分层最短路Magical Girl Haze

Posted 2021-01-03 stolf

https://nanti.jisuanke.com/t/31001

有K次机会可以让一条边的权值变为0，求最短路。

在存储单源最短路的数组上多开一维状态，d[i][k]表示走到序号i的点，且让k条边权值为0时的最短路。
对于每个待更新的点，尝试不置零此边的状态和置零此边的状态，分别压入优先队列去更新其他状态。

另外，此题由于有重边，需要先去重，保留同起始点最短的边。

代码：

#include <iostream>
#include <algorithm>
#include <queue>
#include <cstring>
#include <vector>
#include <cstdio>
using namespace std;
typedef long long ll;
const ll INF = 0x3f3f3f3f3f3f3f3fLL;
const int MAX_V = 100005;
const int MAX_E = 200005;
int N, M, K;
struct Dijkstra {
    struct Edge {
        int to, cost, next;
    } es[MAX_E];
    struct Node {
        int u, k;
        ll d;
        Node(int u, ll d, int k) : u(u), d(d), k(k) {}
        bool operator< (const Node& n) const {
            return d > n.d;
        }
    };
    int head[MAX_V];
    int V, E;
    ll d[MAX_V][15];
    bool vis[MAX_V][15];
    void init(int V) {
        this->V = V;
        this->E = 0;
        memset(head, -1, sizeof head);
    }
    void addEdge(int u, int v, int w) {
        es[E].to = v;
        es[E].cost = w;
        es[E].next = head[u];
        head[u] = E++;
    }
    void dijkstra(int s) {
        priority_queue <Node> Q;
        memset(d, 0x3f, sizeof d);
        memset(vis, 0, sizeof(vis));
        d[s][0] = 0;
        Q.push(Node(s, 0, 0));
        while (!Q.empty()) {
            int u = Q.top().u, k = Q.top().k;
            Q.pop();
            if (vis[u][k])
                continue;
            vis[u][k] = true;
            for (int i = head[u]; i != -1; i = es[i].next) {
                int v = es[i].to, w = es[i].cost;
                if (d[v][k] > d[u][k] + w) {
                    d[v][k] = d[u][k] + w;
                    Q.push(Node(v, d[v][k], k));
                }
                if (k + 1 <= K) {
                    if (d[v][k + 1] > d[u][k]) {
                        d[v][k + 1] = d[u][k];
                        Q.push(Node(v, d[v][k + 1], k + 1));
                    }
                }
            }
        }
    }
} dijk;
struct Elem {
    int u, v, w;
    bool operator< (const Elem& e) const {
        if (u == e.u && v == e.v) {
            return w < e.w;
        }
        if (u == e.u) {
            return v < e.v;
        }
        return u < e.u;
    }
} e[MAX_E];
int main() {
    int T;
    scanf("%d", &T);
    while (T--) {
        scanf("%d%d%d", &N, &M, &K);
        dijk.init(N);
        for (int i = 0; i < M; i++) {
            scanf("%d%d%d", &e[i].u, &e[i].v, &e[i].w);
        }
        sort(e, e + M);
        int preu = 0, prev = 0;
        for (int i = 0; i < M; i++) {
            if (preu != e[i].u || prev != e[i].v) {
                dijk.addEdge(e[i].u, e[i].v, e[i].w);
                preu = e[i].u, prev = e[i].v;
            }
        }
        dijk.dijkstra(1);
        ll ans = INF;
        for (int i = 0; i <= K; i++) {
            ans = min(ans, dijk.d[N][i]);
        }
        printf("%lld
", ans);
    }
}