模板 - Newton 插值

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基于 C++17 标准，实现了环上的 Newton 插值算法

  • 仅在 GCC 下测试过
  • 插入横坐标相同的点会导致除 0 从而 RE

Newton 插值

对给定的点 \((x_0,y_0),(x_1,y_1),\dots,(x_{n-1},y_{n-1})\), Newton 插值得到的结果为

\[ f(x)=f[x_0]+\sum_{i=1}^{n-1}f[x_0,x_1,\dots,x_i]\prod_{k=0}^{i-1}(x-x_i) \]

其中 \(f[x_0,x_1,\dots,x_i]\) 为有限差分，定义如下:

  • \(f[x_i]=y_i\)
  • \(f[x_i,x_{i+1},\dots,x_j]=\dfrac{f[x_i,x_{i+1},\dots,x_{j-1}]-f[x_{i+1},x_{i+2},\dots,x_j]}{x_i-x_j}\)

显然，相比 Lagrange 插值和 Neville 插值，Newton 插值可以做到 \(O(n)\) 的单点插入，且形式更加简单易懂

使用说明

环 T 须有接受 1 个整数的构造函数，T{0} 需为零元，T{1} 需为幺元

时间复杂度

  • 单点插入: \(O(n)\), \(n\) 为已经插入的点的个数
  • 求值: \(O(n)\), \(n\) 为已经插入的点的个数

成员函数列表

NewtonInterp.hview raw
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template <class T>
class NewtonInterp {
  // {(x_0,y_0),(x__1,y_1),...,(x_{n-1},y_{n-1})}
  std::vector<std::pair<T, T>> points;
  // diffs[r][l] = f[x_l,x_{l+1},...,x_r]
  std::vector<std::vector<T>> diffs;
  // (x-x_0)(x-x_1)...(x-x_{n-1})
  std::vector<T> base;
  // f[x_0]+f[x_0,x_1](x-x_0)+...+f[x_0,x_1,...,x_n](x-x_0)(x-x_1)...(x-x_{n-1})
  std::vector<T> fit;

public:
  explicit NewtonInterp() = default;

  // Insert a point
  //! x should be DIFFERENT from points have been inserted
  NewtonInterp &insert(T const &x, T const &y);

  // @return fit
  std::vector<T> coeffs() const;

  // @return f(x)
  T evaluate(T const &x);
};

代码

Show code

NewtonInterp.hppview raw
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template <class T>
class NewtonInterp {
  std::vector<std::pair<T, T>> points;
  std::vector<std::vector<T>> diffs;
  std::vector<T> base;
  std::vector<T> fit;

public:
  explicit NewtonInterp() = default;
  NewtonInterp &insert(T const &x, T const &y) {
    points.emplace_back(x, y);
    size_t n = points.size();
    if (n == 1) {
      base.push_back(1);
    } else {
      size_t m = base.size();
      base.push_back(0);
      for (size_t i = m; i; --i) base[i] = base[i - 1];
      base[0] = 0;
      for (size_t i = 0; i < m; ++i)
        base[i] = base[i] - points[n - 2].first * base[i + 1];
    }
    diffs.emplace_back(points.size());
    diffs[n - 1][n - 1] = y;
    if (n > 1)
      for (size_t i = n - 2; ~i; --i)
        diffs[n - 1][i] = (diffs[n - 2][i] - diffs[n - 1][i + 1]) /
                          (points[i].first - points[n - 1].first);
    fit.push_back(0);
    for (size_t i = 0; i < n; ++i) fit[i] = fit[i] + diffs[n - 1][0] * base[i];
    return *this;
  }
  std::vector<T> coeffs() const { return fit; }
  T evaluate(T const &x) {
    T ans{};
    for (auto it = fit.rbegin(); it != fit.rend(); ++it) ans = ans * x + *it;
    return ans;
  }
};

示例

  • 洛谷 P4463 [集训队互测 2012] calc

    Show code

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    /*
     * @Author: Tifa
     * @Description: From <https://github.com/Tiphereth-A/CP-archives>
     * !!! ATTENEION: All the context below is licensed under a 
     *  GNU Affero General Public License, Version 3.
     *  See <https://www.gnu.org/licenses/agpl-3.0.txt>.
     */
    #include <bits/stdc++.h>
    using ll = long long;
    template <class Tp>
    using vc = std::vector<Tp>;
    template <class Tp>
    using vvc = std::vector<std::vector<Tp>>;
    #define for_(i, l, r, v...) for (ll i = (l), i##e = (r), ##v; i <= i##e; ++i)
    template <typename... Ts>
    void dec(Ts &...x) {
      ((--x), ...);
    }
    template <typename... Ts>
    void inc(Ts &...x) {
      ((++x), ...);
    }
    using namespace std;
    namespace MODINT {
    constexpr int64_t safe_mod(int64_t x, int64_t m) {
      return (x %= m) < 0 ? x + m : x;
    }
    constexpr std::pair<int64_t, int64_t> invgcd(int64_t a, int64_t b) {
      if ((a = safe_mod(a, b)) == 0) return {b, 0};
      int64_t s = b, m0 = 0;
      for (int64_t q = 0, _ = 0, m1 = 1; a;) {
        _ = s - a * (q = s / a);
        s = a;
        a = _;
        _ = m0 - m1 * q;
        m0 = m1;
        m1 = _;
      }
      return {s, m0 + (m0 < 0 ? b / s : 0)};
    }
    template <ptrdiff_t ID = -1>
    class DyMint {
      using self = DyMint<ID>;
      struct Barrett_ {
        uint32_t m_;
        uint64_t im;
        constexpr explicit Barrett_(uint32_t m = 998244353)
          : m_(m), im((uint64_t)(-1) / m + 1) {}
        constexpr uint32_t umod() const { return m_; }
        constexpr uint32_t mul(uint32_t a, uint32_t b) const {
          uint64_t z = a;
          z *= b;
          uint64_t x = (uint64_t)(((__uint128_t)z * im) >> 64);
          uint32_t v = (uint32_t)(z - x * m_);
          return v + (m_ <= v ? m_ : 0);
        }
      };

    protected:
      uint32_t v_;
      static Barrett_ bt_;

    public:
      constexpr static uint32_t mod() { return bt_.umod(); }
      constexpr static void set_mod(uint32_t m) {
        assert(1 <= m);
        bt_ = Barrett_(m);
      }
      constexpr static self raw(uint32_t v) {
        self x;
        x.v_ = v;
        return x;
      }
      constexpr DyMint(): v_(0) {}
      template <class T,
                std::enable_if_t<std::is_integral<T>::value &&
                                 std::is_signed<T>::value> * = nullptr>
      constexpr DyMint(T v): DyMint() {
        int64_t x = (int64_t)(v % (int64_t)mod());
        v_ = (uint32_t)(x + (x < 0 ? mod() : 0));
      }
      template <class T,
                std::enable_if_t<std::is_integral<T>::value &&
                                 std::is_unsigned<T>::value> * = nullptr>
      constexpr DyMint(T v): v_((uint32_t)(v % mod())) {}
      friend std::istream &operator>>(std::istream &is, self &x) {
        int64_t xx;
        is >> xx;
        xx %= mod();
        x.v_ = (uint32_t)(xx + (xx < 0 ? mod() : 0));
        return is;
      }
      friend std::ostream &operator<<(std::ostream &os, const self &x) {
        return os << x.v_;
      }
      constexpr const uint32_t &val() const { return v_; }
      constexpr explicit operator uint32_t() const { return val(); }
      constexpr uint32_t &data() { return v_; }
      constexpr self &operator++() {
        if (++v_ == mod()) v_ = 0;
        return *this;
      }
      constexpr self &operator--() {
        if (!v_) v_ = mod();
        --v_;
        return *this;
      }
      constexpr self operator++(int) {
        self result = *this;
        ++*this;
        return result;
      }
      constexpr self operator--(int) {
        self result = *this;
        --*this;
        return result;
      }
      constexpr self &operator+=(const self &rhs) {
        v_ += rhs.v_;
        if (v_ >= mod()) v_ -= mod();
        return *this;
      }
      constexpr self &operator-=(const self &rhs) {
        v_ -= rhs.v_;
        if (v_ >= mod()) v_ += mod();
        return *this;
      }
      constexpr self &operator*=(const self &rhs) {
        v_ = bt_.mul(v_, rhs.v_);
        return *this;
      }
      constexpr self &operator/=(const self &rhs) {
        return *this = *this * inverse(rhs);
      }
      constexpr self operator+() const { return *this; }
      constexpr self operator-() const { return self() - *this; }
      constexpr friend self pow(self x, uint64_t y) {
        self res(1);
        for (; y; y >>= 1, x *= x)
          if (y & 1) res *= x;
        return res;
      }
      constexpr friend self inverse(const self &x) {
        auto &&_ = invgcd(x.v_, self::mod());
        if (_.first != 1) throw std::runtime_error("Inverse not exist");
        return _.second;
      }
      constexpr friend self operator+(self lhs, const self &rhs) {
        return lhs += rhs;
      }
      constexpr friend self operator-(self lhs, const self &rhs) {
        return lhs -= rhs;
      }
      constexpr friend self operator*(self lhs, const self &rhs) {
        return lhs *= rhs;
      }
      constexpr friend self operator/(self lhs, const self &rhs) {
        return lhs /= rhs;
      }
      constexpr friend bool operator==(const self &lhs, const self &rhs) {
        return lhs.v_ == rhs.v_;
      }
      constexpr friend bool operator!=(const self &lhs, const self &rhs) {
        return lhs.v_ != rhs.v_;
      }
    };
    }  // namespace MODINT
    using mint = MODINT::DyMint<>;
    template <class T>
    class NewtonInterp {
      std::vector<std::pair<T, T>> points;
      std::vector<std::vector<T>> diffs;
      std::vector<T> base;
      std::vector<T> fit;

    public:
      explicit NewtonInterp() = default;
      NewtonInterp &insert(T const &x, T const &y) {
        points.emplace_back(x, y);
        size_t n = points.size();
        if (n == 1) {
          base.push_back(1);
        } else {
          size_t m = base.size();
          base.push_back(0);
          for (size_t i = m; i; --i) base[i] = base[i - 1];
          base[0] = 0;
          for (size_t i = 0; i < m; ++i)
            base[i] = base[i] - points[n - 2].first * base[i + 1];
        }
        diffs.emplace_back(points.size());
        diffs[n - 1][n - 1] = y;
        if (n > 1)
          for (size_t i = n - 2; ~i; --i)
            diffs[n - 1][i] = (diffs[n - 2][i] - diffs[n - 1][i + 1]) /
                              (points[i].first - points[n - 1].first);
        fit.push_back(0);
        for (size_t i = 0; i < n; ++i) fit[i] = fit[i] + diffs[n - 1][0] * base[i];
        return *this;
      }
      std::vector<T> coeffs() const { return fit; }
      T evaluate(T const &x) {
        T ans{};
        for (auto it = fit.rbegin(); it != fit.rend(); ++it) ans = ans * x + *it;
        return ans;
      }
    };
    void solve(int t_ = 0) {
      int k, n, p;
      cin >> k >> n >> p;
      mint::set_mod(p);
      vvc<mint> dp(n + 1, vc<mint>(n * 2 + 2));
      fill(dp[0].begin(), dp[0].end(), 1);
      for_(i, 1, n)
        for_(j, i, n * 2 + 1) dp[i][j] = dp[i - 1][j - 1] * j + dp[i][j - 1];
      mint fact = 1;
      for_(i, 1, n) fact *= i;
      NewtonInterp<mint> ip;
      for_(i, 1, n * 2 + 1) ip.insert(i, dp[n][i]);
      cout << ip.evaluate(k) * fact;
    }
    signed main() {
      std::ios::sync_with_stdio(false);
      std::cin.tie(nullptr);
      std::cerr << std::fixed << std::setprecision(6);
      int i_ = 0;
      solve(i_);
      return 0;
    }