囤题 [补档]

图论

网络流相关

CF 653 D

题目大意

给定一张\(n\)个点, \(m\)条边的有向图(\(n \le 50\), \(m \le 500\)), 每条边都有容量限制. 你要找到至少\(x\)条路径, 使得每条路径点容量都为某个定值\(F\), 且经过任意一条边点所有路径的容量之和小于等于这条边的容量. 求\(F\)的最大值.

题解

我们令原图的边集为\(E = \{ \left< u, v, w \right> \}\)
二分\(F\)的值, 对于一个确定点\(F\), 我们有每条边的最大经过次数. 将这个最大经过次数作为边的容量重新建图, 得到\(E' = \{ \left< u, v, c = \frac w F \right> \}\). 跑一遍最大流, 得到的流量即为可以选出的路径最大数量. 判断若最大流大于等于\(x\), 则对于当前的\(F\), 有可行方案, 否则没有可行方案.

#include <cstdio>
#include <cctype>
#include <cstring>
#include <climits>
#include <deque>
#include <algorithm>

using namespace std;
inline int getInt()
{
	int a = 0, sgn = 1; char c;
	while (! isdigit(c = getchar())) if (c == '-') sgn *= -1;
	while (isdigit(c)) a = a * 10 + c - '0', c = getchar();
	return a * sgn;
}
typedef double D;
typedef long long LL;
const D EPS = 1e-8;
const LL INF = (LL)1e18;
const int N = 50, M = 500;
int n, m, x;
struct Record
{
	int u, v, w;
}rec[M + 7];
struct edge
{
	int v, nxt;
	LL w;
}edg[M * 2 + 7];
int tot, hd[N + 7], dis[N + 7];
inline void addEdge(int u, int v, LL w) { edg[tot].v = v; edg[tot].w = w; edg[tot].nxt = hd[u]; hd[u] = tot ++; }
inline int BFS()
{
	memset(dis, -1, sizeof dis); dis[1] = 0;
	deque<int> que; que.push_back(1);
	while (! que.empty())
	{
		int u = que.front(); que.pop_front();
		if (u == n) return 1;
		for (int p = hd[u]; ~ p; p = edg[p].nxt) if (edg[p].w && dis[edg[p].v] == -1) dis[edg[p].v] = dis[u] + 1, que.push_back(edg[p].v);
	}
	return 0;
}
LL DFS(int u, LL flw)
{
	if (u == n) return flw;
	LL usd = 0;
	for (int p = hd[u]; ~ p; p = edg[p].nxt) if (edg[p].w && dis[edg[p].v] == dis[u] + 1)
	{
		int v = edg[p].v;
		LL cur = DFS(v, min(edg[p].w, flw - usd));
		edg[p].w -= cur; edg[p ^ 1].w += cur;
		usd += cur;
		if (usd == flw) return usd;
	}
	return usd;
}
inline LL flow()
{
	LL sum = 0;
	while (BFS()) sum += DFS(1, INF);
	return sum;
}
int main()
{

#ifndef ONLINE_JUDGE

	freopen("bear.in", "r", stdin);
	freopen("bear.out", "w", stdout);
	
#endif
	
	n = getInt(); m = getInt(); x = getInt();
	for (int i = 0; i < m; ++ i) rec[i]. u = getInt(), rec[i].v = getInt(), rec[i].w = getInt();
	D L = 1e-5, R = 1e6;
	while (R - L > EPS)
	{
		D mid = (L + R) / 2;
		memset(hd, -1, sizeof hd);
		tot = 0; for (int i = 0; i < m; ++ i) addEdge(rec[i].u, rec[i].v, (LL)rec[i].w / mid), addEdge(rec[i].v, rec[i].u, 0);
		if (flow() < x) R = mid; else L = mid;
	}
	printf("%.7lf", L * (D)x);
}

CF 884 F

题目大意

给定一个由小写字母组成的串\(S\)(\(|S| \le 100\)), 每个位置有一个权值\(b_i\), 你要找到一个原串的排列\(T\), 满足对于每个\(i\)都有\(T_i \ne T_{|S| - i + 1}\), 并且使得这个排列的权值最大. 一个排列的权值为\(\sum_k b_k \times [T_k = S_k]\). 求这个权值.

保证存在至少一组合法的排列.

题解

最大费用最大流.

建立\(26\)个节点分别代表\(26\)个字母. 统计每个字母的出现次数, 从源点向每个字母的节点连一条容量为出现次数, 费用为\(0\)的边.

接着我们再建立\(\frac{|S|}2\)个节点, 代表不能相互冲突的一对位置. 考虑从字母向位置连边, 容量显然为\(1\), 费用则比较麻烦: 首先假如当前字母不等于\(S_i\)和\(S_{|S| - i + 1}\), 则费用为\(0\); 假如等于其中的一个, 则费用为\(b\); 假如和这两个位置都相等, 则费用为\(\max(b_i, b_{|S| - i + 1})\).

最后, 从每个位置向汇点连一条容量为\(2\), 费用为\(0\)的边.

跑一遍最大费用的最大流即可.

#include <cstdio>
#include <cstring>
#include <deque>
#include <climits>

using namespace std;
const int N = 100;
const int V = N / 2 + 107;
const int E = 31 * N + 107;
int n;
int cnt[27];
int str[N + 7], b[N + 7];
struct edge
{
	int v, w, c, nxt;
}edg[E];
int hd[V], tot;
int pre[V], rec[V], inQueue[V], dis[V];
inline void addEdge(int u, int v, int w, int c) 
{
	edg[tot].v = v; edg[tot].w = w; edg[tot].c = c; edg[tot].nxt = hd[u]; hd[u] = tot ++;
	edg[tot].v = u; edg[tot].w = 0; edg[tot].c = - c; edg[tot].nxt = hd[v]; hd[v] = tot ++;
}
inline int SPFA()
{
	int t = 26 + n / 2 + 1;
	memset(pre, -1, sizeof pre);
	deque<int> que; que.push_back(0);
	memset(inQueue, 0, sizeof inQueue); inQueue[0] = 1;
	memset(dis, -127, sizeof dis); dis[0] = 0;
	while (! que.empty())
	{
		int u = que.front(); que.pop_front(); inQueue[u] = 0;
		for (int p = hd[u]; ~ p; p = edg[p].nxt) if (edg[p].w && dis[u] + edg[p].c > dis[edg[p].v])
		{
			int v = edg[p].v;
			dis[v] = dis[u] + edg[p].c; pre[v] = u; rec[v] = p;
			if (! inQueue[v]) inQueue[v] = 1, que.push_back(v);
		}
	}
	int mn = INT_MAX;
	for (int p = t; ~ pre[p]; p = pre[p]) mn = min(mn, edg[rec[p]].w);
	for (int p = t; ~ pre[p]; p = pre[p]) 
		edg[rec[p]].w -= mn, edg[rec[p] ^ 1].w += mn;
	return dis[t];
}
inline int flow()
{
	int sum = 0;
	while (1) { int tmp = SPFA(); if (tmp < - (int)2e9) return sum; sum += tmp; }
}
int main()
{

#ifndef ONLINE_JUDGE

	freopen("anti.in", "r", stdin);
	freopen("anti.out", "w", stdout);
	
#endif

	scanf("%d\n", &n);
	for (int i = 1; i <= n; ++ i) ++ cnt[str[i] = (getchar() - 'a' + 1)];
	for (int i = 1; i <= n; ++ i) scanf("%d", b + i);
	memset(hd, -1, sizeof hd); tot = 0;
	for (int i = 1; i <= 26; ++ i) addEdge(0, i, cnt[i], 0);
	for (int i = 1; i <= 26; ++ i) for (int j = 1; j <= n >> 1; ++ j) 
		addEdge(i, 26 + j, 1, max((str[j] == i) * b[j], (str[n - j + 1] == i) * b[n - j + 1]));
	for (int i = 1; i <= n >> 1; ++ i) addEdge(26 + i, 26 + n / 2 + 1, 2, 0);
	printf("%d\n", flow());
}