差分约束 : 区间

区间

分析

\(前缀和 + 差分约束\)

\(令S_i为1\sim i中被选出的数.\)

约束关系

• \(S_i \ge S_{i-1}\)
• \(S_i-S_{i-1}\le 1\Rightarrow S_{i-1}\ge S_i-1\)
• \(S_a-S_b\ge c\)

代码

#include <bits/stdc++.h>
using namespace std;
#define IO ios::sync_with_stdio(false);cin.tie(0); cout.tie(0)
inline int lowbit(int x) { return x & (-x); }
#define ll long long
#define pb push_back
#define PII pair<int, int>
#define x first
#define y second
#define inf 0x3f3f3f3f
const int N = 50010, M = 150010;
int n;
int h[N], e[M], w[M], ne[M], idx;
int dist[N];
int q[N];
bool st[N];

void add(int a, int b, int c) {
    e[idx] = b, ne[idx] = h[a], w[idx] = c, h[a] = idx++;
}

void spfa() {
    memset(dist, -0x3f, sizeof dist);
    dist[0] = 0;
    st[0] = true;
    int hh = 0, tt = 1;
    q[0] = 0;
    while (hh != tt) {
        int t = q[hh++];
        if (hh == N) hh = 0;
        st[t] = false;
        for (int i = h[t]; ~i; i = ne[i]) {
            int j = e[i];
            if (dist[j] < dist[t] + w[i]) {
                dist[j] = dist[t] + w[i];
                if (!st[j]) {
                    q[tt++] = j;
                    if (tt == N) tt = 0;
                    st[j] = true;
                }
            }
        }
    }
}

int main() {
    IO;
    cin >> n;
    memset(h, -1, sizeof h);
    for (int i = 1; i < N; ++i) {
        add(i - 1, i, 0);
        add(i, i - 1, -1);
    }
    for (int i = 0; i < n; ++i) {
        int a, b, c;
        cin >> a >> b >> c;
        a++, b++;
        add(a - 1, b, c);
    }
    spfa();
    cout << dist[50001] << '\n';
    return 0;
}