cp-library
競技プログラミング用 C++ ライブラリ
Hungarian Algorithm（ハンガリアン法）
graph/hungarian.hpp · GitHub で見る
共通ページ
#include "lib/graph/hungarian.hpp"
std::vector<std::vector<int>> a = { {4, 1, 3}, {2, 0, 5} };
Hungarian result(a);
auto cost = result.cost;
auto match = result.row_to_col;
Hungarian maximum(a, true);
| 操作 | 内容・返り値 | 計算量 |
|---|---|---|
Hungarian(a,maximize = false) |
費用行列から最小費用割当を計算。true は最大化 |
\(O(n^2m)\) |
Hungarian(n,m,a,stride = -1,maximize = false) |
平坦な配列から計算。stride は行間隔、省略時は m
|
\(O(n^2m)\) |
cost |
割当の総費用 | \(O(1)\) |
row_to_col[i] / col_to_row[j]
|
行の割当先／列の割当元。未使用の列は -1 | \(O(1)\) |
\(n\le m\) の費用行列について、各行を異なる列へ割り当てます。cost は最小費用、row_to_col[i] は行 i の割当先、col_to_row[j] は列 j の割当元（未使用は -1）です。第 2 引数を true にすると最大費用割当を求めます。
Hungarian<Cost = long long> は整数を扱います。費用・距離・双対変数・それらの加減算・総費用が Cost に収まる必要があります。最大化では費用を符号反転します。広い計算型が必要な場合は Hungarian<__int128_t> などを指定します。
各行の要素が連続する配列・vector<array<T,m>> なども渡せます。平坦な配列は Hungarian(n, m, a.data(), stride = -1, maximize = false) で渡し、stride は隣の行までの要素数（省略時 m）です。入力は変更しません。
時間は \(O(n^2m)\)、追加空間は \(O(n+m)\) です。正方の Monge・逆 Monge 行列は \(O(n^2)\) で処理します。
検証コード
コード
#ifndef HUNGARIAN_HPP
#define HUNGARIAN_HPP
#include <algorithm>
#include <cassert>
#include <cstddef>
#include <iterator>
#include <limits>
#include <numeric>
#include <type_traits>
#include <utility>
#include <vector>
template <class Cost = long long>
struct Hungarian {
static_assert(std::numeric_limits<Cost>::is_integer && std::numeric_limits<Cost>::is_signed);
Cost cost = 0;
std::vector<int> row_to_col, col_to_row;
template <class Matrix>
explicit Hungarian(const Matrix &a, bool maximize = false) {
int n = int(a.size()), m = n ? int(a[0].size()) : 0;
using T = std::remove_cvref_t<decltype(a[0][0])>;
std::vector<const T *> rows(n);
for (int i = 0; i < n; ++i) {
assert(int(a[i].size()) == m);
rows[i] = std::data(a[i]);
}
if (maximize) solve<true>(n, m, rows);
else solve<false>(n, m, rows);
}
template <class T>
Hungarian(int n, int m, const T *a, int stride = -1, bool maximize = false) {
assert(0 <= n && n <= m);
if (stride == -1) stride = m;
assert(stride >= m);
std::vector<const T *> rows(n);
for (int i = 0; i < n; ++i) rows[i] = a + std::size_t(i) * stride;
if (maximize) solve<true>(n, m, rows);
else solve<false>(n, m, rows);
}
private:
template <bool maximize, class T>
void solve(int n, int m, const std::vector<const T *> &a) {
assert(n <= m);
row_to_col.assign(n, -1);
col_to_row.assign(m, -1);
if (!n) return;
auto value = [&](int i, int j) -> Cost {
if constexpr (maximize) return -Cost(a[i][j]);
else return Cost(a[i][j]);
};
auto finish = [&] {
for (int i = 0; i < n; ++i) cost += Cost(a[i][row_to_col[i]]);
};
if (n == m) {
using Wide = std::conditional_t<(sizeof(T) < 8), long long, __int128_t>;
int order = 3;
for (int i = 1; i < n && order; ++i)
for (int j = 1; j < m; ++j) {
Wide x = Wide(a[i - 1][j - 1]) + Wide(a[i][j]);
Wide y = Wide(a[i - 1][j]) + Wide(a[i][j - 1]);
if (x > y) order &= 2;
if (x < y) order &= 1;
if (!order) break;
}
if (order) {
bool reverse = maximize ? bool(order & 1) : !(order & 1);
for (int i = 0; i < n; ++i) {
int j = reverse ? n - 1 - i : i;
row_to_col[i] = j;
col_to_row[j] = i;
}
finish();
return;
}
}
std::vector<Cost> price(m), dist(m);
std::vector<int> columns(m), pred(m);
if (n == m) {
std::vector<T> minimum(a[0], a[0] + m);
std::vector<int> owner(m), count(n);
for (int i = 1; i < n; ++i)
for (int j = 0; j < m; ++j) {
T x = a[i][j];
bool better = maximize ? x > minimum[j] : x < minimum[j];
minimum[j] = better ? x : minimum[j];
owner[j] = better ? i : owner[j];
}
for (int j = 0; j < m; ++j) price[j] = maximize ? -Cost(minimum[j]) : Cost(minimum[j]);
for (int j = m - 1; j >= 0; --j) {
int i = owner[j];
if (!count[i]++) row_to_col[i] = j, col_to_row[j] = i;
}
for (int i = 0; i < n; ++i) if (count[i] == 1) {
Cost delta = std::numeric_limits<Cost>::max();
int matched = row_to_col[i];
for (int j = 0; j < m; ++j)
if (j != matched) delta = std::min(delta, Cost(value(i, j) - price[j]));
price[matched] -= delta;
}
for (int pass = 0; pass < 2; ++pass)
for (int i = 0; i < n; ++i) if (row_to_col[i] == -1) {
int best_col = 0;
Cost best = value(i, 0) - price[0], second = std::numeric_limits<Cost>::max();
for (int j = 1; j < m; ++j) {
Cost x = value(i, j) - price[j];
if (x < best) second = best, best = x, best_col = j;
else if (x == best && col_to_row[best_col] != -1 && col_to_row[j] == -1) second = best, best_col = j;
else second = std::min(second, x);
}
if (best < second) price[best_col] -= second - best;
int old = col_to_row[best_col];
if (old != -1) row_to_col[old] = -1;
row_to_col[i] = best_col;
col_to_row[best_col] = i;
}
} else {
for (int i = 0; i < n; ++i) {
int best_col = 0;
Cost best = value(i, 0);
for (int j = 1; j < m; ++j) {
Cost x = value(i, j);
if (x < best || (x == best && col_to_row[best_col] != -1 && col_to_row[j] == -1)) best = x, best_col = j;
}
if (col_to_row[best_col] == -1) row_to_col[i] = best_col, col_to_row[best_col] = i;
}
}
std::iota(columns.begin(), columns.end(), 0);
for (int root = 0; root < n; ++root) if (row_to_col[root] == -1) {
for (int j = 0; j < m; ++j) dist[j] = value(root, j) - price[j], pred[j] = root;
int scanned = 0, queued = 0, target = -1;
Cost level = 0;
while (target == -1) {
if (scanned == queued) {
level = dist[columns[scanned]];
for (int k = scanned; k < m; ++k) {
int j = columns[k];
if (dist[j] > level) continue;
if (dist[j] < level) level = dist[j], queued = scanned, target = -1;
std::swap(columns[k], columns[queued++]);
if (col_to_row[j] == -1) target = j;
}
if (target != -1) break;
}
int j0 = columns[scanned++], i = col_to_row[j0];
Cost offset = value(i, j0) - price[j0] - level;
for (int k = queued; k < m; ++k) {
int j = columns[k];
Cost candidate = value(i, j) - price[j] - offset;
if (candidate >= dist[j]) continue;
dist[j] = candidate;
pred[j] = i;
if (candidate == level) {
if (col_to_row[j] == -1) { target = j; break; }
std::swap(columns[k], columns[queued++]);
}
}
}
for (int k = 0; k < scanned; ++k) {
int j = columns[k];
price[j] += dist[j] - level;
}
while (target != -1) {
int i = pred[target], next = row_to_col[i];
row_to_col[i] = target;
col_to_row[target] = i;
target = next;
}
}
finish();
}
};
#endif
#line 1 "graph/hungarian.hpp"
#include <algorithm>
#include <cassert>
#include <cstddef>
#include <iterator>
#include <limits>
#include <numeric>
#include <type_traits>
#include <utility>
#include <vector>
template <class Cost = long long>
struct Hungarian {
static_assert(std::numeric_limits<Cost>::is_integer && std::numeric_limits<Cost>::is_signed);
Cost cost = 0;
std::vector<int> row_to_col, col_to_row;
template <class Matrix>
explicit Hungarian(const Matrix &a, bool maximize = false) {
int n = int(a.size()), m = n ? int(a[0].size()) : 0;
using T = std::remove_cvref_t<decltype(a[0][0])>;
std::vector<const T *> rows(n);
for (int i = 0; i < n; ++i) {
assert(int(a[i].size()) == m);
rows[i] = std::data(a[i]);
}
if (maximize) solve<true>(n, m, rows);
else solve<false>(n, m, rows);
}
template <class T>
Hungarian(int n, int m, const T *a, int stride = -1, bool maximize = false) {
assert(0 <= n && n <= m);
if (stride == -1) stride = m;
assert(stride >= m);
std::vector<const T *> rows(n);
for (int i = 0; i < n; ++i) rows[i] = a + std::size_t(i) * stride;
if (maximize) solve<true>(n, m, rows);
else solve<false>(n, m, rows);
}
private:
template <bool maximize, class T>
void solve(int n, int m, const std::vector<const T *> &a) {
assert(n <= m);
row_to_col.assign(n, -1);
col_to_row.assign(m, -1);
if (!n) return;
auto value = [&](int i, int j) -> Cost {
if constexpr (maximize) return -Cost(a[i][j]);
else return Cost(a[i][j]);
};
auto finish = [&] {
for (int i = 0; i < n; ++i) cost += Cost(a[i][row_to_col[i]]);
};
if (n == m) {
using Wide = std::conditional_t<(sizeof(T) < 8), long long, __int128_t>;
int order = 3;
for (int i = 1; i < n && order; ++i)
for (int j = 1; j < m; ++j) {
Wide x = Wide(a[i - 1][j - 1]) + Wide(a[i][j]);
Wide y = Wide(a[i - 1][j]) + Wide(a[i][j - 1]);
if (x > y) order &= 2;
if (x < y) order &= 1;
if (!order) break;
}
if (order) {
bool reverse = maximize ? bool(order & 1) : !(order & 1);
for (int i = 0; i < n; ++i) {
int j = reverse ? n - 1 - i : i;
row_to_col[i] = j;
col_to_row[j] = i;
}
finish();
return;
}
}
std::vector<Cost> price(m), dist(m);
std::vector<int> columns(m), pred(m);
if (n == m) {
std::vector<T> minimum(a[0], a[0] + m);
std::vector<int> owner(m), count(n);
for (int i = 1; i < n; ++i)
for (int j = 0; j < m; ++j) {
T x = a[i][j];
bool better = maximize ? x > minimum[j] : x < minimum[j];
minimum[j] = better ? x : minimum[j];
owner[j] = better ? i : owner[j];
}
for (int j = 0; j < m; ++j) price[j] = maximize ? -Cost(minimum[j]) : Cost(minimum[j]);
for (int j = m - 1; j >= 0; --j) {
int i = owner[j];
if (!count[i]++) row_to_col[i] = j, col_to_row[j] = i;
}
for (int i = 0; i < n; ++i) if (count[i] == 1) {
Cost delta = std::numeric_limits<Cost>::max();
int matched = row_to_col[i];
for (int j = 0; j < m; ++j)
if (j != matched) delta = std::min(delta, Cost(value(i, j) - price[j]));
price[matched] -= delta;
}
for (int pass = 0; pass < 2; ++pass)
for (int i = 0; i < n; ++i) if (row_to_col[i] == -1) {
int best_col = 0;
Cost best = value(i, 0) - price[0], second = std::numeric_limits<Cost>::max();
for (int j = 1; j < m; ++j) {
Cost x = value(i, j) - price[j];
if (x < best) second = best, best = x, best_col = j;
else if (x == best && col_to_row[best_col] != -1 && col_to_row[j] == -1) second = best, best_col = j;
else second = std::min(second, x);
}
if (best < second) price[best_col] -= second - best;
int old = col_to_row[best_col];
if (old != -1) row_to_col[old] = -1;
row_to_col[i] = best_col;
col_to_row[best_col] = i;
}
} else {
for (int i = 0; i < n; ++i) {
int best_col = 0;
Cost best = value(i, 0);
for (int j = 1; j < m; ++j) {
Cost x = value(i, j);
if (x < best || (x == best && col_to_row[best_col] != -1 && col_to_row[j] == -1)) best = x, best_col = j;
}
if (col_to_row[best_col] == -1) row_to_col[i] = best_col, col_to_row[best_col] = i;
}
}
std::iota(columns.begin(), columns.end(), 0);
for (int root = 0; root < n; ++root) if (row_to_col[root] == -1) {
for (int j = 0; j < m; ++j) dist[j] = value(root, j) - price[j], pred[j] = root;
int scanned = 0, queued = 0, target = -1;
Cost level = 0;
while (target == -1) {
if (scanned == queued) {
level = dist[columns[scanned]];
for (int k = scanned; k < m; ++k) {
int j = columns[k];
if (dist[j] > level) continue;
if (dist[j] < level) level = dist[j], queued = scanned, target = -1;
std::swap(columns[k], columns[queued++]);
if (col_to_row[j] == -1) target = j;
}
if (target != -1) break;
}
int j0 = columns[scanned++], i = col_to_row[j0];
Cost offset = value(i, j0) - price[j0] - level;
for (int k = queued; k < m; ++k) {
int j = columns[k];
Cost candidate = value(i, j) - price[j] - offset;
if (candidate >= dist[j]) continue;
dist[j] = candidate;
pred[j] = i;
if (candidate == level) {
if (col_to_row[j] == -1) { target = j; break; }
std::swap(columns[k], columns[queued++]);
}
}
}
for (int k = 0; k < scanned; ++k) {
int j = columns[k];
price[j] += dist[j] - level;
}
while (target != -1) {
int i = pred[target], next = row_to_col[i];
row_to_col[i] = target;
col_to_row[target] = i;
target = next;
}
}
finish();
}
};