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:heavy_check_mark: test/mytest/minplus_conv_triple.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include "my_template.hpp"

#include "random/base.hpp"
#include "convex/minplus_convolution_of_triples.hpp"

void test() {
  FOR(N, 0, 300) {
    vi A(N), B(N), C(N);
    FOR(i, N) A[i] = RNG(1, 1 << 30);
    FOR(i, N) B[i] = RNG(1, 1 << 30);
    FOR(i, N) C[i] = RNG(1, 1 << 30);
    // 愚直
    vi dp(2 * N + 1, infty<ll>);
    dp[0] = 0;
    FOR(i, N) {
      ll a = A[i], b = B[i], c = C[i];
      vi newdp(2 * N + 1, infty<ll>);
      FOR(i, len(dp)) {
        if (i + 0 <= 2 * N) chmin(newdp[i + 0], dp[i] + a);
        if (i + 1 <= 2 * N) chmin(newdp[i + 1], dp[i] + b);
        if (i + 2 <= 2 * N) chmin(newdp[i + 2], dp[i] + c);
      }
      swap(dp, newdp);
    }

    MinPlus_Convolution_of_Triples<ll> X;
    FOR(i, N) X.add(A[i], B[i], C[i]);
    X.solve();
    FOR(K, 2 * N + 1) {
      ll val = X[K];
      vc<int> x = X.restore(K);
      assert(SUM<int>(x) == K);
      ll sm = 0;
      FOR(i, N) {
        assert(0 <= x[i] && x[i] <= 2);
        if (x[i] == 0) sm += A[i];
        if (x[i] == 1) sm += B[i];
        if (x[i] == 2) sm += C[i];
      }
      assert(sm == val && val == dp[K]);
    }
  }
}

void solve() {
  int a, b;
  cin >> a >> b;
  cout << a + b << "\n";
}

signed main() {
  test();
  solve();
  return 0;
}
#line 1 "test/mytest/minplus_conv_triple.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else

// https://codeforces.com/blog/entry/96344
#pragma GCC optimize("Ofast,unroll-loops")
#pragma GCC target("avx2,popcnt")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) \
  for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
  sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 3 "test/mytest/minplus_conv_triple.test.cpp"

#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                     chrono::high_resolution_clock::now().time_since_epoch())
                     .count())
        * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 1 "convex/monotone_minima.hpp"

// select(i,j,k) : (i,j) -> (i,k) を行うかどうか
template <typename F>
vc<int> monotone_minima(int H, int W, F select) {
  vc<int> min_col(H);
  auto dfs = [&](auto& dfs, int x1, int x2, int y1, int y2) -> void {
    if (x1 == x2) return;
    int x = (x1 + x2) / 2;
    int best_y = y1;
    for (int y = y1 + 1; y < y2; ++y) {
      if (select(x, best_y, y)) best_y = y;
    }
    min_col[x] = best_y;
    dfs(dfs, x1, x, y1, best_y + 1);
    dfs(dfs, x + 1, x2, best_y, y2);
  };
  dfs(dfs, 0, H, 0, W);
  return min_col;
}
#line 2 "convex/minplus_convolution.hpp"

template <typename T>
vc<T> minplus_convolution_convex_convex(vc<T>& A, vc<T>& B) {
  int n = len(A), m = len(B);
  if (n == 0 && m == 0) return {};
  vc<T> C(n + m - 1, infty<T>);
  while (n > 0 && A[n - 1] == infty<T>) --n;
  while (m > 0 && B[m - 1] == infty<T>) --m;
  if (n == 0 && m == 0) return C;
  int a = 0, b = 0;
  while (a < n && A[a] == infty<T>) ++a;
  while (b < m && B[b] == infty<T>) ++b;
  C[a + b] = A[a] + B[b];
  for (int i = a + b + 1; i < n + m - 1; ++i) {
    if (b == m - 1 || (a != n - 1 && A[a + 1] + B[b] < A[a] + B[b + 1])) {
      chmin(C[i], A[++a] + B[b]);
    } else {
      chmin(C[i], A[a] + B[++b]);
    }
  }
  return C;
}

template <typename T>
vc<T> minplus_convolution_arbitrary_convex(vc<T>& A, vc<T>& B) {
  int n = len(A), m = len(B);
  if (n == 0 && m == 0) return {};
  vc<T> C(n + m - 1, infty<T>);
  while (m > 0 && B[m - 1] == infty<T>) --m;
  if (m == 0) return C;
  int b = 0;
  while (b < m && B[b] == infty<T>) ++b;

  auto select = [&](int i, int j, int k) -> bool {
    if (i < k) return false;
    if (i - j >= m - b) return true;
    return A[j] + B[b + i - j] >= A[k] + B[b + i - k];
  };
  vc<int> J = monotone_minima(n + m - b - 1, n, select);
  FOR(i, n + m - b - 1) {
    T x = A[J[i]], y = B[b + i - J[i]];
    if (x < infty<T> && y < infty<T>) C[b + i] = x + y;
  }
  return C;
}

template <typename T, bool convA, bool convB>
vc<T> minplus_convolution(vc<T>& A, vc<T>& B) {
  static_assert(convA || convB);
  if constexpr (convA && convB) return minplus_convolution_convex_convex(A, B);
  if constexpr (convA && !convB)
    return minplus_convolution_arbitrary_convex(B, A);
  if constexpr (convB && !convA)
    return minplus_convolution_arbitrary_convex(A, B);
  return {};
}
#line 2 "convex/minplus_convolution_of_triples.hpp"

// https://codeforces.com/contest/436/problem/E
// 長さ 3 の数列 {a[i][0], a[i][1], a[i][2]} たちの畳み込み, O(NlogN)
// 同種の問題:(a_i,b_i) があって、b_i は a_i を取ってからだけ取れる
template <typename T>
struct MinPlus_Convolution_of_Triples {
  int N = 0;
  T sm0 = 0;
  vc<array<T, 3>> dat;
  vc<T> dp1, dp2, dp;
  vc<int> I1, I2;
  bool solved = false;

  void add(T x0, T x1, T x2) { sm0 += x0, dat.eb(array<T, 3>{x0, x1, x2}); }

  void solve() {
    solved = true;
    N = dat.size();
    FOR(i, N) {
      int a = dat[i][1] - dat[i][0], b = dat[i][2] - dat[i][1];
      (a <= b ? I1 : I2).eb(i);
    };
    sort(all(I2), [&](int i, int j) -> bool {
      return dat[i][2] - dat[i][0] < dat[j][2] - dat[j][0];
    });
    solve_1();
    solve_2();
    dp = minplus_convolution<T, true, false>(dp1, dp2);
    for (auto&& x: dp) x += sm0;
  }

  T operator[](int i) { return dp[i]; }

  vc<int> restore(int k) {
    assert(solved);
    int k1 = -1, k2 = -1;
    FOR(i, k + 1) {
      int j = k - i;
      if (0 <= i && i < len(dp1) && 0 <= j && j < len(dp2)
          && dp1[i] + dp2[j] + sm0 == dp[k]) {
        k1 = i, k2 = j;
        break;
      }
    }
    assert(k1 != -1 && k2 != -1);
    vc<int> x(N);
    vc<int> x1 = restore_1(k1);
    vc<int> x2 = restore_2(k2);
    for (int i = 0; i < N; ++i) x[i] = x1[i] + x2[i];
    return x;
  }

private:
  void solve_1() {
    dp1.reserve(len(I1));
    for (int i: I1) {
      dp1.eb(dat[i][1] - dat[i][0]), dp1.eb(dat[i][2] - dat[i][1]);
    }
    sort(all(dp1));
    dp1 = cumsum<T>(dp1);
  }

  vc<int> restore_1(int k) {
    vc<pair<T, int>> A;
    for (int i: I1) {
      A.eb(dat[i][1] - dat[i][0], i);
      A.eb(dat[i][2] - dat[i][1], i);
    }
    nth_element(A.begin(), A.begin() + k, A.end());
    vc<int> x(N);
    FOR(i, k) x[A[i].se]++;
    return x;
  }

  void solve_2() {
    // B-A > C-B のケース
    // 解の構造を考えると、ほとんどすべてで x=0 or x=2 というとりかたになる
    // 既に C-A でソート済
    auto& I = I2;
    int n = len(I);
    dp2.assign(2 * n + 1, infty<T>);
    // 偶数個
    dp2[0] = 0;
    for (int i = 0; i < n; ++i) {
      dp2[2 * i + 2] = dp2[2 * i] + (dat[I[i]][2] - dat[I[i]][0]);
    }
    // 奇数個, prefix からひとつキャンセルする
    T ma = -infty<T>;
    for (int i = 0; i < n; ++i) {
      chmax(ma, dat[I[i]][2] - dat[I[i]][1]);
      chmin(dp2[2 * i + 1], dp2[2 * i + 2] - ma);
    }
    // 奇数個, suffix からひとつ追加する
    T mi = infty<T>;
    for (int i = n - 1; i >= 0; --i) {
      chmin(mi, dat[I[i]][1] - dat[I[i]][0]);
      chmin(dp2[2 * i + 1], dp2[2 * i] + mi);
    }
    return;
  }

  vc<int> restore_2(int k) {
    auto& I = I2;
    int n = len(I);
    vc<int> x(N);
    if (k % 2 == 0) {
      FOR(i, k / 2) x[I[i]] = 2;
      return x;
    }
    pair<T, int> ma = {-infty<T>, -1};
    FOR(i, (k + 1) / 2) {
      if (chmax(ma.fi, dat[I[i]][2] - dat[I[i]][1])) ma.se = I[i];
    }
    if (dp2[k] == dp2[k + 1] - ma.fi) {
      FOR(i, (k + 1) / 2) x[I[i]] = 2;
      x[ma.se]--;
      return x;
    }
    pair<T, int> mi = {infty<T>, -1};
    for (int i = n - 1; i >= k / 2; --i) {
      if (chmin(mi.fi, dat[I[i]][1] - dat[I[i]][0])) mi.se = I[i];
    }
    if (dp2[k] == dp2[k - 1] + mi.fi) {
      FOR(i, (k - 1) / 2) x[I[i]] = 2;
      x[mi.se] = 1;
      return x;
    }
    assert(0);
    return x;
  }
};
#line 6 "test/mytest/minplus_conv_triple.test.cpp"

void test() {
  FOR(N, 0, 300) {
    vi A(N), B(N), C(N);
    FOR(i, N) A[i] = RNG(1, 1 << 30);
    FOR(i, N) B[i] = RNG(1, 1 << 30);
    FOR(i, N) C[i] = RNG(1, 1 << 30);
    // 愚直
    vi dp(2 * N + 1, infty<ll>);
    dp[0] = 0;
    FOR(i, N) {
      ll a = A[i], b = B[i], c = C[i];
      vi newdp(2 * N + 1, infty<ll>);
      FOR(i, len(dp)) {
        if (i + 0 <= 2 * N) chmin(newdp[i + 0], dp[i] + a);
        if (i + 1 <= 2 * N) chmin(newdp[i + 1], dp[i] + b);
        if (i + 2 <= 2 * N) chmin(newdp[i + 2], dp[i] + c);
      }
      swap(dp, newdp);
    }

    MinPlus_Convolution_of_Triples<ll> X;
    FOR(i, N) X.add(A[i], B[i], C[i]);
    X.solve();
    FOR(K, 2 * N + 1) {
      ll val = X[K];
      vc<int> x = X.restore(K);
      assert(SUM<int>(x) == K);
      ll sm = 0;
      FOR(i, N) {
        assert(0 <= x[i] && x[i] <= 2);
        if (x[i] == 0) sm += A[i];
        if (x[i] == 1) sm += B[i];
        if (x[i] == 2) sm += C[i];
      }
      assert(sm == val && val == dp[K]);
    }
  }
}

void solve() {
  int a, b;
  cin >> a >> b;
  cout << a + b << "\n";
}

signed main() {
  test();
  solve();
  return 0;
}
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