w));
    }
    dijkstra(n, s);
    printf("%d %d", d[t], pathCount[t]);
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 100;
const int INF = 1e9;

struct Edge {
    int v, dis;
    Edge(int _v, int _dis) {
        v = _v, dis = _dis;
    }
};

vector<Edge> G[MAXN];
int d[MAXN], pre[MAXN];
bool vis[MAXN];

void dijkstra(int n, int s) {
    fill(d, d + MAXN, INF);
    memset(pre, -1, sizeof(pre));
    memset(vis, false, sizeof(vis));
    d[s] = 0;
    for (int i = 0; i < n; i++) {
        int u = -1, minDis = INF;
        for (int j = 0; j < n; j++) {
            if (!vis[j] && d[j] < minDis) {
                u = j;
                minDis = d[j];
            }
        }
        if (u == -1) {
            return;
        }
        vis[u] = true;
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int dis = G[u][j].dis;
            if (!vis[v] && d[u] + dis < d[v]) {
                d[v] = d[u] + dis;
                pre[v] = u;
            }
        }
    }
}

vector<int> path;

void DFS(int v, int s) {
    if (v == s) {
        path.push_back(v);
        return;
    }
    DFS(pre[v], s);
    path.push_back(v);
}

int main() {
    int n, m, s, t;
    scanf("%d%d%d%d", &n, &m, &s, &t);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        G[v].push_back(Edge(u, w));
    }
    dijkstra(n, s);
    DFS(t, s);
    printf("%d ", d[t]);
    for (int i = 0; i < path.size(); i++) {
        printf("%d", path[i]);
        if (i < (int)path.size() - 1) {
            printf("->");
        }
    }
    return 0;
}
cpp 复制代码 
#include <cstdio>
#include <algorithm>
using namespace std;

const int MAXN = 50;
const int INF = 1e9;

int d[MAXN][MAXN];

void floyd(int n) {
    for (int k = 0; k < n; k++) {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (d[i][k] != INF && d[k][j] != INF && d[i][k] + d[k][j] < d[i][j]) {
                    d[i][j] = d[i][k] + d[k][j];
                }
            }
        }
    }
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    fill(d[0], d[0] + MAXN * MAXN, INF);
    for (int i = 0; i < n; i++) {
        d[i][i] = 0;
    }
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        d[u][v] = d[v][u] = w;
    }
    floyd(n);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (d[i][j] == INF) {
                printf("-1");
            } else {
                printf("%d", d[i][j]);
            }
            if (j < n - 1) {
                printf(" ");
            }
        }
        printf("\n");
    }
    return 0;
}

10.5最小生成树

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 500;
const int INF = 1e9;

struct Edge {
    int v, dis;
    Edge(int _v, int _dis) {
        v = _v, dis = _dis;
    }
};

vector<Edge> G[MAXN];
int d[MAXN];
bool vis[MAXN];

int prim(int n) {
    fill(d, d + MAXN, INF);
    memset(vis, false, sizeof(vis));
    d[0] = 0;
    int weightSum = 0;
    for (int i = 0; i < n; i++) {
        int u = -1, minDis = INF;
        for (int j = 0; j < n; j++) {
            if (!vis[j] && d[j] < minDis) {
                u = j;
                minDis = d[j];
            }
        }
        if (u == -1) {
            return -1;
        }
        vis[u] = true;
        weightSum += d[u];
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int dis = G[u][j].dis;
            if (!vis[v] && dis < d[v]) {
                d[v] = dis;
            }
        }
    }
    return weightSum;
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        G[v].push_back(Edge(u, w));
    }
    int weightSum = prim(n);
    printf("%d", weightSum);
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;

const int MAXN = 10000;

struct Edge {
    int u, v, w;
    Edge(int _u, int _v, int _w) {
        u = _u, v = _v, w = _w;
    }
};

vector<Edge> edges;

bool cmp(Edge a, Edge b) {
    return a.w < b.w;
}

int father[MAXN];

int findFather(int x) {
    int xCopy = x;
    while (father[x] != x) {
        x = father[x];
    }
    int root = x;
    x = xCopy;
    while (father[x] != x) {
        int fatherX = father[x];
        father[x] = root;
        x = fatherX;
    }
    return root;
}

int kruskal(int n, int m) {
    for (int i = 0; i < n; i++) {
        father[i] = i;
    }
    int weightSum = 0, edgeCount = 0;
    sort(edges.begin(), edges.end(), cmp);
    for (int i = 0; i < m; i++) {
        int faU = findFather(edges[i].u);
        int faV = findFather(edges[i].v);
        if (faU != faV) {
            father[faU] = faV;
            weightSum += edges[i].w;
            edgeCount++;
        }
        if (edgeCount == n - 1) {
            break;
        }
    }
    if (edgeCount != n - 1) {
        return -1;
    } else {
        return weightSum;
    }
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        edges.push_back(Edge(u, v, w));
    }
    int weightSum = kruskal(n, m);
    printf("%d", weightSum);
    return 0;
}

10.6拓扑排序

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;

const int MAXN = 100;

vector<int> G[MAXN];
int inDegree[MAXN];
vector<int> topoOrder;

void topoSort(int n) {
    priority_queue<int, vector<int>, greater<int> > q;
    for (int i = 0; i < n; i++) {
        if (inDegree[i] == 0) {
            q.push(i);
        }
    }
    while (!q.empty()) {
        int u = q.top();
        q.pop();
        topoOrder.push_back(u);
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i];
            inDegree[v]--;
            if (inDegree[v] == 0) {
                q.push(v);
            }
        }
        G[u].clear();
    }
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v;
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        inDegree[v]++;
    }
    topoSort(n);
    for (int i = 0; i < topoOrder.size(); i++) {
        printf("%d", topoOrder[i]);
        if (i < (int)topoOrder.size() - 1) {
            printf(" ");
        }
    }
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;

const int MAXN = 100;

vector<int> G[MAXN];
int inDegree[MAXN];

bool topoSort(int n) {
    int vertexCount = 0;
    priority_queue<int, vector<int>, greater<int> > q;
    for (int i = 0; i < n; i++) {
        if (inDegree[i] == 0) {
            q.push(i);
        }
    }
    while (!q.empty()) {
        int u = q.top();
        q.pop();
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i];
            inDegree[v]--;
            if (inDegree[v] == 0) {
                q.push(v);
            }
        }
        G[u].clear();
        vertexCount++;
    }
    return vertexCount == n;
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v;
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        inDegree[v]++;
    }
    printf(topoSort(n) ? "Yes" : "No");
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;

const int MAXN = 100;

vector<int> G[MAXN];
int inDegree[MAXN];
vector<int> topoOrder;

int topoSort(int n) {
    int vertexCount = 0;
    priority_queue<int, vector<int>, greater<int> > q;
    for (int i = 0; i < n; i++) {
        if (inDegree[i] == 0) {
            q.push(i);
        }
    }
    while (!q.empty()) {
        int u = q.top();
        q.pop();
        topoOrder.push_back(u);
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i];
            inDegree[v]--;
            if (inDegree[v] == 0) {
                q.push(v);
            }
        }
        G[u].clear();
        vertexCount++;
    }
    return vertexCount;
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v;
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        G[u].push_back(v);
        inDegree[v]++;
    }
    int learnCount = topoSort(n);
    if (learnCount == n) {
        printf("Yes\n");
        for (int i = 0; i < topoOrder.size(); i++) {
            printf("%d", topoOrder[i]);
            if (i < (int)topoOrder.size() - 1) {
                printf(" ");
            }
        }
    } else {
        printf("No\n%d\n", n - learnCount);
    }
    return 0;
}

10.7关键路径

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
#include <algorithm>
#include <cstring>
using namespace std;

const int MAXN = 100;

struct Edge {
    int v, w;
    Edge(int _v, int _w) {
        v = _v, w = _w;
    }
};

vector<Edge> G[MAXN];
int inDegree[MAXN] = {0};
vector<int> topoOrder;
int ve[MAXN], vl[MAXN];
vector<int> activity[MAXN];

bool topoSort(int n) {
    priority_queue<int, vector<int>, greater<int> > q;
    for (int i = 0; i < n; i++) {
        if (inDegree[i] == 0) {
            q.push(i);
        }
    }
    while (!q.empty()) {
        int u = q.top();
        q.pop();
        topoOrder.push_back(u);
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i].v;
            inDegree[v]--;
            if (inDegree[v] == 0) {
                q.push(v);
            }
            if (ve[u] + G[u][i].w > ve[v]) {
                ve[v] = ve[u] + G[u][i].w;
            }
        }
    }
    return (int)topoOrder.size() == n;
}

int getCriticalPathLength(int n) {
    memset(ve, 0, sizeof(ve));
    if (!topoSort(n)) {
        return -1;
    }
    int maxLength = 0;
    for(int i = 0; i < n; i++) {
        if(ve[i] > maxLength) {
            maxLength = ve[i];
        }
    }
    return maxLength;
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        inDegree[v]++;
    }
    int pathLength = getCriticalPathLength(n);
    if (pathLength == -1) {
        printf("No");
    } else {
        printf("Yes\n%d", pathLength);
    }
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
#include <algorithm>
#include <cstring>
using namespace std;

const int MAXN = 100;

struct Edge {
    int v, w;
    Edge(int _v, int _w) {
        v = _v, w = _w;
    }
};

vector<Edge> G[MAXN];
int inDegree[MAXN] = {0};
vector<int> topoOrder;
int ve[MAXN], vl[MAXN];
vector<int> activity[MAXN];

bool topoSort(int n) {
    priority_queue<int, vector<int>, greater<int> > q;
    for (int i = 0; i < n; i++) {
        if (inDegree[i] == 0) {
            q.push(i);
        }
    }
    while (!q.empty()) {
        int u = q.top();
        q.pop();
        topoOrder.push_back(u);
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i].v;
            inDegree[v]--;
            if (inDegree[v] == 0) {
                q.push(v);
            }
            if (ve[u] + G[u][i].w > ve[v]) {
                ve[v] = ve[u] + G[u][i].w;
            }
        }
    }
    return (int)topoOrder.size() == n;
}

int getCriticalPath(int n) {
    memset(ve, 0, sizeof(ve));
    if (!topoSort(n)) {
        return -1;
    }
    int maxLength = 0;
    for(int i = 0; i < n; i++) {
        if(ve[i] > maxLength) {
            maxLength = ve[i];
        }
    }
    fill(vl, vl + n, maxLength);

    for (int i = (int)topoOrder.size() - 1; i >= 0; i--) {
        int u = topoOrder[i];
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int w = G[u][j].w;
            if (vl[v] - w < vl[u]) {
                vl[u] = vl[v] - w;
            }
        }
    }

    for (int u = 0; u < n; u++) {
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i].v;
            int w = G[u][i].w;
            int e = ve[u], l = vl[v] - w;
            if (e == l) {
                activity[u].push_back(v);
            }
        }
    }
    return maxLength;
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        inDegree[v]++;
    }
    if (getCriticalPath(n) == -1) {
        printf("No");
    } else {
        printf("Yes\n");
        for (int i = 0; i < n; i++) {
            sort(activity[i].begin(), activity[i].end());
            for (int j = 0; j < activity[i].size(); j++) {
                printf("%d %d\n", i, activity[i][j]);
            }
        }
    }
    return 0;
}

cpp 复制代码 
#include <cstdio>
#include <vector>
#include <queue>
#include <algorithm>
#include <cstring>
using namespace std;

const int MAXN = 100;

struct Edge {
    int v, w;
    Edge(int _v, int _w) {
        v = _v, w = _w;
    }
};

vector<Edge> G[MAXN];
int inDegree[MAXN] = {0}, inDegreeOrigin[MAXN] = {0};
vector<int> topoOrder;
int ve[MAXN], vl[MAXN];
vector<int> activity[MAXN];

bool topoSort(int n) {
    priority_queue<int, vector<int>, greater<int> > q;
    for (int i = 0; i < n; i++) {
        if (inDegree[i] == 0) {
            q.push(i);
        }
    }
    while (!q.empty()) {
        int u = q.top();
        q.pop();
        topoOrder.push_back(u);
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i].v;
            inDegree[v]--;
            if (inDegree[v] == 0) {
                q.push(v);
            }
            if (ve[u] + G[u][i].w > ve[v]) {
                ve[v] = ve[u] + G[u][i].w;
            }
        }
    }
    return (int)topoOrder.size() == n;
}

int getCriticalPath(int n) {
    memset(ve, 0, sizeof(ve));
    if (!topoSort(n)) {
        return -1;
    }
    int maxLength = 0;
    for(int i = 0; i < n; i++) {
        if(ve[i] > maxLength) {
            maxLength = ve[i];
        }
    }
    fill(vl, vl + n, maxLength);

    for (int i = (int)topoOrder.size() - 1; i >= 0; i--) {
        int u = topoOrder[i];
        for (int j = 0; j < G[u].size(); j++) {
            int v = G[u][j].v;
            int w = G[u][j].w;
            if (vl[v] - w < vl[u]) {
                vl[u] = vl[v] - w;
            }
        }
    }

    for (int u = 0; u < n; u++) {
        for (int i = 0; i < G[u].size(); i++) {
            int v = G[u][i].v;
            int w = G[u][i].w;
            int e = ve[u], l = vl[v] - w;
            if (e == l) {
                activity[u].push_back(v);
            }
        }
    }
    return maxLength;
}

vector<int> criticalPath;
void printCriticalPath(int u) {
    if(activity[u].size() == 0) {
        criticalPath.push_back(u);
        for(int i = 0; i < criticalPath.size(); i++) {
            printf("%d", criticalPath[i]);
            if(i < criticalPath.size() - 1) {
                printf("->");
            } else {
                printf("\n");
            }
        }
        criticalPath.pop_back();
        return;
    }
    criticalPath.push_back(u);
    sort(activity[u].begin(), activity[u].end());
    for(int i = 0; i < activity[u].size(); i++) {
        printCriticalPath(activity[u][i]);
    }
    criticalPath.pop_back();
}

int main() {
    int n, m;
    scanf("%d%d", &n, &m);
    int u, v, w;
    for (int i = 0; i < m; i++) {
        scanf("%d%d%d", &u, &v, &w);
        G[u].push_back(Edge(v, w));
        inDegree[v]++;
        inDegreeOrigin[v]++;
    }
    if (getCriticalPath(n) == -1) {
        printf("No");
    } else {
        printf("Yes\n");
        for(int i = 0; i < n; i++) {
            if(inDegreeOrigin[i] == 0 && activity[i].size() != 0) {
                printCriticalPath(i);
            }
        }
    }
    return 0;
}
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