library_for_cpp

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:heavy_check_mark: 最大流
(Max_Flow/Max_Flow.hpp)

Outline

最大流問題を解く.

Contents

Constructor

template<typename Cap>
Max_Flow F(int N = 0)

order

int F.order()

size

int F.size()

add_arc

int F.add_arc(const int u, const int v, const Cap cap)

add_vertex

int F.add_vertex()

get_flow

vector<vector<tuple<int, int, Cap>>> F.get_flow()

max_flow

Cap F.max_flow(const V source, const V target)
Cap F.max_flow(const V source, const V target, Cap flow_limit)

min_cut

vector<bool> F.min_cut(const int s)

Depends on

Required by

Verified with

Code

#pragma once

#include"../template/template.hpp"

namespace max_flow {
    template<typename Cap>
    struct Arc {
        int id, source, target;
        Cap cap, flow;
        bool direction;
        Arc* rev;

        Arc(int id, int source, int target, Cap cap, Cap flow, bool direction):
            id(id), source(source), target(target), cap(cap), flow(flow), direction(direction), rev(nullptr) {}

        inline bool is_flowable() const { return flow < cap; }
        inline Cap remain() const { return cap - flow; }

        void push(Cap d) {
            flow += d;
            rev->flow -= d;
        }
    };

    template<typename Cap>
    struct Max_Flow {
        private:
        using V = int;
        vector<vector<Arc<Cap>*>> adj_out;
        vector<Arc<Cap>*> arcs;
        vector<int> level, iter;

        bool bfs(const V s, const V t) {
            level.assign(order(), -1);
            queue<V> Q;

            Q.push(s);
            level[s] = 0;

            while (!Q.empty()) {
                V v = Q.front(); Q.pop();

                for (Arc<Cap>* arc: adj_out[v]) {
                    unless(arc->is_flowable() && level[arc->target] == -1) { continue; }

                    level[arc->target] = level[v] + 1;
                    if (arc->target == t) { return true; }

                    Q.push(arc->target);
                }
            }

            return false;
        }

        Cap dfs(const V v, const V t, const Cap f) {
            if (v == t) { return f; }

            for (int i = iter[v]; i < adj_out[v].size(); ++i) {
                Arc<Cap>* arc = adj_out[v][i];

                unless(arc->is_flowable() && level[v] < level[arc->target]) { continue; }

                Cap d = (f == -1) ? dfs(arc->target, t, arc->remain()) : dfs(arc->target, t, min(f, arc->remain()));

                unless(d > 0) { continue; }

                arc->push(d);
                return d;
            }

            return 0;
        }

        public:
        Max_Flow(int N = 0) {
            adj_out.resize(N);
            level.resize(N);
            iter.resize(N);
        }

        ~Max_Flow() {
            for (Arc<Cap>* arc : arcs) {
                if (arc->rev) { delete arc->rev; }
                delete arc;
            }
        }

        inline int order() const { return adj_out.size(); }
        inline int size() const { return arcs.size(); }

        int add_arc(const V u, const V v, const Cap cap) {
            int m = size();

            Arc<Cap>* arc = new Arc<Cap>(m, u, v, cap, 0, true);
            Arc<Cap>* rev_arc = new Arc<Cap>(m, v, u, 0, 0, false);

            arc->rev = rev_arc;
            rev_arc->rev = arc;

            adj_out[u].emplace_back(arc);
            adj_out[v].emplace_back(rev_arc);
            arcs.emplace_back(arc);

            return m;
        }

        int add_vertex() {
            adj_out.emplace_back({});
            level.emplace_back(-1);
            iter.emplace_back(0);

            return order() - 1;
        }

        Cap max_flow(const V source, const V target) {
            Cap flow = 0;
            while (bfs(source, target)) {
                iter.assign(order(), 0);
                Cap f;
                while ((f = dfs(source, target, -1)) > 0) {
                    flow += f;
                }
            }

            return flow;
        }

        Cap max_flow(const V source, const V target, Cap flow_limit) {
            Cap flow = 0;
            while (flow < flow_limit && bfs(source, target)) {
                iter.assign(order(), 0);
                Cap f;
                while ((f = dfs(source, target, -1)) > 0) {
                    flow += f;
                }
            }

            return flow;
        }

        vector<vector<tuple<int, V, Cap>>> get_flow() const {
            vector<vector<tuple<int, V, Cap>>> flow(order());

            for (Arc<Cap>* arc: arcs) {
                flow[arc->source].emplace_back(
                    arc->id,
                    arc->target,
                    arc->flow
                );
            }

            return flow;
        }

        vector<bool> min_cut(const int s) {
            vector<bool> group(order(), true); group[s] = false;

            deque<V> Q{s};

            while (!Q.empty()) {
                V x = Q.front(); Q.pop_front();

                for (Arc<Cap>* arc: adj_out[x]) {
                    V y = arc->target;
                    unless(arc->is_flowable() && group[y]) { continue; }

                    group[y] = false;
                    Q.emplace_back(y);
                }
            }

            return group;
        }
    };
}
#line 2 "Max_Flow/Max_Flow.hpp"

#line 2 "template/template.hpp"

using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <concepts>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <optional>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility
#line 2 "template/utility.hpp"

using ll = long long;

// a ← max(a, b) を実行する. a が更新されたら, 返り値が true.
template<typename T, typename U>
inline bool chmax(T &a, const U b){
    return (a < b ? a = b, 1: 0);
}

// a ← min(a, b) を実行する. a が更新されたら, 返り値が true.
template<typename T, typename U>
inline bool chmin(T &a, const U b){
    return (a > b ? a = b, 1: 0);
}

// a の最大値を取得する.
template<typename T>
inline T max(const vector<T> &a){
    if (a.empty()) throw invalid_argument("vector is empty.");

    return *max_element(a.begin(), a.end());
}

// vector<T> a の最小値を取得する.
template<typename T>
inline T min(const vector<T> &a){
    if (a.empty()) throw invalid_argument("vector is empty.");

    return *min_element(a.begin(), a.end());
}

// vector<T> a の最大値のインデックスを取得する.
template<typename T>
inline size_t argmax(const vector<T> &a){
    if (a.empty()) throw std::invalid_argument("vector is empty.");

    return distance(a.begin(), max_element(a.begin(), a.end()));
}

// vector<T> a の最小値のインデックスを取得する.
template<typename T>
inline size_t argmin(const vector<T> &a){
    if (a.empty()) throw invalid_argument("vector is empty.");

    return distance(a.begin(), min_element(a.begin(), a.end()));
}
#line 61 "template/template.hpp"

// math
#line 2 "template/math.hpp"

// 演算子
template<typename T>
T add(const T &x, const T &y) { return x + y; }

template<typename T>
T sub(const T &x, const T &y) { return x - y; }

template<typename T>
T mul(const T &x, const T &y) { return x * y; }

template<typename T>
T neg(const T &x) { return -x; }

template<integral T>
T bitwise_and(const T &x, const T &y) { return x & y; }

template<integral T>
T bitwise_or(const T &x, const T &y) { return x | y; }

template<integral T>
T bitwise_xor(const T &x, const T &y) { return x ^ y; }

// 除算に関する関数

// floor(x / y) を求める.
template<integral T, integral U>
auto div_floor(T x, U y){
    return x / y - ((x % y != 0) && ((x < 0) != (y < 0)));
}

// ceil(x / y) を求める.
template<integral T, integral U>
auto div_ceil(T x, U y){
    return x / y + ((x % y != 0) && ((x < 0) == (y < 0)));
}

// x を y で割った余りを求める.
template<integral T, integral U>
auto safe_mod(T x, U y){
    auto q = div_floor(x, y);
    return x - q * y ;
}

// x を y で割った商と余りを求める.
template<integral T, integral U>
auto divmod(T x, U y){
    auto q = div_floor(x, y);
    return make_pair(q, x - q * y);
}

// 四捨五入を求める.
template<integral T, integral U>
auto round(T x, U y){
    auto [q, r] = divmod(x, y);
    if (y < 0) return (r <= div_floor(y, 2)) ? q + 1 : q;
    return (r >= div_ceil(y, 2)) ? q + 1 : q;
}

// 奇数かどうか判定する.
template<integral T>
bool is_odd(const T &x) { return x % 2 != 0; }

// 偶数かどうか判定する.
template<integral T>
bool is_even(const T &x) { return x % 2 == 0; }

// m の倍数かどうか判定する.
template<integral T, integral U>
bool is_multiple(const T &x, const U &m) { return x % m == 0; }

// 正かどうか判定する.
template<typename T>
bool is_positive(const T &x) { return x > 0; }

// 負かどうか判定する.
template<typename T>
bool is_negative(const T &x) { return x < 0; }

// ゼロかどうか判定する.
template<typename T>
bool is_zero(const T &x) { return x == 0; }

// 非負かどうか判定する.
template<typename T>
bool is_non_negative(const T &x) { return x >= 0; }

// 非正かどうか判定する.
template<typename T>
bool is_non_positive(const T &x) { return x <= 0; }

// 指数に関する関数

// x の y 乗を求める.
ll intpow(ll x, ll y){
    ll a = 1;
    while (y){
        if (y & 1) { a *= x; }
        x *= x;
        y >>= 1;
    }
    return a;
}

// x の y 乗を z で割った余りを求める.
template<typename T, integral U>
T modpow(T x, U y, T z) {
    T a = 1;
    while (y) {
        if (y & 1) { (a *= x) %= z; }

        (x *= x) %= z;
        y >>= 1;
    }

    return a;
}

template<typename T>
T sum(const vector<T> &X) {
    T y = T(0);
    for (auto &&x: X) { y += x; }
    return y;
}

template<typename T>
T gcd(const T x, const T y) {
    return y == 0 ? x : gcd(y, x % y);
}

// a x + b y = gcd(a, b) を満たす整数の組 (a, b) に対して, (x, y, gcd(a, b)) を求める.
template<integral T>
tuple<T, T, T> Extended_Euclid(T a, T b) {
    T s = 1, t = 0, u = 0, v = 1;
    while (b) {
        auto [q, r] = divmod(a, b);
        a = b;
        b = r;
        tie(s, t) = make_pair(t, s - q * t);
        tie(u, v) = make_pair(v, u - q * v);
    }

    return make_tuple(s, u, a);
}

// floor(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll isqrt(const ll &N) { 
    if (N <= 0) { return 0; }

    ll x = sqrtl(N);
    while ((x + 1) * (x + 1) <= N) { x++; }
    while (x * x > N) { x--; }

    return x;
}

// floor(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll floor_sqrt(const ll &N) { return isqrt(N); }

// ceil(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll ceil_sqrt(const ll &N) {
    ll x = isqrt(N);
    return x * x == N ? x : x + 1;
}
#line 64 "template/template.hpp"

// inout
#line 1 "template/inout.hpp"
// 入出力
template<class... T>
void input(T&... a){ (cin >> ... >> a); }

void print(){ cout << "\n"; }

template<class T, class... Ts>
void print(const T& a, const Ts&... b){
    cout << a;
    (cout << ... << (cout << " ", b));
    cout << "\n";
}

template<typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &P){
    is >> P.first >> P.second;
    return is;
}

template<typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &P){
    os << P.first << " " << P.second;
    return os;
}

template<typename T>
vector<T> vector_input(int N, int index){
    vector<T> X(N+index);
    for (int i=index; i<index+N; i++) cin >> X[i];
    return X;
}

template<typename T>
istream &operator>>(istream &is, vector<T> &X){
    for (auto &x: X) { is >> x; }
    return is;
}

template<typename T>
ostream &operator<<(ostream &os, const vector<T> &X){
    int s = (int)X.size();
    for (int i = 0; i < s; i++) { os << (i ? " " : "") << X[i]; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const unordered_set<T> &S){
    int i = 0;
    for (T a: S) {os << (i ? " ": "") << a; i++;}
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const set<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const unordered_multiset<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const multiset<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
std::vector<T> input_vector(size_t n, size_t offset = 0) {
    std::vector<T> res;
    // 最初に必要な全容量を確保（再確保を防ぐ）
    res.reserve(n + offset);
    // offset 分をデフォルト値で埋める（特別 indexed 用）
    res.assign(offset, T());
    
    for (size_t i = 0; i < n; ++i) {
        T el;
        if (!(std::cin >> el)) break;
        res.push_back(std::move(el));
    }
    return res;
}
#line 67 "template/template.hpp"

// macro
#line 2 "template/macro.hpp"

// マクロの定義
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if
#define unless(cond) if (!(cond))
#define until(cond) while (!(cond))
#define loop while (true)

// オーバーロードマクロ
#define overload2(_1, _2, name, ...) name
#define overload3(_1, _2, _3, name, ...) name
#define overload4(_1, _2, _3, _4, name, ...) name
#define overload5(_1, _2, _3, _4, _5, name, ...) name

// 繰り返し系
#define rep1(n) for (ll i = 0; i < n; i++)
#define rep2(i, n) for (ll i = 0; i < n; i++)
#define rep3(i, a, b) for (ll i = a; i < b; i++)
#define rep4(i, a, b, c) for (ll i = a; i < b; i += c)
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)

#define foreach1(x, a) for (auto &&x: a)
#define foreach2(x, y, a) for (auto &&[x, y]: a)
#define foreach3(x, y, z, a) for (auto &&[x, y, z]: a)
#define foreach4(x, y, z, w, a) for (auto &&[x, y, z, w]: a)
#define foreach(...) overload5(__VA_ARGS__, foreach4, foreach3, foreach2, foreach1)(__VA_ARGS__)
#line 70 "template/template.hpp"

// bitop
#line 2 "template/bitop.hpp"

// 非負整数 x の bit legnth を求める.
ll bit_length(ll x) {
    if (x == 0) { return 0; }
    return (sizeof(long) * CHAR_BIT) - __builtin_clzll(x);
}

// 非負整数 x の popcount を求める.
ll popcount(ll x) { return __builtin_popcountll(x); }

// 正の整数 x に対して, floor(log2(x)) を求める.
ll floor_log2(ll x) { return bit_length(x) - 1; }

// 正の整数 x に対して, ceil(log2(x)) を求める.
ll ceil_log2(ll x) { return bit_length(x - 1); }

// x の第 k ビットを取得する
int get_bit(ll x, int k) { return (x >> k) & 1; }

// x のビット列を取得する.
// k はビット列の長さとする.
vector<int> get_bits(ll x, int k) {
    vector<int> bits(k);
    rep(i, k) {
        bits[i] = x & 1;
        x >>= 1;
    }

    return bits;
}

// x のビット列を取得する.
vector<int> get_bits(ll x) { return get_bits(x, bit_length(x)); }
#line 73 "template/template.hpp"

// exception
#line 2 "template/exception.hpp"

class NotExist: public exception {
    private:
    string message;

    public:
    NotExist() : message("求めようとしていたものは存在しません.") {}

    const char* what() const noexcept override {
        return message.c_str();
    }
};
#line 4 "Max_Flow/Max_Flow.hpp"

namespace max_flow {
    template<typename Cap>
    struct Arc {
        int id, source, target;
        Cap cap, flow;
        bool direction;
        Arc* rev;

        Arc(int id, int source, int target, Cap cap, Cap flow, bool direction):
            id(id), source(source), target(target), cap(cap), flow(flow), direction(direction), rev(nullptr) {}

        inline bool is_flowable() const { return flow < cap; }
        inline Cap remain() const { return cap - flow; }

        void push(Cap d) {
            flow += d;
            rev->flow -= d;
        }
    };

    template<typename Cap>
    struct Max_Flow {
        private:
        using V = int;
        vector<vector<Arc<Cap>*>> adj_out;
        vector<Arc<Cap>*> arcs;
        vector<int> level, iter;

        bool bfs(const V s, const V t) {
            level.assign(order(), -1);
            queue<V> Q;

            Q.push(s);
            level[s] = 0;

            while (!Q.empty()) {
                V v = Q.front(); Q.pop();

                for (Arc<Cap>* arc: adj_out[v]) {
                    unless(arc->is_flowable() && level[arc->target] == -1) { continue; }

                    level[arc->target] = level[v] + 1;
                    if (arc->target == t) { return true; }

                    Q.push(arc->target);
                }
            }

            return false;
        }

        Cap dfs(const V v, const V t, const Cap f) {
            if (v == t) { return f; }

            for (int i = iter[v]; i < adj_out[v].size(); ++i) {
                Arc<Cap>* arc = adj_out[v][i];

                unless(arc->is_flowable() && level[v] < level[arc->target]) { continue; }

                Cap d = (f == -1) ? dfs(arc->target, t, arc->remain()) : dfs(arc->target, t, min(f, arc->remain()));

                unless(d > 0) { continue; }

                arc->push(d);
                return d;
            }

            return 0;
        }

        public:
        Max_Flow(int N = 0) {
            adj_out.resize(N);
            level.resize(N);
            iter.resize(N);
        }

        ~Max_Flow() {
            for (Arc<Cap>* arc : arcs) {
                if (arc->rev) { delete arc->rev; }
                delete arc;
            }
        }

        inline int order() const { return adj_out.size(); }
        inline int size() const { return arcs.size(); }

        int add_arc(const V u, const V v, const Cap cap) {
            int m = size();

            Arc<Cap>* arc = new Arc<Cap>(m, u, v, cap, 0, true);
            Arc<Cap>* rev_arc = new Arc<Cap>(m, v, u, 0, 0, false);

            arc->rev = rev_arc;
            rev_arc->rev = arc;

            adj_out[u].emplace_back(arc);
            adj_out[v].emplace_back(rev_arc);
            arcs.emplace_back(arc);

            return m;
        }

        int add_vertex() {
            adj_out.emplace_back({});
            level.emplace_back(-1);
            iter.emplace_back(0);

            return order() - 1;
        }

        Cap max_flow(const V source, const V target) {
            Cap flow = 0;
            while (bfs(source, target)) {
                iter.assign(order(), 0);
                Cap f;
                while ((f = dfs(source, target, -1)) > 0) {
                    flow += f;
                }
            }

            return flow;
        }

        Cap max_flow(const V source, const V target, Cap flow_limit) {
            Cap flow = 0;
            while (flow < flow_limit && bfs(source, target)) {
                iter.assign(order(), 0);
                Cap f;
                while ((f = dfs(source, target, -1)) > 0) {
                    flow += f;
                }
            }

            return flow;
        }

        vector<vector<tuple<int, V, Cap>>> get_flow() const {
            vector<vector<tuple<int, V, Cap>>> flow(order());

            for (Arc<Cap>* arc: arcs) {
                flow[arc->source].emplace_back(
                    arc->id,
                    arc->target,
                    arc->flow
                );
            }

            return flow;
        }

        vector<bool> min_cut(const int s) {
            vector<bool> group(order(), true); group[s] = false;

            deque<V> Q{s};

            while (!Q.empty()) {
                V x = Q.front(); Q.pop_front();

                for (Arc<Cap>* arc: adj_out[x]) {
                    V y = arc->target;
                    unless(arc->is_flowable() && group[y]) { continue; }

                    group[y] = false;
                    Q.emplace_back(y);
                }
            }

            return group;
        }
    };
}
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