【題目連結】
- 點選打開連結
【思路要點】
- 打表可得,若各個位置 i i i 上權值的期望是關于 i i i 的一 / 二次函數,則經過一次洗牌後,各個位置 i i i 上權值的期望依然是關于 i i i 的一 / 二次函數。
- 暴力計算牌堆底的三項,然後插值即可。
- 時間複雜度 O ( M + Q ) O(M+Q) O(M+Q) 。
【代碼】
#include<bits/stdc++.h> using namespace std; const int MAXN = 5; const int P = 998244353; typedef long long ll; typedef long double ld; typedef unsigned long long ull; template <typename T> void chkmax(T &x, T y) {x = max(x, y); } template <typename T> void chkmin(T &x, T y) {x = min(x, y); } template <typename T> void read(T &x) { x = 0; int f = 1; char c = getchar(); for (; !isdigit(c); c = getchar()) if (c == '-') f = -f; for (; isdigit(c); c = getchar()) x = x * 10 + c - '0'; x *= f; } template <typename T> void write(T x) { if (x < 0) x = -x, putchar('-'); if (x > 9) write(x / 10); putchar(x % 10 + '0'); } template <typename T> void writeln(T x) { write(x); puts(""); } int power(int x, int y) { if (y == 0) return 1; int tmp = power(x, y / 2); if (y % 2 == 0) return 1ll * tmp * tmp % P; else return 1ll * tmp * tmp % P * x % P; } namespace Lagrange { const int MAXN = 5; int n, x[MAXN], y[MAXN], a[MAXN]; int p[MAXN], q[MAXN]; void work() { memset(p, 0, sizeof(p)); memset(q, 0, sizeof(q)); memset(a, 0, sizeof(a)); p[0] = 1; for (int i = 1; i <= n; i++) { for (int j = i - 1; j >= 0; j--) { p[j + 1] = (p[j + 1] + p[j]) % P; p[j] = (P - 1ll * p[j] * x[i] % P) % P; } } for (int i = 1; i <= n; i++) { memset(q, 0, sizeof(q)); for (int j = n - 1; j >= 0; j--) q[j] = (p[j + 1] + 1ll * q[j + 1] * x[i]) % P; int now = 1; for (int j = 1; j <= n; j++) if (j != i) now = 1ll * now * (x[i] - x[j]) % P; now = power((P + now) % P, P - 2); for (int j = 0; j <= n; j++) q[j] = 1ll * q[j] * now % P; for (int j = 0; j <= n; j++) a[j] = (a[j] + 1ll * q[j] * y[i]) % P; } } int get(int x) { int ans = 0, now = 1; for (int i = 0; i <= n; i++) { ans = (ans + 1ll * now * a[i]) % P; now = 1ll * now * x % P; } return ans; } } void update(int &x, int y) { x += y; if (x >= P) x -= P; } int n, m, type, mid, inv[MAXN]; void work(int x, int y, int now, int p) { if (now >= 4) return; int px = 1ll * x * inv[now] % P; int py = 1ll * (y - mid) * inv[now] % P; if (px) { update(Lagrange :: y[now], 1ll * p * px % P * Lagrange :: get(x) % P); work(x - 1, y, now + 1, 1ll * p * px % P); } if (py) { update(Lagrange :: y[now], 1ll * p * py % P * Lagrange :: get(y) % P); work(x, y - 1, now + 1, 1ll * p * py % P); } } int main() { freopen("landlords.in", "r", stdin); freopen("landlords.out", "w", stdout); read(n), read(m), read(type); Lagrange :: n = 3; Lagrange :: a[type] = 1; for (int i = 1; i <= 3; i++) { Lagrange :: x[i] = n - i + 1; inv[i] = power(n - i + 1, P - 2); Lagrange :: y[i] = 0; } for (int i = 1; i <= m; i++) { read(mid); work(mid, n, 1, 1); Lagrange :: work(); for (int i = 1; i <= 3; i++) Lagrange :: y[i] = 0; } int q; read(q); while (q--) { int x; read(x); writeln(Lagrange :: get(x)); } return 0; }