【BZOJ4013】【HNOI2015】实验比较（树形DP，组合）

组合）"/>

【BZOJ4013】【HNOI2015】实验比较（树形DP，组合）

Description

.php?id=4013

Solution

先将相等的都丢进一个并查集，又因为有条件：“小 D 都最多只记住了某一张质量不比 i 差的另一张图片 Ki”，我们对大小关系进行建图后是一棵树(如果是森林，我们新建一个点连接所有根节点即可)。
设 d p [ u ] [ i ] dp[u][i] dp[u][i]表示 u u u的子树内分成了 i i i段的方案数（将相同的元素看做一段）。
对于 u u u的每个子节点，我们有：
d p [ u ] [ i ] = ∑ j , k d p ′ [ u ] [ j ] × d p [ v ] [ k ] × ( i − 1 j − 1 ) ( j − 1 j + k − i ) dp[u][i] = \sum_{j,k} dp'[u][j]\times dp[v][k] \times \binom{i-1}{j-1}\binom{j-1}{j+k-i} dp[u][i]=∑j,k​dp′[u][j]×dp[v][k]×(j−1i−1​)(j+k−ij−1​)
( d p ′ dp' dp′为上一个子节点转移后的结果)
答案为 ∑ d p [ r o o t ] [ i ] \sum dp[root][i] ∑dp[root][i]

Code

/************************************************* Au: Hany01* Date: Sep 15th, 2018* Prob: BZOJ4013 HNOI2015 实验比较* Email: hany01dxx@gmail & hany01@foxmail* Inst: Yali High School
************************************************/#include<bits/stdc++.h>using namespace std;typedef long long LL;
typedef long double LD;
typedef pair<int, int> PII;
#define rep(i, j) for (register int i = 0, i##_end_ = (j); i < i##_end_; ++ i)
#define For(i, j, k) for (register int i = (j), i##_end_ = (k); i <= i##_end_; ++ i)
#define Fordown(i, j, k) for (register int i = (j), i##_end_ = (k); i >= i##_end_; -- i)
#define Set(a, b) memset(a, b, sizeof(a))
#define Cpy(a, b) memcpy(a, b, sizeof(a))
#define x first
#define y second
#define pb(a) push_back(a)
#define mp(a, b) make_pair(a, b)
#define SZ(a) ((int)(a).size())
#define ALL(a) a.begin(), a.end()
#define INF (0x3f3f3f3f)
#define INF1 (2139062143)
#define debug(...) fprintf(stderr, __VA_ARGS__)
#define y1 wozenmezhemecaiatemplate <typename T> inline bool chkmax(T &a, T b) { return a < b ? a = b, 1 : 0; }
template <typename T> inline bool chkmin(T &a, T b) { return b < a ? a = b, 1 : 0; }inline int read() {static int _, __; static char c_;for (_ = 0, __ = 1, c_ = getchar(); c_ < '0' || c_ > '9'; c_ = getchar()) if (c_ == '-') __ = -1;for ( ; c_ >= '0' && c_ <= '9'; c_ = getchar()) _ = (_ << 1) + (_ << 3) + (c_ ^ 48);return _ * __;
}const int maxn = 105, MOD = 1e9 + 7;int C[maxn][maxn], n, m, sz[maxn], ans, dp[maxn][maxn], deg[maxn], tmp[maxn], beg[maxn], nex[maxn], v[maxn], e, tot;
PII E[maxn];struct DSU {int fa[maxn];inline void init(int n) { For(i, 1, n) fa[i] = i; }int find(int x) { return x == fa[x] ? x : fa[x] = find(fa[x]); }inline void merge(int x, int y) { if ((x = find(x)) ^ (y = find(y))) fa[x] = y; }
}dsu;inline void add(int uu, int vv) { v[++ e] = vv, nex[e] = beg[uu], beg[uu] = e, ++ deg[vv]; }
inline int ad(int x, int y) { if ((x += y) >= MOD) return x - MOD; return x; }void DFS(int u) {sz[u] = dp[u][1] = 1;for (register int p = beg[u]; p; p = nex[p]) {DFS(v[p]);For(i, 1, sz[u] + sz[v[p]]) tmp[i] = 0;For(j, 1, sz[u]) For(k, 1, sz[v[p]]) For(i, max(j, k), j + k)tmp[i] = ad(tmp[i], (LL)dp[u][j] * dp[v[p]][k] % MOD * C[i - 1][j - 1] % MOD * C[j - 1][j + k - i] % MOD);sz[u] += sz[v[p]];For(i, 1, sz[u]) dp[u][i] = tmp[i];}
}int main()
{
#ifdef hany01freopen("bzoj4013.in", "r", stdin);freopen("bzoj4013.out", "w", stdout);
#endifstatic char ty;static int x, y;dsu.init(n = read()), m = read();For(i, 0, n) {C[i][0] = C[i][i] = 1;For(j, 1, i - 1) C[i][j] = ad(C[i - 1][j], C[i - 1][j - 1]);}For(i, 1, m) {x = read(), scanf("%c", &ty), y = read();if (ty == '=') dsu.merge(x, y);else E[++ tot] = mp(x, y);}int cnt = 0; For(i, 1, n) if (dsu.find(i) == i) ++ cnt;if (tot >= cnt) { puts("0"); return 0; }For(i, 1, tot) add(dsu.find(E[i].x), dsu.find(E[i].y));For(i, 1, n) if (!deg[i] && dsu.fa[i] == i) add(0, i);DFS(0);For(i, 1, n + 1) ans = ad(ans, dp[0][i]);printf("%d\n", ans);return 0;
}