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:heavy_check_mark: ABC317-G Rearranging
(Test/Manual/abc317_g.test.cpp)

左の列から順番に条件を満たすように配置を変えることを考える。

順列を構成することを「完全マッチングを一つ抜き去る」と言い換える。

$i$ 行目に要素 $j$ が残り何個あるか？を辺容量として二部マッチングを構成する。

計算量は全ての辺容量が1である $2n + 2$ 頂点 $O(nm)$ 辺の最大流問題を $m$ 回解くので $O(n^{\frac{3}{2}}m^2)$

Tips: https://37zigen.com/noga-alon-edge-coloring-bipartite/ を使うともっと効率的に解ける。いつか盆栽に植えたいねぇ

Depends on

Code

#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/courses/lesson/2/ITP1/1/ITP1_1_A"

#include "../../Src/Template/IOSetting.hpp"
#include "../../Src/Graph/Flow/Dinic.hpp"

#include <cassert>
#include <iostream>
#include <vector>

/*
 * AtCoder Beginner Contest 317 - G Rearranging
 * https://atcoder.jp/contests/abc317/submissions/48902760
 */

void solve() {
    using namespace zawa;
    SetFastIO();
    int n, m; std::cin >> n >> m;
    // i行目に残り要素jが何個残っているか？
    std::vector remain(n, std::vector<int>(n)); for (int i{} ; i < n ; i++) for (int j{} ; j < m ; j++) {
        int a; std::cin >> a;
        remain[i][a - 1]++;
    }
    std::vector ans(n, std::vector<int>(m));
    for (int col{} ; col < m ; col++) {
        Dinic<int> mf(2 * n + 2);
        for (int i{} ; i < n ; i++) {
            mf.addEdge(2 * n, i, 1);
            mf.addEdge(n + i, 2 * n + 1, 1);
        }
        std::vector id(n, std::vector<int>(n));
        for (int i{} ; i < n ; i++) for (int j{} ; j < n ; j++) {
            id[i][j] = mf.addEdge(i, n + j, (bool)remain[i][j]);
        }
        assert(mf.flow(2 * n, 2 * n + 1) == n);
        for (int i{} ; i < n ; i++) for (int j{} ; j < n ; j++) {
            if (mf.flowed(id[i][j]) == 1) {
                ans[i][col] = j + 1;
                remain[i][j]--;
            }
        }
    }
    std::cout << "Yes" << '\n';
    for (int i{} ; i < n ; i++) {
        for (int j{} ; j < m ; j++) {
            std::cout << ans[i][j] << (j + 1 == m ? '\n' : ' ');
        }
    }
}

int main() {
#ifdef ATCODER
    solve();
#else
    std::cout << "Hello World" << '\n';
#endif
}
#line 1 "Test/Manual/abc317_g.test.cpp"
#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/courses/lesson/2/ITP1/1/ITP1_1_A"

#line 2 "Src/Template/IOSetting.hpp"

#line 2 "Src/Template/TypeAlias.hpp"

#include <cstdint>
#include <cstddef>

namespace zawa {

using i16 = std::int16_t;
using i32 = std::int32_t;
using i64 = std::int64_t;
using i128 = __int128_t;

using u8 = std::uint8_t;
using u16 = std::uint16_t;
using u32 = std::uint32_t;
using u64 = std::uint64_t;

using usize = std::size_t;

} // namespace zawa
#line 4 "Src/Template/IOSetting.hpp"

#include <iostream>
#include <iomanip>

namespace zawa {

void SetFastIO() {
    std::cin.tie(nullptr)->sync_with_stdio(false);
}

void SetPrecision(u32 dig) {
    std::cout << std::fixed << std::setprecision(dig);
}

} // namespace zawa
#line 2 "Src/Graph/Flow/Dinic.hpp"

#line 2 "Src/Utility/U32Pair.hpp"

#line 4 "Src/Utility/U32Pair.hpp"

#include <functional>
#line 7 "Src/Utility/U32Pair.hpp"

namespace zawa {

class U32Pair {
private:
    static constexpr u32 SHIFT{32};
    static constexpr u32 MASK{static_cast<u32>((1LL << SHIFT) - 1)};
    u64 value_{};
public:
    constexpr U32Pair() {}
    constexpr U32Pair(u32 first, u32 second) {
        value_ = (static_cast<u64>(first) << SHIFT) | second;
    }
    constexpr u32 first() const noexcept {
        return static_cast<u32>(value_ >> SHIFT);
    }
    constexpr u32 second() const noexcept {
        return static_cast<u32>(value_ & MASK);
    }
    constexpr u64 combined() const noexcept {
        return value_;
    }
    constexpr U32Pair& operator=(const U32Pair& rhs) {
        value_ = rhs.value_;
        return *this;
    }
    friend constexpr bool operator==(const U32Pair& lhs, const U32Pair& rhs) {
        return lhs.value_ == rhs.value_;
    }
    friend constexpr bool operator!=(const U32Pair& lhs, const U32Pair& rhs) {
        return lhs.value_ != rhs.value_;
    }
    friend constexpr bool operator<(const U32Pair& lhs, const U32Pair& rhs) {
        return lhs.value_ < rhs.value_;
    }
    friend constexpr bool operator<=(const U32Pair& lhs, const U32Pair& rhs) {
        return lhs.value_ <= rhs.value_;
    }
    friend constexpr bool operator>(const U32Pair& lhs, const U32Pair& rhs) {
        return lhs.value_ > rhs.value_;
    }
    friend constexpr bool operator>=(const U32Pair& lhs, const U32Pair& rhs) {
        return lhs.value_ >= rhs.value_;
    }
    friend std::ostream& operator<<(std::ostream& os, const U32Pair& pair) {
        os << '(' << pair.first() << ',' << pair.second() << ')';
        return os;
    }
};

struct U32PairHash {
    usize operator()(const U32Pair& pair) const noexcept {
        return std::hash<u64>{}(pair.combined());
    }
};

} // namespace zawa
#line 5 "Src/Graph/Flow/Dinic.hpp"

#include <algorithm>
#include <cassert>
#include <limits>
#include <type_traits>
#include <vector>
#include <queue>

namespace zawa {

template <class Cap> 
class Dinic {
private:
    static_assert(std::is_signed_v<Cap>, "Cap must be signed");
    usize n_{}, m_{};
    static constexpr u32 invalid() noexcept {
        return std::numeric_limits<u32>::max();
    }
public:
    inline usize size() const noexcept {
        return n_;
    }
    inline usize edgeNumber() const noexcept {
        return m_;
    }
private:
    struct Edge {
        u32 to{}, rev{};
        Cap residual{};
        Edge() = default;
        Edge(u32 to, u32 rev, const Cap& residual) 
            : to{to}, rev{rev}, residual{residual} {}
    };

    std::vector<std::vector<Edge>> g_;
    std::vector<U32Pair> edges_;
    std::vector<u32> label_, cur_;

    bool dualStep(u32 s, u32 t) {
        std::fill(label_.begin(), label_.end(), invalid());
        label_[s] = 0;
        std::queue<u32> queue{ { s } };
        while (queue.size()) {
            u32 v{queue.front()};
            queue.pop();
            for (const Edge& e : g_[v]) if (e.residual > 0) {
                if (label_[e.to] > label_[v] + 1) {
                    label_[e.to] = label_[v] + 1;
                    if (e.to == t) return true;
                    queue.emplace(e.to);
                }
            }
        }
        return false;
    }

    bool admissible(u32 v, const Edge& e) const noexcept {
        return e.residual > 0 and label_[v] + 1 == label_[e.to];
    }

    inline void flow(Edge& e, Cap f) {
        e.residual -= f;
        g_[e.to][e.rev].residual += f;
    }

    Cap dfs(u32 v, u32 t, Cap up) {
        if (v == t) return up;
        Cap res{};
        for (u32& i{cur_[v]} ; i < g_[v].size() ; i++) {
            if (!admissible(v, g_[v][i])) continue;
            Cap f{dfs(g_[v][i].to, t, std::min(g_[v][i].residual, up - res))};
            if (f == 0) continue;
            flow(g_[v][i], f);
            res += f;
            if (res == up) return res;
        }
        return res;
    }

    Cap primalStep(u32 s, u32 t) {
        std::fill(cur_.begin(), cur_.end(), 0u);
        cur_[t] = g_[t].size();
        Cap res{};
        while (true) {
            Cap f{dfs(s, t, std::numeric_limits<Cap>::max())};
            if (f == 0) break;
            res += f;
        }
        return res;
    }

    const Edge& edge(u32 i) const noexcept {
        return g_[edges_[i].first()][edges_[i].second()];
    }
    const Edge& reverse(u32 i) const noexcept {
        const Edge& e{edge(i)};
        return g_[e.to][e.rev];
    }

public:
    Dinic() = default;
    Dinic(u32 n, u32 m = 0u) 
        : n_{n}, m_{m}, g_(n), edges_{}, label_(n), cur_(n) {
        g_.shrink_to_fit();
        label_.shrink_to_fit();
        cur_.shrink_to_fit();
        edges_.reserve(m);
    }

    u32 addEdge(u32 u, u32 v, const Cap& cap) {
        assert(u < size());
        assert(v < size());
        u32 id{static_cast<u32>(g_[u].size())};
        u32 revId{u == v ? id + 1 : static_cast<u32>(g_[v].size())};
        u32 res{static_cast<u32>(edges_.size())};
        edges_.emplace_back(u, id);
        g_[u].emplace_back(v, revId, cap);
        g_[v].emplace_back(u, id, Cap{});
        m_++;
        return res;
    }

    const Cap& flowed(u32 id) const noexcept {
        assert(id < edgeNumber());
        return reverse(id).residual;
    }
    const Cap& residual(u32 id) const noexcept {
        assert(id < edgeNumber());
        return edge(id).residual;
    }
    const Cap& capacity(u32 id) const noexcept {
        assert(id < edgeNumber());
        return edge(id).residual + reverse(id).residual;
    }
    const u32& from(u32 id) const noexcept {
        assert(id < edgeNumber());
        return edges_[id].first();
    }
    const u32& to(u32 id) const noexcept {
        assert(id < edgeNumber());
        return edge(id).to;
    }

    Cap flow(u32 s, u32 t) {
        assert(s < size());
        assert(t < size()); 
        Cap res{};
        while (dualStep(s, t)) {
            res += primalStep(s, t);
        }
        return res;
    }

    std::vector<bool> cut(u32 s) {
        std::vector<bool> res(size());
        res[s] = true;
        std::queue<u32> queue{ { s } };
        while (queue.size()) {
            u32 v{queue.front()};
            queue.pop();
            for (const auto& e : g_[v]) if (e.residual > 0 and !res[e.to]) {
                res[e.to] = true;
                queue.emplace(e.to);
            }
        }
        return res;
    }    
};

} // namespace zawa
#line 5 "Test/Manual/abc317_g.test.cpp"

#line 9 "Test/Manual/abc317_g.test.cpp"

/*
 * AtCoder Beginner Contest 317 - G Rearranging
 * https://atcoder.jp/contests/abc317/submissions/48902760
 */

void solve() {
    using namespace zawa;
    SetFastIO();
    int n, m; std::cin >> n >> m;
    // i行目に残り要素jが何個残っているか？
    std::vector remain(n, std::vector<int>(n)); for (int i{} ; i < n ; i++) for (int j{} ; j < m ; j++) {
        int a; std::cin >> a;
        remain[i][a - 1]++;
    }
    std::vector ans(n, std::vector<int>(m));
    for (int col{} ; col < m ; col++) {
        Dinic<int> mf(2 * n + 2);
        for (int i{} ; i < n ; i++) {
            mf.addEdge(2 * n, i, 1);
            mf.addEdge(n + i, 2 * n + 1, 1);
        }
        std::vector id(n, std::vector<int>(n));
        for (int i{} ; i < n ; i++) for (int j{} ; j < n ; j++) {
            id[i][j] = mf.addEdge(i, n + j, (bool)remain[i][j]);
        }
        assert(mf.flow(2 * n, 2 * n + 1) == n);
        for (int i{} ; i < n ; i++) for (int j{} ; j < n ; j++) {
            if (mf.flowed(id[i][j]) == 1) {
                ans[i][col] = j + 1;
                remain[i][j]--;
            }
        }
    }
    std::cout << "Yes" << '\n';
    for (int i{} ; i < n ; i++) {
        for (int j{} ; j < m ; j++) {
            std::cout << ans[i][j] << (j + 1 == m ? '\n' : ' ');
        }
    }
}

int main() {
#ifdef ATCODER
    solve();
#else
    std::cout << "Hello World" << '\n';
#endif
}
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