题意:
有一个n行m列的网格图,起点在格点(n+1,1), 终点在(1,m+1),有k个格点是不能走的障碍点。求从起点到终点的路线有多少种?(只能向上或向右)
思路:
从数据范围看当n,m<=1000时我们可以用动态规划解决:
dp[n+1][1]=1;
dp[i][j]=dp[i+1][j]+dp[i][j-1]
当n,m超过1000时,我们发现k很小,所以可以用组合数和容斥解决:
从(x,y)到(x2,y2)的路线方式数目为C(x-x2,x-x2+y2-y)(既有(x-x2+y2-y)步,其中选择了(x-x2)步向上走);
代码:
#include <bits/stdc++.h> #include <cstdio> #include <cstring> #include <string> #include <algorithm> #include <cmath> #include <vector> #include <queue> #include <iostream> using namespace std; typedef long long ll; const ll inf=998244353; ll d[1005][1005], dp[1005][1005]; ll qpow(ll n,ll m) { ll z=1; while(m) { if(m&1) { z=z*n%inf; } n=n*n%inf; m>>=1; } return z; } ll C(ll n,ll m) { if(m-n<n) { n=m-n; } ll x=1, y=1; for(ll i=1, j=m;i<=n;i++,j--) { x=x*j%inf; y=y*i%inf; } x=x*qpow(y,inf-2)%inf; return x; } struct w { ll x, y; }w[5]; ll ans=0; ll n, m, k; void rongchi(int i,int flag,ll x,ll y, ll v,int o) { if(i==k) { v=v*C(x-1,x-1+m+1-y)%inf; //printf("v=%lld %d %lld\n",v,flag,o); ans=(ans+v*flag+inf)%inf; } else { if(w[i].x<=x&&w[i].y>=y) { ll vi=v*C(x-w[i].x,x-w[i].x+w[i].y-y)%inf; rongchi(i+1,-flag,w[i].x,w[i].y,vi,o+1); } rongchi(i+1,flag,x,y,v,o); } } bool cmp(struct w a,struct w b) { if(a.x!=b.x) { return a.x>b.x; } return a.y<b.y; } int main() { scanf("%lld%lld%lld",&n,&m,&k); if(n<=1000&&m<=1000) { for(int i=0;i<k;i++) { int x, y; scanf("%d%d",&x,&y); d[x][y]=1; } dp[n+1][1]=1; for(int i=n+1;i>=1;i--) { for(int j=1;j<=m+1;j++) { if(d[i][j]==1) { continue; } dp[i][j]=(dp[i][j]+dp[i+1][j]+dp[i][j-1])%inf; } } cout << dp[1][m+1] << endl; } else { for(int i=0;i<k;i++) { scanf("%lld%lld",&w[i].x,&w[i].y); } sort(w,w+k,cmp); rongchi(0,1,n+1,1,1,0); cout << ans << endl; } return 0; }