Luogu3177 [HAOI2015]树上染色

题目蓝链

Description

给定一棵有\(n\)个节点的树，初始全为白色。你要在里面找到\(k\)个点，并把它们染成黑色。要使得染完色后，黑点两两之间的距离加上白点两两之间的距离的和最大

Solution

我们可以设\(dp[i][j]\)表示以\(i\)为根的子树中选择\(j\)个节点对全局答案的最大贡献

然后对于每一个非叶子节点，枚举儿子子树进行转移

\[dp[u][j + k] \leftarrow dp[u][j] + dp[v][k] + val \]

其中\(val\)为以\(v\)为根的子树内的点通过对全局答案的贡献

至于时间复杂度，我们可以发现任意一对点只会在它们LCA处被计算一次，所以复杂度为\(\mathcal{O}(n^2)\)

Code

#include <bits/stdc++.h>

using namespace std;

#define fst first
#define snd second
#define mp make_pair
#define squ(x) ((LL)(x) * (x))
#define debug(...) fprintf(stderr, __VA_ARGS__)

typedef long long LL;
typedef pair<int, int> pii;

template<typename T> inline bool chkmax(T &a, const T &b) { return a < b ? a = b, 1 : 0; }
template<typename T> inline bool chkmin(T &a, const T &b) { return a > b ? a = b, 1 : 0; }

inline int read() {
	int sum = 0, fg = 1; char c = getchar();
	for (; !isdigit(c); c = getchar()) if (c == '-') fg = -1;
	for (; isdigit(c); c = getchar()) sum = (sum << 3) + (sum << 1) + (c ^ 0x30);
	return fg * sum;
}

const int maxn = 2e3 + 10;

int Begin[maxn], Next[maxn << 1], To[maxn << 1], w[maxn << 1], e;
inline void link(int x, int y, int z) { To[++e] = y, Next[e] = Begin[x], Begin[x] = e, w[e] = z; }

int n, m, sz[maxn];
LL dp[maxn][maxn];

inline void dfs(int now, int f) {
	int lim = 1;
	for (int i = Begin[now]; i; i = Next[i]) {
		int son = To[i];
		if (son == f) continue;
		dfs(son, now);
		for (int j = min(lim, m); ~j; j--)
			for (int k = min(sz[son], m); ~k; k--)
				if (j + k <= m)
					chkmax(dp[now][j + k], dp[now][j] + dp[son][k] + (LL) w[i] * (k * (m - k) + (sz[son] - k) * (n - m - sz[son] + k)));
		lim += sz[son];
	}
	sz[now] = lim;
}

int main() {
#ifdef xunzhen
	freopen("tree.in", "r", stdin);
	freopen("tree.out", "w", stdout);
#endif

	n = read(), m = read();
	chkmin(m, n - m);
	for (int i = 1; i < n; i++) {
		int x = read(), y = read(), z = read();
		link(x, y, z), link(y, x, z);
	}

	dfs(1, 0);

	printf("%lld\n", dp[1][m]);

	return 0;
}