[NOI2019]斗主地 [拉格朗日插值, 找规律, 组合数学]

Posted 2022-07-08 qq62c30ac77b2a7

​​传送门​​

首先有暴力的30分做法, 设 f[i][j] 为还是i, j 个的概率

然后 v[i] 为当前的数列

然后你会毒瘤地发现，type = 1 时它是个等差的，type = 2时是一个二阶等差的

也就是说，我们只需要维护前3项，然后暴力插出后面的就可以了

每次分成A 和 n - A 两堆的时候，我们需要插出 A+1,A+2, A+3的值，然后类似上面的 v 解决当前的前3项

现在唯一的问题是 f 数组，显然不能 n^2 处理

发现分子和分母是定下来的

分母是  

分子是  

然后两堆的取法是一个组合数 

---

30 pts (没想到矩阵快速幂啊)

#include<bits/stdc++.h>
#define N 105
using namespace std;
typedef long long ll;
int read()
  int cnt = 0, f = 1; char ch = 0;
  while(!isdigit(ch)) ch = getchar(); if(ch == -) f = -1;
  while(isdigit(ch)) cnt = (cnt << 1) + (cnt << 3) + (ch-0), ch = getchar();
  return cnt * f;

const int Mod = 998244353;
int n, m, type;
ll f[N][N], v[N], a[N], b[N];
ll add(ll a, ll b) return (a+b) % Mod;
ll mul(ll a, ll b) return (a*b) % Mod;
ll power(ll a, ll b)
  ll ans = 1;
  for(;b;b>>=1)
    if(b&1) ans = mul(ans, a);
    a = mul(a, a);
   return ans;

int main()
  n = read(), m = read(), type = read();
  for(int i=1; i<=n; i++) v[i] = (type == 1 ? i : mul(i, i));
  for(int Case=1; Case<=m; Case++)
    int A = read(); 
    for(int i=1; i<=A; i++) a[i] = v[i];
    for(int i=1; i<=n-A; i++) b[i] = v[i + A];
    memset(f, 0, sizeof(f));
    f[A][n-A] = 1;
    for(int i=A; i>=0; i--)
      for(int j=n-A; j>=0; j--)
        ll inv = power(i + j, Mod - 2);
        if(i) f[i-1][j] = add(f[i-1][j], mul(f[i][j], mul(inv, i)));
        if(j) f[i][j-1] = add(f[i][j-1], mul(f[i][j], mul(inv, j)));
      
     memset(v, 0, sizeof(v));
    for(int i=n; i>=1; i--)
      ll inv = power(i, Mod - 2);
      for(int j=min(i, A); j>=0; j--)
        int k = i - j; if(k > n - A) break;
        if(j) v[i] = add(v[i], mul(f[j][k], mul(a[j], mul(inv, j))));
        if(k) v[i] = add(v[i], mul(f[j][k], mul(b[k], mul(inv, k))));
      
    
  
  int q = read();
  for(int i=1; i<=q; i++)
    int x = read(); printf("%lld\\n", v[x]);
   return 0;

---

100 pts

#include<bits/stdc++.h>
#define N 10000050
using namespace std;
typedef long long ll;
int read()
  int cnt = 0, f = 1; char ch = 0;
  while(!isdigit(ch)) ch = getchar(); if(ch == -) f = -1;
  while(isdigit(ch)) cnt = (cnt << 1) + (cnt << 3) + (ch-0), ch = getchar();
  return cnt * f;

const int Mod = 998244353;
int n, m, type;
ll add(ll a, ll b) return (a+b) % Mod;
ll mul(ll a, ll b) return (a*b) % Mod;
ll power(ll a, ll b)
  ll ans = 1;
  for(;b;b>>=1)
    if(b&1) ans = mul(ans, a);
    a = mul(a, a);
   return ans;

ll fac[N], inv[N];
void prework()
  fac[0] = fac[1] = inv[0] = inv[1] = 1;
  for(int i=2; i<=n; i++) fac[i] = mul(fac[i-1], i);
  inv[n] = power(fac[n], Mod - 2);
  for(int i=n-1; i>=2; i--) inv[i] = mul(inv[i+1], i+1);

ll f[5], g[5], h[5], pre[5], suf[5];
ll C(int n, int m) return mul(fac[n], mul(inv[n-m], inv[m]));
ll Get(int A, int B, int i, int j)
  if(A < i || B < j) return 0;
  ll ans = power(mul(fac[A + B], inv[i + j]), Mod - 2);
  ans = mul(ans, C(A + B - i - j, A - i));
  ans = mul(ans, mul(mul(fac[A], inv[i]), mul(fac[B], inv[j])));
  return ans;

const int up = 3;
ll lagrange(ll *y, int n)
  if(n <= up) return y[n]; pre[0] = suf[up + 1] = 1;
  for(int i=1; i<=up; i++) pre[i] = mul(pre[i-1], n - i);
  for(int i=up; i>=1; i--) suf[i] = mul(suf[i+1], n - i);
  ll ans = 0;
  for(int i=1; i<=up; i++)
    ll tmp = mul(y[i], mul(mul(inv[i-1], inv[up-i]), mul(pre[i-1], suf[i+1])));
    ans = ((up - i) & 1) ? add(ans, Mod - tmp) : add(ans, tmp);
   return ans;

int main()
  freopen("landlords.in","r",stdin);
  freopen("landlords.out","w",stdout);
  n = read(), m = read(), type = read(); prework();
  for(int i=1; i<=3; i++) f[i] = (type == 1 ? i : mul(i, i));
  for(int Case = 1; Case <= m; Case ++)
    int A = read();
    swap(f, g);
    for(int i=1; i<=3; i++) h[i] = lagrange(g, A + i);
    memset(f, 0, sizeof(f));
    for(int i=0; i<=3; i++)
      for(int j=0; i+j <= 3; j++)
        if(i + j == 0) continue;
        ll tmp = Get(A, n - A, i, j);
        ll inv = power(i + j, Mod - 2);
        f[i + j] = add(f[i + j], mul(tmp, mul(g[i], mul(i, inv))));
        f[i + j] = add(f[i + j], mul(tmp, mul(h[j], mul(j, inv))));
      
    
  
  int q = read();
  while(q--)
    int x = read(); printf("%lld\\n", lagrange(f, x));
   return 0;