[多项式学习]Lagrange 插值

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非常简单的算法，但是要注意避免让复杂度升到 $\Theta(n^2\log m)$。

容易验证，$n$ 次多项式 $f(x)=\sum\limits{i}y_i\prod\limits{j\ne i}\dfrac{x-x_j}{x_i-x_j}$ 过 $n+1$ 个点 $(x_i,y_i)$。

注意到如果直接求解复杂度是 $\Theta(n^2\log m)$ 的，进行优化：$f(x)=\sum\limitsiy_i\dfrac{\prod\limits{j\ne i}(x-xj)}{\prod\limits{j\ne i}(x_i-x_j)}$.

当 $xi=i$ 时容易写出：$f(x)=\sum\limits_iy_i\dfrac{\prod\limits{j}(x-j)}{(-1)^{n-i}(x-i)\cdot(n-i)!\cdot(i-1)!}$.注意到当 $i=x$ 时分母会出现 $0$，此时该项取 $y_i$ 即可。

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/**
 * @file P4781 【模板】拉格朗日插值 II.cpp
 * @author Kuriyama Mirai (hermione_granger@foxmail.com)
 * @brief auto 记得加 & 和 &&！
 * @link https://www.luogu.com.cn/record/68162446
 * @date 2022-01-27
 * 
 * @copyright Copyright (c) 2022
 * 
 */
#include <cstdio>
#include <cstdlib>
#include <array>
#include <algorithm>
#include <unordered_map>
#include <vector>

namespace mirai {

constexpr int MAXN = 2005;
constexpr int MOD = 998244353;
typedef long long ll;

template <typename ll = long long>
ll pow(ll a, ll b, ll mod = MOD) {
  ll res = 1;
  while (b) {
    if (b & 1) {
      res = res * a % mod;
    }
    a = a * a % mod;
    b >>= 1;
  }
  return res;
}

template <typename ll = long long>
inline ll inv(ll a, ll mod = MOD) {
  return pow(a, mod - 2, mod);
}

// quick Positive MODulo.
template <typename ll = long long>
inline ll pmod(ll a, ll mod = MOD) {
  return a >= mod ? a - mod : a;
}
// quick Negative MODulo.
template <typename ll = long long>
inline ll nmod(ll a, ll mod = MOD) {
  return a < 0 ? a + mod : a;
}

template <typename ll = long long, std::size_t deg = MAXN, ll mod = MOD>
class poly {
 public:
  std::vector<ll> coe;
  const ll mod_ = mod;
  poly(ll val = 0) {
    coe.resize(deg + 1);
    coe[0] = val;
  }
  
  std::size_t degree() const {
    return deg;
  }
  ll operator[](std::size_t x) const {
    return coe[x];
  }
  ll& operator[](std::size_t x) {
    return coe[x];
  }
  ll at(std::size_t x) const {
    return coe.at(x);
  }
  ll& at(std::size_t x) {
    return coe.at(x);
  }
  
  ll operator()(ll const& x) const {
    ll res = coe[0], now = x;
    for (std::size_t i = 1; i <= deg; ++i) {
      res = (res + now * coe[i]) % mod;
      now = now * x % mod;
    }
    return now;
  }
  
  poly<ll, deg, mod>& operator=(poly<ll, deg, mod> const& x) {
    coe = x.coe;
    return *this;
  }
  poly<ll, deg, mod>& operator+=(poly<ll, deg, mod> const& x) {
    *this = *this + x;
    return *this;
  }
  poly<ll, deg, mod>& operator-=(poly<ll, deg, mod> const& x) {
    *this = *this - x;
    return *this;
  }
};

template <typename ll, std::size_t deg, ll mod>
inline std::size_t degree(poly<ll, deg, mod> const& x) {
  return x.degree();
}

template <typename ll, std::size_t deg, ll mod>
poly<ll, deg, mod> operator+(poly<ll, deg, mod> const& x, poly<ll, deg, mod> const& y) {
  poly<ll, deg, mod> res;
  for (std::size_t i = 0; i <= deg; ++i) {
    res[i] = pmod(x[i] + y[i], mod);
  }
  return res;
}

template <typename ll, std::size_t deg, ll mod>
poly<ll, deg, mod> operator-(poly<ll, deg, mod> const& x, poly<ll, deg, mod> const& y) {
  poly<ll, deg, mod> res;
  for (std::size_t i = 0; i <= deg; ++i) {
    res[i] = nmod(x[i] - y[i], mod);
  }
  return res;
}

template <typename ll, std::size_t deg, ll mod>
poly<ll, deg, mod> derivative(poly<ll, deg, mod> x) {
  for (std::size_t i = 0; i < deg; ++i) {
    x[i] = (x[i + 1] * (i + 1)) % mod;
  }
  x[deg] = 0;
  return x;
}

// The constant will be set to 0.
template <typename ll, std::size_t deg, ll mod>
poly<ll, deg, mod> indefinite_integral(poly<ll, deg, mod> x) {
  for (std::size_t i = deg; i > 0; --i) {
    x[i] = (x[i - 1] * inv(i - 1, mod)) % mod;
  }
}
template <typename ll, std::size_t deg, ll mod>
inline poly<ll, deg, mod> integral(poly<ll, deg, mod> x) {
  return indefinite_integral(x);
}

template <typename ll, std::size_t deg, ll mod>
inline poly<ll, deg, mod> plug_in(poly<ll, deg, mod> const& f, ll x) {
  return f(x);
}

template <typename ll = long long, ll mod = MOD>
ll lagrange_polynomial_with_value(std::vector<std::pair<ll, ll>> const& points, ll const& x) {
  ll res = 0;
  for (auto &&i : points) {
    ll now = 1, now2 = 1;
    for (auto &&j : points) {
      if (j.first == i.first) {
        if (j.second != i.second) {
          throw "Two distinct points with one x-value!";
        } else {
          continue;
        }
      }
      now = now * nmod(x - j.first, mod) % mod;
      now2 = now2 * nmod(i.first - j.first, mod) % mod;
    }
    res = (res + i.second * now % mod * inv(now2, mod)) % mod;
  }
  return res;
}


int main(int argc, char** argv) {
  int n;
  long long val;
  std::scanf("%d%lld", &n, &val);
  std::vector<std::pair<long long, long long>> points;
  points.resize(n);
  for (auto &i : points) {
    std::scanf("%lld%lld", &i.first, &i.second);
  }
  std::printf("%lld\n", lagrange_polynomial_with_value(points, val));
  return 0;
}

}

int main(int argc, char** argv) {
  return mirai::main(argc, argv);
}