【Foreign】登山 [DP][数学]

  1 #include<iostream>
  2 #include<string>
  3 #include<algorithm>
  4 #include<cstdio>
  5 #include<cstring>
  6 #include<cstdlib>
  7 #include<cmath>
  8 using namespace std;
  9 typedef long long s64;
 10 
 11 const int ONE = 5000005;
 12 const int MOD = 1e9 + 7;
 13 
 14 int n, m;
 15 int x, y;
 16 int fac[ONE], inv[ONE];
 17 int f[ONE];
 18 
 19 struct point
 20 {
 21         int x, y;
 22 }a[ONE];
 23 bool cmp(const point &a, const point &b)
 24 {
 25         if(a.x != b.x) return a.x < b.x;
 26         return a.y < b.y;
 27 }
 28 
 29 int get()
 30 {
 31         int res=1,Q=1;  char c;
 32         while( (c=getchar())<48 || c>57)
 33         if(c==‘-‘)Q=-1;
 34         if(Q) res=c-48; 
 35         while((c=getchar())>=48 && c<=57) 
 36         res=res*10+c-48;
 37         return res*Q; 
 38 }
 39 
 40 int Quickpow(int a, int b)
 41 {
 42         int res = 1;
 43         while(b)
 44         {
 45             if(b & 1) res = (s64)res * a % MOD;
 46             a = (s64)a * a % MOD;
 47             b >>= 1;
 48         }
 49         return res;
 50 }
 51 
 52 void Deal_first()
 53 {
 54         fac[0] = 1;
 55         for(int i = 1; i <= 2 * n; i++)
 56             fac[i] = (s64)fac[i - 1] * i % MOD;
 57         inv[2 * n] = Quickpow(fac[2 * n], MOD - 2);
 58         for(int i = 2 * n - 1; i >= 0; i--)
 59             inv[i] = (s64)inv[i + 1] * (i + 1) % MOD;
 60 }
 61 
 62 int C(int n, int m)
 63 {
 64         if(n < 0 || m < 0) return 0;
 65         return (s64)fac[n] * inv[m] % MOD * inv[n - m] % MOD;
 66 }
 67 
 68 void Modit(int &a)
 69 {
 70         if(a < 0) a += MOD;
 71         if(a >= MOD) a -= MOD;
 72 }
 73 
 74 int Ways(point a, point b)
 75 {
 76         if(n < 0 || m < 0) return 0;
 77         return C(b.y - a.y + b.x - a.x, b.y - a.y);
 78 }
 79 
 80 int Getit(point a, point b)
 81 {
 82         return Ways(a, b) - Ways(a, (point){b.y - 1, b.x + 1});
 83 }
 84 
 85 int main()
 86 {
 87         n = get();    m = get();
 88         Deal_first();
 89 
 90         for(int i = 1; i <= m; i++)
 91             a[i].x = get(), a[i].y = get();
 92 
 93         a[++m] = (point){n, n};
 94         sort(a + 1, a + m + 1, cmp);
 95 
 96         for(int i = 1; i <= m; i++)
 97         {
 98             Modit(f[i] = Getit((point){0, 0}, a[i]));
 99             for(int j = 1; j < i; j++)
100                 Modit(f[i] -= (s64)f[j] * Getit(a[j], a[i]) % MOD);
101         }
102 
103         printf("%d", f[m]);
104 }