[容斥原理][Meet-in-middle][Matrix-Tree定理] SRM551 Div1 Hard SweetFruits

$Solution$

将原问题分成两个子问题。

• 从所有甜的水果中选出$k$个水果作为真甜的，要求他们甜度之和不超过$maxSweetness$的方案数$cnt_k$
• 令$n$为甜的水果数目，求出组成的树的方案数$tree_k$使得$n$个甜的水果中恰有$k$个水果为真甜的。
  $$ans=\sum_{k=0}^ncnt_ktree_k$$

第一个问题Meet-in-middle可以$O(2^{n\over2}n)$解决。
第二个问题让真甜的水果和任意水果连边，让半甜的水果只能和不甜的水果连边，苦的水果和任意水果连边。这样求出其生成树的方案数$T_k=\sum_{i=0}^k{k\choose i}tree_i$。二项式反演一下就能求到$tree_k=\sum_{i=0}^k(-1)^{k-i}{k\choose i}T_i$了。时间复杂度$\mathcal{O}(n^4)$。

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#line 5 "SweetFruits.cpp"
#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
const int N = 90;
const int M = 21;
const int MOD = 1000000007;

int lim, n, m, ans;
int a[N];
int tree[N], T[N];
int C[N][N];
vector<int> l1[N], l2[N];
struct Matrix {
    int a[N][N];
    inline int *operator [](int x) {
        return a[x];
    }
    inline void Clear(void) {
        memset(a, 0, sizeof a);
    }
};
Matrix A;

inline void Add(int &x, int a) {
    x = (x + a) % MOD;
}
inline int Pow(int a, int b) {
    int c = 1;
    while (b) {
        if (b & 1) c = (ll)c * a % MOD;
        b >>= 1; a = (ll)a * a % MOD;
    }
    return c;
}
inline int Inv(int x) {
    return Pow(x, MOD - 2);
}
inline int Det(Matrix a, int n) {
    int f = 0;
    for (int i = 1; i <= n; i++) {
        int k = 0;
        for (int j = i; j <= n; j++)
            if (a[j][i]) { k = j; break; }
        if (i != k) {
            for (int j = i; j <= n; j++)
                swap(a[i][j], a[k][j]);
            f ^= 1;
        }
        for (int j = i + 1; j <= n; j++) {
            int t = (ll)Inv(a[i][i]) * a[j][i] % MOD;
            for (int k = i; k <= n; k++)
                Add(a[j][k], MOD - (ll)t * a[i][k] % MOD);
        }
    }
    int res = 1;
    for (int i = 1; i <= n; i++)
        res = (ll)res * a[i][i] % MOD;
    if (f) res = (MOD - res) % MOD;
    return res;
}

class SweetFruits {

public:
    inline void Pre(void) {
        int mid = n >> 1, U;
        U = 1 << mid;
        for (int S = 0; S < U; S++) {
            int c1 = 0, co = 0;
            for (int i = 1; i <= mid; i++)
                if ((S >> i - 1) & 1) {
                    ++c1; co += a[i];
                }
            l1[c1].push_back(co);
        }
        U = 1 << (n - mid);
        for (int S = 0; S < U; S++) {
            int c1 = 0, co = 0;
            for (int i = mid + 1; i <= n; i++)
                if ((S >> i - mid - 1) & 1) {
                    ++c1; co += a[i];
                }
            l2[c1].push_back(co);
        }
        for (int i = 0; i <= mid; i++)
            sort(l1[i].begin(), l1[i].end());
        for (int i = 0; i <= n - mid; i++)
            sort(l2[i].begin(), l2[i].end());
        for (int i = 0; i <= n + m; i++) {
            C[i][0] = 1;
            for (int j = 1; j <= i; j++)
                C[i][j] = (C[i - 1][j - 1] + C[i - 1][j]) % MOD;
        }
    }
    inline int Count(int k) {
        int res = 0;
        for (int x = 0; x <= k; x++) {
            int y = k - x, p2 = l2[y].size() - 1;
            for (int u: l1[x]) {
                while (~p2 && l2[y][p2] + u > lim) --p2;
                Add(res, p2 + 1);
            }
        }
        return res;
    }
    inline void AddEdge(int from, int to) {
        A[from][to] = MOD - 1; A[to][from] = MOD - 1;
        ++A[from][from]; ++A[to][to];
    }
    inline int Tree(int k) {
        A.Clear();
        for (int i = 1; i <= n + m; i++)
            for (int j = i + 1; j <= n + m; j++)
                if (j <= k || j > n)
                    AddEdge(i, j);
        T[k] = Det(A, n + m - 1);
        for (int i = 0; i <= k; i++)
            if ((k - i) & 1) Add(tree[k], MOD - (ll)C[k][i] * T[i] % MOD);
            else Add(tree[k], (ll)C[k][i]* T[i] % MOD);
        return tree[k];
    }
    int countTrees(vector <int> sweetness, int maxSweetness) {
        lim = maxSweetness;
        for (int x: sweetness)
            if (x >= 0) a[++n] = x;
            else ++m;
        Pre(); ans = 0;
        for (int k = 0; k <= n; k++)
            Add(ans, (ll)Count(k) * Tree(k) % MOD);
        return ans;
    }

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public:
    void run_test(int Case) { if ((Case == -1) || (Case == 0)) test_case_0(); if ((Case == -1) || (Case == 1)) test_case_1(); if ((Case == -1) || (Case == 2)) test_case_2(); if ((Case == -1) || (Case == 3)) test_case_3(); if ((Case == -1) || (Case == 4)) test_case_4(); }
private:
    template <typename T> string print_array(const vector<T> &V) { ostringstream os; os << "{ "; for (typename vector<T>::const_iterator iter = V.begin(); iter != V.end(); ++iter) os << '\"' << *iter << "\","; os << " }"; return os.str(); }
    void verify_case(int Case, const int &Expected, const int &Received) { cerr << "Test Case #" << Case << "..."; if (Expected == Received) cerr << "PASSED" << endl; else { cerr << "FAILED" << endl; cerr << "\tExpected: \"" << Expected << '\"' << endl; cerr << "\tReceived: \"" << Received << '\"' << endl; } }
    void test_case_0() { int Arr0[] = {1, 2, -1, 3}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arg1 = 3; int Arg2 = 3; verify_case(0, Arg2, countTrees(Arg0, Arg1)); }
    void test_case_1() { int Arr0[] = {1, 2, -1, 3}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arg1 = 5; int Arg2 = 7; verify_case(1, Arg2, countTrees(Arg0, Arg1)); }
    void test_case_2() { int Arr0[] = {-1, -1, 2, 5, 5}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arg1 = 6; int Arg2 = 20; verify_case(2, Arg2, countTrees(Arg0, Arg1)); }
    void test_case_3() { int Arr0[] = {2, 6, 8, 4, 1, 10, -1, -1, -1, -1}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arg1 = 15; int Arg2 = 17024000; verify_case(3, Arg2, countTrees(Arg0, Arg1)); }
    void test_case_4() { int Arr0[] = {1078451, -1, 21580110, 8284711, -1, 4202301, 3427559, 8261270, -1, 16176713,
                                                                         22915672, 24495540, 19236, 5477666, 12280316, 3305896, 17917887, 564911, 22190488, 21843923,
                                                                         23389728, 14641920, 9590140, 12909561, 20405638, 100184, 23336457, 12780498, 18859535, 23180993,
                                                                         10278898, 5753075, 21250919, 17563422, 10934412, 22557980, 24895749, 7593671, 10834579, 5606562}; vector <int> Arg0(Arr0, Arr0 + (sizeof(Arr0) / sizeof(Arr0[0]))); int Arg1 = 245243285; int Arg2 = 47225123; verify_case(4, Arg2, countTrees(Arg0, Arg1)); }

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};

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int main(void) {
    SweetFruits ___test;
    ___test.run_test(-1);
    system("pause");
}
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