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:x: geo/max_norm_sum.hpp

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Code

#include "geo/base.hpp"
#include "geo/angle_sort.hpp"

// ベクトルの列が与えられる. 部分列を選んで,和の norm を最小化する.
// 総和の座標の 2 乗和が SM でオーバーフローしないように注意せよ.
// https://atcoder.jp/contests/abc139/tasks/abc139_f
// https://codeforces.com/contest/1841/problem/F
template <typename SM, typename T>
pair<SM, vc<int>> max_norm_sum(vc<Point<T>> dat) {
  auto I = angle_sort(dat);
  {
    vc<int> J;
    for (auto&& i: I) {
      if (dat[i].x != 0 || dat[i].y != 0) J.eb(i);
    }
    swap(I, J);
  }
  dat = rearrange(dat, I);
  const int N = len(dat);

  if (N == 0) { return {0, {}}; }
  SM ANS = 0;
  pair<int, int> LR = {0, 0};

  int L = 0, R = 1;
  Point<T> c = dat[0];
  auto eval = [&]() -> SM { return SM(c.x) * c.x + SM(c.y) * c.y; };
  if (chmax(ANS, eval())) LR = {L, R};

  while (L < N) {
    Point<T>&A = dat[L], &B = dat[R % N];
    if (R - L < N && (A.det(B) > 0 || (A.det(B) == 0 && A.dot(B) > 0))) {
      c = c + B;
      R++;
      if (chmax(ANS, eval())) LR = {L, R};
    } else {
      c = c - A;
      L++;
      if (chmax(ANS, eval())) LR = {L, R};
    }
  }
  vc<int> ids;
  FOR(i, LR.fi, LR.se) { ids.eb(I[i % N]); }
  return {ANS, ids};
}
#line 2 "geo/base.hpp"
template <typename T>
struct Point {
  T x, y;

  Point() : x(0), y(0) {}

  template <typename A, typename B>
  Point(A x, B y) : x(x), y(y) {}

  template <typename A, typename B>
  Point(pair<A, B> p) : x(p.fi), y(p.se) {}

  Point operator+(Point p) const { return {x + p.x, y + p.y}; }
  Point operator-(Point p) const { return {x - p.x, y - p.y}; }
  bool operator==(Point p) const { return x == p.x && y == p.y; }
  bool operator!=(Point p) const { return x != p.x || y != p.y; }
  Point operator-() const { return {-x, -y}; }
  Point operator*(T t) const { return {x * t, y * t}; }
  Point operator/(T t) const { return {x / t, y / t}; }

  bool operator<(Point p) const {
    if (x != p.x) return x < p.x;
    return y < p.y;
  }
  T dot(Point other) { return x * other.x + y * other.y; }
  T det(Point other) { return x * other.y - y * other.x; }

  double norm() { return sqrtl(x * x + y * y); }
  double angle() { return atan2(y, x); }

  Point rotate(double theta) {
    static_assert(!is_integral<T>::value);
    double c = cos(theta), s = sin(theta);
    return Point{c * x - s * y, s * x + c * y};
  }
};

#ifdef FASTIO
template <typename T>
void rd(Point<T>& p) {
  fastio::rd(p.x), fastio::rd(p.y);
}
template <typename T>
void wt(Point<T>& p) {
  fastio::wt(p.x);
  fastio::wt(' ');
  fastio::wt(p.y);
}
#endif

// A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1
template <typename T>
int ccw(Point<T> A, Point<T> B, Point<T> C) {
  T x = (B - A).det(C - A);
  if (x > 0) return 1;
  if (x < 0) return -1;
  return 0;
}

template <typename REAL, typename T>
REAL dist(Point<T> A, Point<T> B) {
  A = A - B;
  T p = A.dot(A);
  return sqrt(REAL(p));
}

// ax+by+c
template <typename T>
struct Line {
  T a, b, c;

  Line(T a, T b, T c) : a(a), b(b), c(c) {}
  Line(Point<T> A, Point<T> B) {
    a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x;
  }
  Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  template <typename U>
  U eval(Point<U> P) {
    return a * P.x + b * P.y + c;
  }

  template <typename U>
  T eval(U x, U y) {
    return a * x + b * y + c;
  }

  // 同じ直線が同じ a,b,c で表現されるようにする
  void normalize() {
    static_assert(is_same_v<T, int> || is_same_v<T, long long>);
    T g = gcd(gcd(abs(a), abs(b)), abs(c));
    a /= g, b /= g, c /= g;
    if (b < 0) { a = -a, b = -b, c = -c; }
    if (b == 0 && a < 0) { a = -a, b = -b, c = -c; }
  }

  bool is_parallel(Line other) { return a * other.b - b * other.a == 0; }
  bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; }
};

template <typename T>
struct Segment {
  Point<T> A, B;

  Segment(Point<T> A, Point<T> B) : A(A), B(B) {}
  Segment(T x1, T y1, T x2, T y2)
      : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  bool contain(Point<T> C) {
    static_assert(is_integral<T>::value);
    T det = (C - A).det(B - A);
    if (det != 0) return 0;
    return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0;
  }

  Line<T> to_Line() { return Line(A, B); }
};

template <typename REAL>
struct Circle {
  Point<REAL> O;
  REAL r;
  Circle(Point<REAL> O, REAL r) : O(O), r(r) {}
  Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {}
  template <typename T>
  bool contain(Point<T> p) {
    REAL dx = p.x - O.x, dy = p.y - O.y;
    return dx * dx + dy * dy <= r * r;
  }
};

template <typename T>
struct Polygon {
  vc<Point<T>> points;
  T a;

  template <typename A, typename B>
  Polygon(vc<pair<A, B>> pairs) {
    for (auto&& [a, b]: pairs) points.eb(Point<T>(a, b));
    build();
  }
  Polygon(vc<Point<T>> points) : points(points) { build(); }

  int size() { return len(points); }

  template <typename REAL>
  REAL area() {
    return a * 0.5;
  }

  template <enable_if_t<is_integral<T>::value, int> = 0>
  T area_2() {
    return a;
  }

  bool is_convex() {
    FOR(j, len(points)) {
      int i = (j == 0 ? len(points) - 1 : j - 1);
      int k = (j == len(points) - 1 ? 0 : j + 1);
      if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false;
    }
    return true;
  }

private:
  void build() {
    a = 0;
    FOR(i, len(points)) {
      int j = (i + 1 == len(points) ? 0 : i + 1);
      a += points[i].det(points[j]);
    }
    if (a < 0) {
      a = -a;
      reverse(all(points));
    }
  }
};
#line 2 "geo/angle_sort.hpp"

#line 4 "geo/angle_sort.hpp"

// 偏角ソートに対する argsort

template <typename T>
vector<int> angle_sort(vector<Point<T>>& P) {
  vector<int> lower, origin, upper;
  const Point<T> O = {0, 0};
  FOR(i, len(P)) {
    if (P[i] == O) origin.eb(i);
    elif ((P[i].y < 0) || (P[i].y == 0 && P[i].x > 0)) lower.eb(i);
    else upper.eb(i);
  }
  sort(all(lower), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  sort(all(upper), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  auto& I = lower;
  I.insert(I.end(), all(origin));
  I.insert(I.end(), all(upper));
  return I;
}

// 偏角ソートに対する argsort

template <typename T>
vector<int> angle_sort(vector<pair<T, T>>& P) {
  vc<Point<T>> tmp(len(P));
  FOR(i, len(P)) tmp[i] = Point<T>(P[i]);
  return angle_sort<T>(tmp);
}
#line 3 "geo/max_norm_sum.hpp"

// ベクトルの列が与えられる. 部分列を選んで,和の norm を最小化する.
// 総和の座標の 2 乗和が SM でオーバーフローしないように注意せよ.
// https://atcoder.jp/contests/abc139/tasks/abc139_f
// https://codeforces.com/contest/1841/problem/F
template <typename SM, typename T>
pair<SM, vc<int>> max_norm_sum(vc<Point<T>> dat) {
  auto I = angle_sort(dat);
  {
    vc<int> J;
    for (auto&& i: I) {
      if (dat[i].x != 0 || dat[i].y != 0) J.eb(i);
    }
    swap(I, J);
  }
  dat = rearrange(dat, I);
  const int N = len(dat);

  if (N == 0) { return {0, {}}; }
  SM ANS = 0;
  pair<int, int> LR = {0, 0};

  int L = 0, R = 1;
  Point<T> c = dat[0];
  auto eval = [&]() -> SM { return SM(c.x) * c.x + SM(c.y) * c.y; };
  if (chmax(ANS, eval())) LR = {L, R};

  while (L < N) {
    Point<T>&A = dat[L], &B = dat[R % N];
    if (R - L < N && (A.det(B) > 0 || (A.det(B) == 0 && A.dot(B) > 0))) {
      c = c + B;
      R++;
      if (chmax(ANS, eval())) LR = {L, R};
    } else {
      c = c - A;
      L++;
      if (chmax(ANS, eval())) LR = {L, R};
    }
  }
  vc<int> ids;
  FOR(i, LR.fi, LR.se) { ids.eb(I[i % N]); }
  return {ANS, ids};
}
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