「CF894E」 Ralph and Mushrooms

传送门
Luogu

解题思路

首先我们要发现：在同一个强连通分量里的所有边都是可以无限走的。
那么我们就有了思路：先缩点，再跑拓扑排序。
那么问题就是 \(\text{DP}\) 状态如何初始化。
我们首先考虑一条原始边权为 \(c\) 的边，无限走可以刷出多少贡献：
假设我们走 \(t\) 次就可以把这条边刷完，那么 \(t\) 应该是满足下面这个式子的最大整数：

\[\frac{t(t+1)}{2}< c \]

解得：

\[t=\left\lfloor\sqrt{2t+\frac{1}{4}}-\frac{1}{2}\right\rfloor \]

那么我们的贡献就是：

\[\begin{aligned}sum&=\sum_{i=0}^{t}\left(c-\sum_{j=0}^{i}j\right)\\&=(t+1)c-\sum_{i=0}^{t}\frac{i(i+1)}{2}\\&=(t+1)c-\frac{1}{2}\left(\sum_{i=0}^{t}i^2+\sum_{i=0}^{t}i\right)\\&=(t+1)c-\frac{1}{2}\left(\frac{t(t+1)(2t+1)}{6}+\frac{t(t+1)}{2}\right)\\&=(t+1)c-\frac{t(t+1)(t+2)}{6}\end{aligned} \]

于是我们就解决了这道题，最后输出 \(\max\limits_{1\le i\le col}\{f[i]\}\) 即可。

细节注意事项

• 由于 \(\text{tarjan}\) 缩点时对边的操作不方便，可以在外部处理

参考代码

#include <algorithm>
#include <iostream>
#include <cstring>
#include <cstdlib>
#include <cstdio>
#include <cctype>
#include <cmath>
#include <ctime>
#include <queue>
#define rg register
#define pii pair < int, int >
using namespace std;
template < typename T > inline void read(T& s) {
	s = 0; int f = 0; char c = getchar();
	while (!isdigit(c)) f |= (c == '-'), c = getchar();
	while (isdigit(c)) s = s * 10 + (c ^ 48), c = getchar();
	s = f ? -s : s;
}

typedef long long LL;
const int _ = 1000010;
const LL INF = 1ll << 60;

int tot, head[_], nxt[_], ver[_]; LL w[_];
inline void Add_edge(int u, int v, LL d)
{ nxt[++tot] = head[u], head[u] = tot, ver[tot] = v, w[tot] = d; }

int n, m, s, x[_], y[_]; LL c[_];
int num, dfn[_], low[_];
int top, st[_], col, co[_];
int dgr[_]; LL val[_], f[_];

inline void tarjan(int u) {
	dfn[u] = low[u] = ++num, st[++top] = u;
	for (rg int i = head[u]; i; i = nxt[i]) {
		int v = ver[i];
		if (!dfn[v])
			tarjan(v), low[u] = min(low[u], low[v]);
		else
			if (!co[v]) low[u] = min(low[u], dfn[v]);
	}
	if (low[u] == dfn[u]) {
		++col;
		do co[st[top]] = col;
		while (st[top--] != u);
	}
}

inline LL calc(LL c) {
	LL t = sqrt(c * 2 + 0.25) - 0.5;
	return (t + 1) * c - t * (t + 1) * (t + 2) / 6;
}

inline void rebuild() {
	for (rg int i = 1; i <= m; ++i)
		if (co[x[i]] == co[y[i]])
			val[co[x[i]]] += calc(c[i]);
	memset(head, tot = 0, sizeof head);	
	for (rg int i = 1; i <= m; ++i)
		if (co[x[i]] != co[y[i]])
			Add_edge(co[x[i]], co[y[i]], c[i] + val[co[y[i]]]), ++dgr[co[y[i]]];
}

inline LL toposort() {
	LL _max = 0;
	static queue < int > Q;
	for (rg int i = 1; i <= col; ++i) {
		if (dgr[i] == 0) Q.push(i); f[i] = -INF;
	}
	f[co[s]] = val[co[s]];
	while (!Q.empty()) {
		int u = Q.front(); Q.pop();
		for (rg int v, i = head[u]; i; i = nxt[i]) {
			if (!--dgr[v = ver[i]]) Q.push(v);
			f[v] = max(f[v], f[u] + w[i]);
		}
	}
	for (rg int i = 1; i <= col; ++i) _max = max(_max, f[i]);
	return _max;
}

int main() {
#ifndef ONLINE_JUDGE
	freopen("in.in", "r", stdin);
#endif
	read(n), read(m);
	for (rg int i = 1; i <= m; ++i)
		read(x[i]), read(y[i]), read(c[i]), Add_edge(x[i], y[i], c[i]);
	read(s);
	for (rg int i = 1; i <= n; ++i) if (!dfn[i]) tarjan(i);
	rebuild();
	printf("%lld\n", toposort());
	return 0;
}

完结撒花 \(qwq\)