Add Lagrange Interpolation and modular int class

Added one test

Author: simonfallon

code/data-structures/seg_tree_lazy_prop.cpp

@@ -1,4 +1,3 @@
-#include "../c++/template.cpp"
 const int N = 1e5 + 10;
 int t[N << 2], lazy[N << 2];
 struct stree{
@@ -12,6 +11,14 @@ struct stree{
       lazy[v] = 0;
     }
   }
+  void build(int v, int tl, int tr, vi& a){
+    if(tl == tr){
+      t[v] = a[tl]; return;
+    }
+    int tm = (tl + tr) >> 1;
+    build(v << 1, tl, tm, a), build((v << 1) | 1, tm+1, tr, a);
+    t[v] = oper(t[v << 1], t[(v << 1) | 1]);
+  }
   void upd(int v, int tl, int tr){
     if(tl > r || tr < l) return;
     if(l <= tl && tr <= r){

code/math/__Test/LagrangeInterpolation_Test.cpp

@@ -0,0 +1,93 @@
+// https://codeforces.com/gym/104505/problem/A
+#include <bits/stdc++.h>
+using namespace std;
+
+#define forn(i,n) for(int i=0; i<n; ++i)
+#define for1(i,n) for(int i=1; i<=n; ++i)
+#define fore(i,l,r) for(int i=l; i<=r; ++i)
+#define fored(i,l,r) for(int i=r; i>=l; --i)
+#define el '\n'
+#define sz(v) int(v.size())
+
+typedef long long ll;
+const int mod = 1e9 + 7;
+template <class T>
+T bpow(T b, int e) {
+  T a(1);
+  do{
+    if(e & 1) a *= b;
+    b *= b;
+  }while(e >>= 1);
+  return a;
+}
+struct mint {
+  int x;
+  mint(): x(0){}
+  mint(ll v) : x((v % mod + mod) % mod) {} // be careful of negative numbers!
+  // Helpers to shorten code
+  #define add(a, b) a + b >= mod ? a + b - mod : a + b
+  #define sub(a, b) a < b ? a + mod - b : a - b
+  #define yo *this
+  #define cmint const mint&
+  
+  mint &operator += (cmint o) { return x = add(x, o.x), yo; }
+  mint &operator -= (cmint o) { return x = sub(x, o.x), yo; }
+  mint &operator *= (cmint o) { return x = ll(x) * o.x % mod, yo; }
+  mint &operator /= (cmint o) { return yo *= bpow(o, mod-2); }
+
+  mint operator + (cmint b) const { return mint(yo) += b; }
+  mint operator - (cmint b) const { return mint(yo) -= b; }
+  mint operator * (cmint b) const { return mint(yo) *= b; }
+  mint operator / (cmint b) const { return mint(yo) /= b; }
+
+  mint operator - () const { return mint() - mint(yo); }
+  bool operator == (cmint b) const { return x == b.x; }
+  bool operator != (cmint b) const { return x != b.x; }
+
+  friend ostream& operator << (ostream &os, cmint p) { return os << p.x; }
+  friend istream& operator >> (istream &is, mint &p) { return is >> p.x; }
+};
+
+const int N = 1e6;
+mint f[N], fr[N];
+void initC(){
+  if(f[0] == 1) return; // Already precalculated
+  f[0] = 1;
+  for1(i, N-1) f[i] = f[i-1] * i;
+  fr[N-1] = bpow(f[N-1], mod-2);
+  fored(i, 1, N-1) fr[i-1] = fr[i] * i;
+}
+struct LagrangePol {
+  int n;
+  vector<mint> y, den, l, r;
+  LagrangePol(vector<mint> f): n(sz(f)), y(f), den(n), l(n), r(n){// f[i] := f(i) 
+    // Calcula interpol. pol P in O(n) := deg(P) = sz(v)-1 
+    initC();
+    forn(i, n) {
+      den[i] = fr[n-1-i] * fr[i];
+      if((n-1-i) & 1) den[i] = -den[i];
+    }
+  }
+  mint eval(mint x){ // Evaluate LagrangePoly P(x) in O(n)
+    l[0] = r[n-1] = 1;
+    for1(i, n-1) l[i] = l[i-1] * (x - i + 1);
+    fored(i, 0, n-2) r[i] = r[i+1] * (x - i - 1);
+    mint ans = 0;
+    forn(i, n) ans += l[i] * r[i] * y[i] * den[i];
+    return ans;
+  }
+};
+
+int main() {
+  ios_base::sync_with_stdio(0); cin.tie(0);
+  mint n;  int k;
+  cin >> n >> k;
+  vector<mint> num(2*k + 10);
+  auto den = num;
+  for1(i, sz(num)-1) {
+    num[i] = num[i-1] + bpow(mint(i), k) * bpow(n-i+1, k);
+    den[i] = den[i-1] + bpow(n-i+1, k);
+  }
+  LagrangePol p_num(num), p_den(den);
+  cout << p_num.eval(n) / p_den.eval(n) << el;
+}

code/math/lagrange_interpolation.cpp

@@ -0,0 +1,31 @@
+#include "mint.cpp"
+const int N = 1e6;
+mint f[N], fr[N];
+void initC(){
+  if(f[0] == 1) return; // Already precalculated
+  f[0] = 1;
+  for1(i, N-1) f[i] = f[i-1] * i;
+  fr[N-1] = bpow(f[N-1], mod-2);
+  fored(i, 1, N-1) fr[i-1] = fr[i] * i;
+}
+// mint C(int n, int k) { return k<0 || k>n ? 0 : f[n] * fr[k] * fr[n-k]; }
+struct LagrangePol {
+  int n;
+  vector<mint> y, den, l, r;
+  LagrangePol(vector<mint> f): n(sz(f)), y(f), den(n), l(n), r(n){// f[i] := f(i) 
+    // Calcula interpol. pol P in O(n) := deg(P) = sz(v)-1 
+    initC();
+    forn(i, n) {
+      den[i] = fr[n-1-i] * fr[i];
+      if((n-1-i) & 1) den[i] = -den[i];
+    }
+  }
+  mint eval(mint x){ // Evaluate LagrangePoly P(x) in O(n)
+    l[0] = r[n-1] = 1;
+    for1(i, n-1) l[i] = l[i-1] * (x - i + 1);
+    fored(i, 0, n-2) r[i] = r[i+1] * (x - i - 1);
+    mint ans = 0;
+    forn(i, n) ans += l[i] * r[i] * y[i] * den[i];
+    return ans;
+  }
+};

code/math/mint.cpp

@@ -0,0 +1,38 @@
+typedef long long ll;
+const int mod = 1e9 + 7;
+template <class T>
+T bpow(T b, int e) {
+  T a(1);
+  do{
+    if(e & 1) a *= b;
+    b *= b;
+  }while(e >>= 1);
+  return a;
+}
+struct mint {
+  int x;
+  mint(): x(0){}
+  mint(ll v) : x((v % mod + mod) % mod) {} // be careful of negative numbers!
+  // Helpers to shorten code
+  #define add(a, b) a + b >= mod ? a + b - mod : a + b
+  #define sub(a, b) a < b ? a + mod - b : a - b
+  #define yo *this
+  #define cmint const mint&
+  
+  mint &operator += (cmint o) { return x = add(x, o.x), yo; }
+  mint &operator -= (cmint o) { return x = sub(x, o.x), yo; }
+  mint &operator *= (cmint o) { return x = ll(x) * o.x % mod, yo; }
+  mint &operator /= (cmint o) { return yo *= bpow(o, mod-2); }
+
+  mint operator + (cmint b) const { return mint(yo) += b; }
+  mint operator - (cmint b) const { return mint(yo) -= b; }
+  mint operator * (cmint b) const { return mint(yo) *= b; }
+  mint operator / (cmint b) const { return mint(yo) /= b; }
+
+  mint operator - () const { return mint() - mint(yo); }
+  bool operator == (cmint b) const { return x == b.x; }
+  bool operator != (cmint b) const { return x != b.x; }
+
+  friend ostream& operator << (ostream &os, cmint p) { return os << p.x; }
+  friend istream& operator >> (istream &is, mint &p) { return is >> p.x; }
+};