每日一题 - 240410 - 路径

• 路径

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• TAG - $模板题、算法-【图论-最短路-dijkstra】$
• 时间复杂度 - $O(M\ast \log M)$

//
#include <bits/stdc++.h>
using namespace std;
#define int long long 
#define endl '\n'
#define data da

const int N = 10005;
struct A {
    int idx, data;
    bool operator < (const A& in) const {
        return in.data < data;
    }
};
vector<A> v[N];
int dis[N];
bool used[N];

int gcd(int a, int b) {             //
    return a % b == 0 ? b : gcd(b, a % b);
}

int lcm(int a, int b) {
    return a * b / gcd(a, b);
}

void dijkstra() {
    priority_queue<A> q;
    q.push((A){1, 0});

    memset(dis, 0x3f, sizeof(dis));
    dis[1] = 0;

    while (!q.empty()) {
        int x = q.top().idx;
        q.pop();

        if (used[x]) continue;
        used[x] = 1;

        for (int i = 0; i < v[x].size(); i++) {
            auto [y, data] = v[x][i];
            if (dis[y] > dis[x] + data) {
                dis[y] = dis[x] + data;
                q.push((A){y, dis[y]});
            }
        }
    }
}

void solve() {
    for (int i = 1; i <= 2021; i++) {
        for (int j = i + 1; j <= min(i + 21, 2021LL); j++) {
            int tt = lcm(i, j);
            v[i].push_back(A{j, tt});
            v[j].push_back(A{i, tt});
        }
    }
    dijkstra();

    cout << dis[2021] << endl;
}

signed main() {
    int t = 1;
    // scanf("%d", &t);
    while (t--) solve();
    return 0;
}

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实现细节

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参考示意图

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参考链接

• Dijkstra算法时间复杂度分析

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作者 | 乐意奥AI