combinatorial_opt/test/convex_sum.test.cpp

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• Last update: 2022-01-08 20:23:44+09:00
• Problem: https://yukicoder.me/problems/no/1495

Depends on

• 引数総和に関する等式制約下の互いに独立な一引数離散凸関数和の最小化 (combinatorial_opt/convex_sum.hpp)

Code

#include "../convex_sum.hpp"
#include <cassert>
#include <iostream>
#define PROBLEM "https://yukicoder.me/problems/no/1495"
using namespace std;

long long solve() {
    using lint = long long;
    int N, M, K;
    cin >> N >> M >> K;
    vector<lint> a(N), b(N), c(N);
    while (M--) {
        int x;
        lint y;
        cin >> x >> y;
        x--;
        a[x]++;
        b[x] -= y * 2;
        c[x] += y * y;
    }
    vector<Quadratic> costs;
    for (int i = 0; i < N; i++) costs.emplace_back(a[i], b[i], c[i], 0, K);
    auto sol =
        MinConvexSumUnderLinearConstraint<Quadratic, lint, 1LL << 60>(vector<lint>(N, 1), costs, K);
    return sol.first;
}

int main() {
    cin.tie(nullptr), ios::sync_with_stdio(false);
    int T;
    cin >> T;
    while (T--) cout << solve() << '\n';
}

#line 2 "combinatorial_opt/convex_sum.hpp"
#include <cassert>
#include <utility>
#include <vector>

// ax^2 + bx + c (convex), lb <= x <= ub
struct Quadratic {
    using Int = long long;
    Int a, b, c, lb, ub;
    Quadratic(Int a, Int b, Int c, Int lb, Int ub) : a(a), b(b), c(c), lb(lb), ub(ub) {}
    Int slope(Int s) const noexcept {
        if (a == 0) return b <= s ? ub : lb;
        auto ret = (s + a - b) / (a * 2);
        return ret > ub ? ub : ret < lb ? lb : ret;
    }
    Int eval(Int x) const noexcept { return (a * x + b) * x + c; }
    // f(x) - f(x - 1)
    Int next_cost(Int x) const noexcept { return 2 * a * x - a + b; }
};

// x^3 - ax, x \geq 0 (convex)
struct Cubic {
    int a, lb, ub;
    Cubic(int a, int ub) : a(a), lb(0), ub(ub) {}
    int slope(long long s) const noexcept {
        int lo = lb, hi = ub + 1;
        while (hi - lo > 1) {
            int x = (lo + hi) / 2;
            (next_cost(x) <= s ? lo : hi) = x;
        }
        return lo;
    }
    long long eval(long long x) const noexcept { return (x * x - a) * x; }
    // f(x) - f(x - 1)
    long long next_cost(long long x) const noexcept { return 3 * x * x - 3 * x + 1 - a; }
};

// \minimize $\sum_i \sum_j^{k_i} f_i(x_{ij})$
// https://codeforces.com/contest/1344/problem/D
// https://yukicoder.me/problems/no/1495
// return: (y, [[(x_i, # of such x_i), ... ], ...])
template <typename F, typename Int, Int INF>
std::pair<Int, std::vector<std::vector<std::pair<Int, Int>>>>
MinConvexSumUnderLinearConstraint(const std::vector<Int> &k, const std::vector<F> &f, Int C) {
    assert(k.size() == f.size());
    assert(k.size() > 0);
    Int lbsum = 0, ubsum = 0;
    for (auto func : f) lbsum += func.lb, ubsum += func.ub;
    if (lbsum > C or ubsum < C) return {};

    const int N = k.size();
    Int few = -INF, enough = INF;
    while (enough - few > 1) {
        auto slope = few + (enough - few) / 2;
        Int cnt = 0;
        for (int i = 0; i < N; i++) {
            auto tmp = f[i].slope(slope);
            cnt += tmp * k[i];
            if (cnt >= C) break;
        }
        (cnt >= C ? enough : few) = slope;
    }

    std::vector<std::vector<std::pair<Int, Int>>> ret(N);
    std::vector<int> additional;
    Int ctmp = 0;
    Int sol = 0;
    for (int i = 0; i < N; i++) {
        auto xlo = f[i].slope(few);
        auto xhi = f[i].slope(few + 1);
        ctmp += k[i] * xlo;
        ret[i].emplace_back(xlo, k[i]);
        if (xlo < xhi) additional.push_back(i);
        sol += k[i] * f[i].eval(xlo);
    }
    sol += (C - ctmp) * (few + 1);
    while (additional.size()) {
        int i = additional.back();
        additional.pop_back();
        Int add = C - ctmp > k[i] ? k[i] : C - ctmp;
        auto x = ret[i][0].first;
        if (add) {
            ret[i][0].second -= add;
            if (ret[i][0].second == 0) ret[i].pop_back();
            ret[i].emplace_back(x + 1, add);
            ctmp += add;
        }
    }
    return {sol, ret};
}
#line 3 "combinatorial_opt/test/convex_sum.test.cpp"
#include <iostream>
#define PROBLEM "https://yukicoder.me/problems/no/1495"
using namespace std;

long long solve() {
    using lint = long long;
    int N, M, K;
    cin >> N >> M >> K;
    vector<lint> a(N), b(N), c(N);
    while (M--) {
        int x;
        lint y;
        cin >> x >> y;
        x--;
        a[x]++;
        b[x] -= y * 2;
        c[x] += y * y;
    }
    vector<Quadratic> costs;
    for (int i = 0; i < N; i++) costs.emplace_back(a[i], b[i], c[i], 0, K);
    auto sol =
        MinConvexSumUnderLinearConstraint<Quadratic, lint, 1LL << 60>(vector<lint>(N, 1), costs, K);
    return sol.first;
}

int main() {
    cin.tie(nullptr), ios::sync_with_stdio(false);
    int T;
    cin >> T;
    while (T--) cout << solve() << '\n';
}

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