#1110398 (C++23) No.3148 Min-Cost Destruction of Parentheses

提出ソース
結果

問題	No.3148 Min-Cost Destruction of Parentheses
コンテスト
ユーザー	hitonanode
提出日時	2025-08-05 00:16:44
言語	C++23 (gcc 13.3.0 + boost 1.87.0)
結果	AC
実行時間	92 ms / 4,000 ms
コード長	23,165 bytes
コンパイル時間	3,305 ms
コンパイル使用メモリ	258,476 KB
実行使用メモリ	20,604 KB
最終ジャッジ日時	2025-08-05 00:16:50
合計ジャッジ時間	6,349 ms
ジャッジサーバーID （参考情報）	judge1 / judge2
このコードへのチャレンジ
（要ログイン）
ファイルパターン	結果
sample	AC * 3
other	AC * 31
権限があれば一括ダウンロードができます
ソースコード

raw source code
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <numeric>
#include <optional>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template <typename T> bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; }
template <typename T> bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; }
const std::vector<std::pair<int, int>> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <class T1, class T2> T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); }
template <class T1, class T2> std::pair<T1, T2> operator+(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first + r.first, l.second + r.second); }
template <class T1, class T2> std::pair<T1, T2> operator-(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first - r.first, l.second - r.second); }
template <class T> std::vector<T> sort_unique(std::vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <class T> int arglb(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); }
template <class T> int argub(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); }
template <class IStream, class T> IStream &operator>>(IStream &is, std::vector<T> &vec) { for (auto &v : vec) is >> v; return is; }

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec);
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr);
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const pair<T, U> &pa);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa);
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp);
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp);
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl);

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
template <class... T> std::istream &operator>>(std::istream &is, std::tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; }
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; }
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m";
#define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl
#define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr)
#else
#define dbg(x) ((void)0)
#define dbgif(cond, x) ((void)0)
#endif


// Sorted set of integers [0, n)
// Space complexity: (64 / 63) n + O(log n) bit
class fast_set {
    static constexpr int B = 64;

    int n;
    int cnt;
    std::vector<std::vector<uint64_t>> _d;

    static int bsf(uint64_t x) { return __builtin_ctzll(x); }
    static int bsr(uint64_t x) { return 63 - __builtin_clzll(x); }

public:
    // 0 以上 n_ 未満の整数が入れられる sorted set を作成
    fast_set(int n_) : n(n_), cnt(0) {
        do { n_ = (n_ + B - 1) / B, _d.push_back(std::vector<uint64_t>(n_)); } while (n_ > 1);
    }

    bool contains(int i) const {
        assert(0 <= i and i < n);
        return (_d.front().at(i / B) >> (i % B)) & 1;
    }

    void insert(int i) {
        assert(0 <= i and i < n);
        if (contains(i)) return;
        ++cnt;
        for (auto &vec : _d) {
            bool f = vec.at(i / B);
            vec.at(i / B) |= 1ULL << (i % B), i /= B;
            if (f) break;
        }
    }

    void erase(int i) {
        assert(0 <= i and i < n);
        if (!contains(i)) return;
        --cnt;
        for (auto &vec : _d) {
            vec.at(i / B) &= ~(1ULL << (i % B)), i /= B;
            if (vec.at(i)) break;
        }
    }

    // i 以上の最小要素 なければ default_val
    int next(int i, const int default_val) const {
        assert(0 <= i and i <= n);

        for (auto itr = _d.cbegin(); itr != _d.cend(); ++itr, i = i / B + 1) {
            if (i / B >= int(itr->size())) break;

            if (auto d = itr->at(i / B) >> (i % B); d) {
                i += bsf(d);
                while (itr != _d.cbegin()) i = i * B + bsf((--itr)->at(i));
                return i;
            }
        }

        return default_val;
    }
    int next(const int i) const { return next(i, n); }

    // i 以下の最小要素 なければ default_val
    int prev(int i, int default_val = -1) const {
        assert(-1 <= i and i < n);

        for (auto itr = _d.cbegin(); itr != _d.cend() and i >= 0; ++itr, i = i / B - 1) {
            if (auto d = itr->at(i / B) << (B - 1 - i % B); d) {
                i += bsr(d) - (B - 1);
                while (itr != _d.cbegin()) i = i * B + bsr((--itr)->at(i));
                return i;
            }
        }

        return default_val;
    }

    // return minimum element (if exists) or `n` (empty)
    int min() const { return next(0); }
    // return maximum element (if exists) or `-1` (empty)
    int max() const { return prev(n - 1); }
    int size() const { return cnt; }
    bool empty() const { return cnt == 0; }

    void clear() {
        if (!cnt) return;
        cnt = 0;
        auto rec = [&](auto &&self, int d, int x) -> void {
            if (d) {
                for (auto m = _d.at(d).at(x); m;) {
                    int i = bsf(m);
                    m -= 1ULL << i, self(self, d - 1, x * B + i);
                }
            }
            _d.at(d).at(x) = 0;
        };
        rec(rec, _d.size() - 1, 0);
    }
};

#include <cassert>
#include <limits>

// Rational number + {infinity(1 / 0), -infiity(-1 / 0), nan(0 / 0)} （有理数）
// Verified: Yandex Cup 2022 Final E https://contest.yandex.com/contest/42710/problems/K
template <class Int, bool AutoReduce = false> struct Rational {
    Int num, den; // den >= 0

    static constexpr Int my_gcd(Int a, Int b) {
        // return __gcd(a, b);
        if (a < 0) a = -a;
        if (b < 0) b = -b;
        while (a and b) {
            if (a > b) {
                a %= b;
            } else {
                b %= a;
            }
        }
        return a + b;
    }

    constexpr Rational(Int num = 0, Int den = 1) : num(num), den(den) { normalize(); }
    constexpr void normalize() noexcept {
        if constexpr (AutoReduce) { // reduction
            Int g = my_gcd(num, den);
            if (g) num /= g, den /= g;
        } else {
            if (den == 0) {
                if (num > 1) num = 1;
                if (num < -1) num = -1;
            }
        }
        if (den < 0) num = -num, den = -den; // denominator >= 0
    }

    constexpr bool is_finite() const noexcept { return den != 0; }
    constexpr bool is_infinite_or_nan() const noexcept { return den == 0; }

    constexpr Rational operator+(const Rational &r) const noexcept {
        if (is_infinite_or_nan() and r.is_infinite_or_nan()) return Rational(num + r.num, 0);
        return Rational(num * r.den + den * r.num, den * r.den);
    }
    constexpr Rational operator-(const Rational &r) const noexcept {
        if (is_infinite_or_nan() and r.is_infinite_or_nan()) return Rational(num - r.num, 0);
        return Rational(num * r.den - den * r.num, den * r.den);
    }
    constexpr Rational operator*(const Rational &r) const noexcept {
        return Rational(num * r.num, den * r.den);
    }
    constexpr Rational operator/(const Rational &r) const noexcept {
        return Rational(num * r.den, den * r.num);
    }
    constexpr Rational &operator+=(const Rational &r) noexcept { return *this = *this + r; }
    constexpr Rational &operator-=(const Rational &r) noexcept { return *this = *this - r; }
    constexpr Rational &operator*=(const Rational &r) noexcept { return *this = *this * r; }
    constexpr Rational &operator/=(const Rational &r) noexcept { return *this = *this / r; }
    constexpr Rational operator-() const noexcept { return Rational(-num, den); }
    constexpr Rational abs() const noexcept { return Rational(num > 0 ? num : -num, den); }

    constexpr Int floor() const {
        assert(is_finite());
        if (num > 0) {
            return num / den;
        } else {
            return -((-num + den - 1) / den);
        }
    }

    constexpr bool operator==(const Rational &r) const noexcept {
        if (is_infinite_or_nan() or r.is_infinite_or_nan()) {
            return num == r.num and den == r.den;
        } else {
            return num * r.den == r.num * den;
        }
    }

    constexpr bool operator!=(const Rational &r) const noexcept { return !(*this == r); }

    constexpr bool operator<(const Rational &r) const noexcept {
        if (is_infinite_or_nan() and r.is_infinite_or_nan())
            return num < r.num;
        else if (is_infinite_or_nan()) {
            return num < 0;
        } else if (r.is_infinite_or_nan()) {
            return r.num > 0;
        } else {
            return num * r.den < den * r.num;
        }
    }
    constexpr bool operator<=(const Rational &r) const noexcept {
        return (*this == r) or (*this < r);
    }
    constexpr bool operator>(const Rational &r) const noexcept { return r < *this; }
    constexpr bool operator>=(const Rational &r) const noexcept {
        return (r == *this) or (r < *this);
    }

    constexpr explicit operator double() const noexcept { return (double)num / (double)den; }
    constexpr explicit operator long double() const noexcept {
        return (long double)num / (long double)den;
    }
    template <class OStream> constexpr friend OStream &operator<<(OStream &os, const Rational &x) {
        return os << x.num << '/' << x.den;
    }
};

template <class Int> struct std::numeric_limits<Rational<Int, false>> {
    static constexpr Rational<Int, false> max() noexcept {
        return std::numeric_limits<Int>::max();
    }
    static constexpr Rational<Int, false> min() noexcept {
        return std::numeric_limits<Int>::min();
    }
    static constexpr Rational<Int, false> lowest() noexcept {
        return std::numeric_limits<Int>::lowest();
    }
};

using rat = Rational<lint, false>;

#include <algorithm>
#include <numeric>
#include <utility>
#include <vector>

// UnionFind Tree (0-indexed), based on size of each disjoint set
struct UnionFind {
    std::vector<int> par, cou;
    UnionFind(int N = 0) : par(N), cou(N, 1) { iota(par.begin(), par.end(), 0); }
    int find(int x) { return (par[x] == x) ? x : (par[x] = find(par[x])); }
    bool unite(int x, int y) {
        x = find(x), y = find(y);
        if (x == y) return false;
        if (cou[x] < cou[y]) std::swap(x, y);
        par[y] = x, cou[x] += cou[y];
        return true;
    }
    int count(int x) { return cou[find(x)]; }
    bool same(int x, int y) { return find(x) == find(y); }
    std::vector<std::vector<int>> groups() {
        std::vector<std::vector<int>> ret(par.size());
        for (int i = 0; i < int(par.size()); ++i) ret[find(i)].push_back(i);
        ret.erase(std::remove_if(ret.begin(), ret.end(),
                                 [&](const std::vector<int> &v) { return v.empty(); }),
                  ret.end());
        return ret;
    }
};


// "01 on Tree"
// just idea: 木構造を持っておいて部分木毎に解く？
// とりあえず部分木毎に目的関数は出せる？
// モノイドを載せる？
using T = long long;
std::pair<std::vector<int>, T>
MinimizeTreePopInversion(const std::vector<std::vector<int>> &to, const std::vector<T> &xs,
                        const std::vector<T> &ys, int root) {
    // for (auto x : dxs) assert(x >= 0);

    struct Vec {
        T x, y;
        Vec(T x, T y) : x(x), y(y) {}
        Vec() : x(0), y(0) {}
        bool operator<(const Vec &r) const {
            if (x == 0 and y == 0) return false;
            if (r.x == 0 and r.y == 0) return true;
            if (x == 0 and r.x == 0) return y < r.y;
            if (x == 0) return false;
            if (r.x == 0) return true;
            return y * r.x < x * r.y; // be careful of overflow
        }
        bool operator>(const Vec &r) const { return r < *this; }

        void operator+=(const Vec &r) { x += r.x, y += r.y; }
    };

    const int N = to.size();

    std::vector<Vec> first_slope(N);

    std::vector<int> par(N, -1);

    using Pque = std::priority_queue<Vec, std::vector<Vec>, std::greater<Vec>>;
    auto rec = [&](auto &&self, int now, int prv) -> Pque {
        std::vector<Pque> chs;

        for (int nxt : to[now]) {
            if (nxt == prv) continue;
            assert(par[nxt] == -1);
            par[nxt] = now;
            chs.emplace_back(self(self, nxt, now));
        }

        const Vec v(xs.at(now), ys.at(now));
        if (chs.empty()) {
            first_slope[now] = v;
            Pque pq;
            pq.emplace(v);
            return pq;
        } else {
            Vec first = v;

            const int idx =
                std::max_element(chs.begin(), chs.end(),
                                 [](const auto &a, const auto &b) { return a.size() < b.size(); }) -
                chs.begin();
            std::swap(chs[idx], chs.front());

            for (int i = 1; i < (int)chs.size(); ++i) {
                // while (!chs[i].empty() and chs[i].top() < first) {
                //     first += chs[i].top();
                //     chs[i].pop();
                // }
                while (!chs[i].empty()) {
                    chs.front().emplace(chs[i].top());
                    chs[i].pop();
                }
            }

            while (!chs.front().empty() and chs.front().top() < first) {
                first += chs.front().top();
                chs.front().pop();
            }

            first_slope[now] = first;

            chs.front().emplace(first);

            return std::move(chs.front());
        }
    };

    rec(rec, root, -1);

    std::priority_queue<std::pair<Vec, int>, std::vector<std::pair<Vec, int>>,
                        std::greater<std::pair<Vec, int>>>
        pq;

    pq.emplace(first_slope.at(root), root);

    std::vector<int> order;
    long long ret = 0, y = 0;
    while (!pq.empty()) {
        const int idx = pq.top().second;
        order.emplace_back(idx);
        pq.pop();
        ret += y * xs.at(idx);
        y += ys.at(idx);

        for (int nxt : to.at(idx)) {
            if (nxt == par.at(idx)) continue;
            pq.emplace(first_slope.at(nxt), nxt);
        }
    }

    return {order, ret};
}

lint solve() {
    int N;
    string str;
    cin >> N >> str;
    vector<vector<int>> child(N + 1);
    {
        vector<int> stk{0};
        int openid = 1;
        for (auto c : str) {
            if (c == '(') {
                stk.push_back(openid++);
            } else {
                int v = stk.back();
                stk.pop_back();
                if (stk.size()) child.at(stk.back()).push_back(v);
            }
        }
    }

    vector<long long> A(N);
    cin >> A;
    A.insert(A.begin(), 0);
    dbg(A);
    vector<long long> dx(A.size(), 1);
    dx.at(0) = 0;

    auto seq = MinimizeTreePopInversion(child, A, dx, 0).first;
    dbg(seq);
    reverse(ALL(seq));
    {
        lint dy = 0, ans = 0;
        for (int i : seq) {
            if (i == 0) break;
            dy += A.at(i);
            ans += dy;
        }

        return ans;
    }
    // dbg(res + accumulate(ALL(A), 0LL));
    
    // return res + accumulate(ALL(A), 0LL);

    using T = tuple<rat, int, lint>; // (傾き, 要素数, 重みつき和)
    using S = multiset<T>;
    auto rec = [&](auto &&self, int now) -> S {
        if (child.at(now).empty()) {
            // vector<int> ret = {A.at(now)};
            S ans;
            ans.emplace(rat(A.at(now), 1), 1, 0);
            return ans;
        }

        vector<S> children;

        for (int nxt : child.at(now)) {
            auto r = self(self, nxt);
            children.emplace_back(std::move(r));
        }
        int max_idx = 0;
        FOR(i, 1, children.size()) {
            if (children.at(i).size() > children.at(max_idx).size()) max_idx = i;
        }
        std::swap(children.at(max_idx), children.front());
        FOR(i, 1, children.size()) {
            for (auto x : children.at(i)) { children.front().emplace(x); }
        }
        children.resize(1);

        int addn = 1;
        int addsum = A.at(now);
        lint addws = 0;

        while (children.front().size()) {
            auto [slope, n, ws] = *prev(children.front().end());
            if (slope >= rat(addsum, addn)) {
                children.front().erase(prev(children.front().end()));

                auto sumr = slope * n;
                const lint sum = sumr.num / sumr.den;

                addws += addn * sum + ws;
                addn += n;
                addsum += sum;
            } else {
                break;
            }
        }
        children.front().emplace(rat(addsum, addn), addn, addws);

        return std::move(children.front());
    };

    auto ret = rec(rec, 0);
    vector<T> retv;
    for (auto x : ret) { retv.emplace_back(x); }
    reverse(ALL(retv));
    dbg(retv);

    lint ans = 0;

    dbg(retv);
    lint cs = 0;
    int ndone = 0;
    REP(t, retv.size()) {
        auto [r, n, ws] = retv.at(t);
        const rat sumr = r * n;
        const lint sum = sumr.num / sumr.den;

        ans += ws + ndone * sum;
        ndone += n;
        cs += sum;
    }
    dbg(ans);

    return ans;
}

int main() {
    cout << solve() << '\n';
    // int N;
    // string S;
    // cin >> N >> S;
    // vector<int> A(N);
    // cin >> A;
    // dbg(make_tuple(S, A));

    // fast_set fs(N * 2);
    // REP(i, N * 2) fs.insert(i);

    // vector<int> indices(N * 2, -1);
    // vector<int> id2left(N, -1), id2right(N, -1);

    // {
    //     int openid = 0;
    //     vector<int> stk;
    //     REP(i, N * 2) {
    //         if (S.at(i) == '(') {
    //             indices.at(i) = openid;
    //             id2left.at(openid) = i;
    //             stk.push_back(openid);
    //             openid++;
    //         } else {
    //             indices.at(i) = stk.back();
    //             id2right.at(stk.back()) = i;
    //             stk.pop_back();
    //         }
    //     }
    // }
    // dbg(indices);

    // priority_queue<pint> pq;
    // REP(i, N * 2 - 1) {
    //     if (S.at(i) == '(' and S.at(i + 1) == ')') {
    //         const int idx = indices.at(i);
    //         pq.emplace(A.at(idx), idx);
    //     }
    // }

    // lint ret = 0;
    // lint X = 0;
    // while (pq.size()) {
    //     auto [a, idx] = pq.top();
    //     pq.pop();
    //     const int left = id2left.at(idx);
    //     const int right = id2right.at(idx);
    //     if (!fs.contains(left)) continue;
    //     X += a;
    //     ret += X;

    //     fs.erase(left);
    //     fs.erase(right);
    //     const int ll = fs.prev(left), rr = fs.next(right);
    //     if (ll >= 0 and rr < N * 2 and indices.at(ll) == indices.at(rr)) {
    //         const int idx2 = indices.at(ll);
    //         pq.emplace(A.at(idx2), idx2);
    //     }
    // }

    // cout << ret << '\n';
}
yukicoder

結果

ソースコード
