51Nod-1486 大大走格子

Posted 2021-01-01 xfanx

类型为含有多个禁止位置的路线问题

h行w列矩阵，从左上角出发，只能向右或下走，且有n个禁止位置，求到右下角的路线数
(1 leq h, w leq 10^5, 1 leq n leq 2000)

容斥，对禁止位置排序，考虑到达第i个禁止位置的不经过其他禁止位置的路线，它可以通过容斥，用前i-1个更新。
复杂度(O(n^2))

#include <cstdio>
#include <cstring>
#include <algorithm>
#define MOD 1000000007
using namespace std;
typedef long long LL;

int h, w, n;
const int maxn = 2e3 + 10;
const int maxhw = 2e5 + 10;
LL ans[maxn];
int fact[maxhw], inv[maxhw], INV[maxhw];
struct node {
    int x; int y;
} pn[maxn];

bool cmp(const node& A, const node& B) {
    return A.x == B.x ? A.y < B.y : A.x < B.x;
}

void init() {
    fact[0] = fact[1] = 1;
    inv[0] = inv[1] = 1;
    INV[0] = INV[1] = 1;
    for (int i = 2; i < maxhw; i++) {
        fact[i] = (LL)fact[i - 1] * i % MOD;
        inv[i] = (LL)(MOD - MOD / i) * inv[MOD % i] % MOD;
        INV[i] = (LL)inv[i] * INV[i - 1] % MOD;
    }
}

int main() {
    init();
    scanf("%d%d%d", &h, &w, &n);
    for (int i = 0; i < n; i++) {
        scanf("%d%d", &pn[i].x, &pn[i].y);
    }
    sort(pn, pn + n, cmp);
    pn[n].x = h; pn[n].y = w; 
    for (int i = 0; i <= n; i++) {
        int x = pn[i].x; int y = pn[i].y;
        ans[i] = (LL)fact[x + y - 2] * INV[x - 1] % MOD * INV[y - 1] % MOD;
        for (int j = 0; j < i; j++) {
            int nx = pn[j].x, ny = pn[j].y;
            if (nx > x || ny > y) continue;
            ans[i] = (ans[i] - ans[j]  
                        * fact[x - nx + y - ny] % MOD 
                        * INV[x - nx] % MOD 
                        * INV[y - ny] % MOD
                    ) % MOD;
        }
    }
    printf("%lld
", (ans[n] + MOD) % MOD);
    return 0;
}