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最短路

Posted on Edited on In

dijkstra 基于贪心思想，利用了三角不等式，因为全局最小值不可能再被其他节点更新

dijkstra 用于解决单源最短路问题

disjstra 用于解决非负权图问题

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#include<bits/stdc++.h>
using namespace std;
const int inf = 0x3f3f3f3f;
const int maxn = 1e6 + 100;
typedef pair<int,int> pii;

struct node {
    int v;
    int w;
};
vector<node> e[maxn];
int dis[maxn];
bool vis[maxn];
int pre[maxn];

void dijkstra(int s) {
    priority_queue<pii, vector<pii>, greater<pii>> q;
    memset(dis, inf, sizeof(dis));
    dis[s] = 0;
    q.push({dis[s], s});
    while(!q.empty()) {
        pii t = q.top();
        q.pop();
        int u = t.second;
        if(vis[u])
            continue;
        vis[u] = true;
        for(node x:e[u]) {
            int v = x.v;
            int w = x.w;
            if(dis[v]>dis[u]+w) {
                dis[v] = dis[u] + w;
                pre[v] = u;
                q.push({dis[v], v});
            }
        }
    }
}

int main() {
    int n, m, s;
    cin >> n >> m >> s;
    for (int i = 0;i<n;i++) {
        pre[i] = -1;
    }
    for (int i = 0; i < m; i++) {
        int u, v, w;
        cin >> u >> v >> w;
        e[u].push_back((node){v, w});
        // e[v].push_back((node){u, w});
    }
    dijkstra(s);
    for (int i = 1; i <= n;i++) {
        cout << dis[i] << " ";
    }
    cout << endl;
}

最短路计数，假设一张无权图，边长为1

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void dijkstra(int s) {
    priority_queue<pii,vector<pii>,greater<pii>> q;
    memset(dis, inf, sizeof(dis));
    dis[s] = 0;
    q.push({0, s});
    cnt[s] = 1;
    while(!q.empty()) {
        pii tmp = q.top();
        int u = tmp.second;
        q.pop();
        if(vis[u])
            continue;
        vis[u] = true;
        for (int v:e[u]) {
            if(dis[v]>dis[u]+1) {
                dis[v] = dis[u] + 1;
                cnt[v] = cnt[u];
                if(!vis[v]) {
                    q.push({dis[v],v});
                }
            } else if(dis[v] == dis[u]+1) {
                cnt[v] = (cnt[v] + cnt[u]) % mod;
            }
        }
    }
}

SPFA
基于队列优化的Bellman-Ford算法