[容斥原理][DP] SRM 498 Div1 Hard FoxJumping

Solution

对不能走(+ai,+ai)容斥，设dpp,q表示走了p步不合法的，和为10q的方案数。

同时DP出x,y两个方向上走了p步，长度为q的方案数xp,q,yp,q。因为此时x,y独立，最后计算的时候直接乘起来就好了。

那最后答案就是∑p=0R∑q=0⌊min(m,n)10⌋(−1)p(Rp)xR−p,n−10pyR−p,m−10qdpp,q

x,y的DP显然可以前缀和优化，刚开始暴力n3T掉以后还打算FFT。

// BEGIN CUT HERE

// END CUT HERE
#line 5 "FoxJumping.cpp"
#include <bits/stdc++.h>
using namespace std;

const int N = 888;
const int MOD = 10007;

int dp[N << 1]
;
int x[N << 1]
, y[N << 1]
;
int gx[N << 1]
, gy[N << 1]
;
int C[N << 1][N << 1];
int n, m, mn, dx, dy, R, cnt, ans;
int a
;

class FoxJumping {

public:
inline void Add(int &x, int a) {
x = (x + a) % MOD;
}
int getCount(int Tx, int Ty, int Mx, int My, int _R, vector <int> bad) {
n = Tx; m = Ty; dx = Mx; dy = My; R = _R; mn = min(m, n);
bad.push_back(0); cnt = bad.size();
for (int i = 0; i < cnt; i++) a[i + 1] = bad[i];
sort(a + 1, a + cnt + 1);
cnt = unique(a + 1, a + cnt + 1) - a - 1;

ans = 0;
memset(x, 0, sizeof x);
memset(y, 0, sizeof y);
memset(dp, 0, sizeof dp);

for (int i = 0; i <= R; 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;
}

x[0][0] = 1;
for (int i = 0; i <= R; i++)
for (int j = 0; j <= n; j++) {
gx[i][j + 1] = (gx[i][j] + x[i][j]) % MOD;
Add(x[i + 1][j], (gx[i][j + 1] - gx[i][max(j + 1 - dx - 1, 0)] + MOD) % MOD);
}
y[0][0] = 1;
for (int i = 0; i <= R; i++)
for (int j = 0; j <= m; j++) {
gy[i][j + 1] = (gy[i][j] + y[i][j]) % MOD;
Add(y[i + 1][j], (gy[i][j + 1] - gy[i][max(j + 1 - dy - 1, 0)] + MOD) % MOD);
}
dp[0][0] = 1;
for (int i = 0; i <= R; i++)
for (int j = 0; j * 10 <= mn; j++)
for (int k = 1; k <= cnt; k++)
if (j * 10 + a[k] <= mn) Add(dp[i + 1][j + a[k] / 10], dp[i][j]);
else break;

for (int p = 0; p <= R; p++)
for (int q = 0; 10 * q <= mn; q++) {
int res = x[R - p][n - 10 * q] * y[R - p][m - 10 * q] % MOD * dp[p][q] % MOD * C[R][p] % MOD;
if (p & 1) Add(ans, MOD - res);
else Add(ans, res);
}
return ans;
}

// BEGIN CUT HERE
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(); if ((Case == -1) || (Case == 5)) test_case_5(); }
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 Arg0 = 2; int Arg1 = 2; int Arg2 = 1; int Arg3 = 1; int Arg4 = 2; int Arr5[] = {}; vector <int> Arg5(Arr5, Arr5 + (sizeof(Arr5) / sizeof(Arr5[0]))); int Arg6 = 1; verify_case(0, Arg6, getCount(Arg0, Arg1, Arg2, Arg3, Arg4, Arg5)); }
void test_case_1() { int Arg0 = 2; int Arg1 = 2; int Arg2 = 1; int Arg3 = 1; int Arg4 = 3; int Arr5[] = {}; vector <int> Arg5(Arr5, Arr5 + (sizeof(Arr5) / sizeof(Arr5[0]))); int Arg6 = 6; verify_case(1, Arg6, getCount(Arg0, Arg1, Arg2, Arg3, Arg4, Arg5)); }
void test_case_2() { int Arg0 = 10; int Arg1 = 10; int Arg2 = 10; int Arg3 = 10; int Arg4 = 1; int Arr5[] = {}; vector <int> Arg5(Arr5, Arr5 + (sizeof(Arr5) / sizeof(Arr5[0]))); int Arg6 = 1; verify_case(2, Arg6, getCount(Arg0, Arg1, Arg2, Arg3, Arg4, Arg5)); }
void test_case_3() { int Arg0 = 10; int Arg1 = 10; int Arg2 = 10; int Arg3 = 10; int Arg4 = 1; int Arr5[] = {10}; vector <int> Arg5(Arr5, Arr5 + (sizeof(Arr5) / sizeof(Arr5[0]))); int Arg6 = 0; verify_case(3, Arg6, getCount(Arg0, Arg1, Arg2, Arg3, Arg4, Arg5)); }
void test_case_4() { int Arg0 = 11; int Arg1 = 11; int Arg2 = 11; int Arg3 = 11; int Arg4 = 2; int Arr5[] = {10}; vector <int> Arg5(Arr5, Arr5 + (sizeof(Arr5) / sizeof(Arr5[0]))); int Arg6 = 140; verify_case(4, Arg6, getCount(Arg0, Arg1, Arg2, Arg3, Arg4, Arg5)); }
void test_case_5() { int Arg0 = 123; int Arg1 = 456; int Arg2 = 70; int Arg3 = 80; int Arg4 = 90; int Arr5[] = {30, 40, 20, 10, 50}; vector <int> Arg5(Arr5, Arr5 + (sizeof(Arr5) / sizeof(Arr5[0]))); int Arg6 = 6723; verify_case(5, Arg6, getCount(Arg0, Arg1, Arg2, Arg3, Arg4, Arg5)); }

// END CUT HERE

};

// BEGIN CUT HERE
int main(void) {
FoxJumping ___test;
___test.run_test(-1);
system("pause");
}
// END CUT HERE