51Nod-1486 大大走格子

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;
}