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#include "graph/others/bipartite-graph-edge-coloring.hpp"
二部グラフの辺彩色を求めるアルゴリズム. 辺彩色とは, 与えられたグラフの辺に色を付けることを指す. このとき隣接する(頂点を共有する)辺は全て異なる色になるようにする.
一般グラフの辺彩色数は, そのグラフの最大次数 $D$ または $D+1$ に一致する. 特に, 二部グラフに限定すると, その辺彩色数は 最大次数 $D$ に一致する.
まず, 与えられたグラフに対して最大次数 $D$ を維持したまま辺を追加したり縮約したりすることで$D-$正則グラフにする. $D$ が偶数の場合, オイラー閉路を求めて, 偶数番目に通った辺, 奇数番目に通った辺に分けて, 再帰的に解く. $D$ が奇数の場合, 完全マッチングを求めてそれらの辺を同じ色で塗った上で消して, 再帰的に解く. これらを繰り返すことで実際に $D$ 色の辺彩色を構成できる.
add_edge(a, b)
: $a$ から $b$ に辺を張る. $a$ は二部グラフの左側, $b$ は右側の頂点を指す.build()
: 二部グラフの辺彩色を返す. 同じ色に塗るべき辺の番号が同じ配列に格納される. 辺の番号は add_edge()
を呼び出した順に 0-indexed で付与される.$O(M \sqrt N \log D)$
$N$: 頂点数, $M$: 辺の本数, $D$: 頂点の最大次数
#pragma once
#include "../../structure/union-find/union-find.hpp"
#include "../flow/bipartite-flow.hpp"
#include "eulerian-trail.hpp"
/**
* @brief Bipartite Graph Edge Coloring(二部グラフの辺彩色)
*
* @see https://ei1333.hateblo.jp/entry/2020/08/25/015955
*/
struct BipariteGraphEdgeColoring {
private:
vector< vector< int > > ans;
vector< int > A, B;
int L, R;
struct RegularGraph {
int k{}, n{};
vector< int > A, B;
};
RegularGraph g;
static UnionFind contract(valarray< int > °, int k) {
using pi = pair< int, int >;
priority_queue< pi, vector< pi >, greater<> > que;
for(int i = 0; i < (int) deg.size(); i++) {
que.emplace(deg[i], i);
}
UnionFind uf(deg.size());
while(que.size() > 1) {
auto p = que.top();
que.pop();
auto q = que.top();
que.pop();
if(p.first + q.first > k) continue;
p.first += q.first;
uf.unite(p.second, q.second);
que.emplace(p);
}
return uf;
}
RegularGraph build_k_regular_graph() {
valarray< int > deg[2];
deg[0] = valarray< int >(L);
deg[1] = valarray< int >(R);
for(auto &p : A) deg[0][p]++;
for(auto &p : B) deg[1][p]++;
int k = max(deg[0].max(), deg[1].max());
/* step 1 */
UnionFind uf[2];
uf[0] = contract(deg[0], k);
uf[1] = contract(deg[1], k);
vector< int > id[2];
int ptr[] = {0, 0};
id[0] = vector< int >(L);
id[1] = vector< int >(R);
for(int i = 0; i < L; i++) if(uf[0].find(i) == i) id[0][i] = ptr[0]++;
for(int i = 0; i < R; i++) if(uf[1].find(i) == i) id[1][i] = ptr[1]++;
/* step 2 */
int N = max(ptr[0], ptr[1]);
deg[0] = valarray< int >(N);
deg[1] = valarray< int >(N);
/* step 3 */
vector< int > C, D;
C.reserve(N * k);
D.reserve(N * k);
for(int i = 0; i < (int) A.size(); i++) {
int u = id[0][uf[0].find(A[i])];
int v = id[1][uf[1].find(B[i])];
C.emplace_back(u);
D.emplace_back(v);
deg[0][u]++;
deg[1][v]++;
}
int j = 0;
for(int i = 0; i < N; i++) {
while(deg[0][i] < k) {
while(deg[1][j] == k) ++j;
C.emplace_back(i);
D.emplace_back(j);
++deg[0][i];
++deg[1][j];
}
}
return {k, N, C, D};
}
void rec(const vector< int > &ord, int k) {
if(k == 0) {
return;
} else if(k == 1) {
ans.emplace_back(ord);
return;
} else if((k & 1) == 0) {
EulerianTrail< false > et(g.n + g.n);
for(auto &p : ord) et.add_edge(g.A[p], g.B[p] + g.n);
auto paths = et.enumerate_eulerian_trail();
vector< int > path;
for(auto &ps : paths) {
for(auto &e : ps) path.emplace_back(ord[e]);
}
vector< int > beet[2];
for(int i = 0; i < (int) path.size(); i++) {
beet[i & 1].emplace_back(path[i]);
}
rec(beet[0], k / 2);
rec(beet[1], k / 2);
} else {
BipartiteFlow flow(g.n, g.n);
for(auto &i : ord) flow.add_edge(g.A[i], g.B[i]);
flow.max_matching();
vector< int > beet;
ans.emplace_back();
for(auto &i : ord) {
if(flow.match_l[g.A[i]] == g.B[i]) {
flow.match_l[g.A[i]] = -1;
ans.back().emplace_back(i);
} else {
beet.emplace_back(i);
}
}
rec(beet, k - 1);
}
}
public:
explicit BipariteGraphEdgeColoring() : L(0), R(0) {}
void add_edge(int a, int b) {
A.emplace_back(a);
B.emplace_back(b);
L = max(L, a + 1);
R = max(R, b + 1);
}
vector< vector< int > > build() {
g = build_k_regular_graph();
vector< int > ord(g.A.size());
iota(ord.begin(), ord.end(), 0);
rec(ord, g.k);
vector< vector< int > > res;
for(int i = 0; i < (int) ans.size(); i++) {
res.emplace_back();
for(auto &j : ans[i]) if(j < (int)A.size()) res.back().emplace_back(j);
}
return res;
}
};
#line 2 "graph/others/bipartite-graph-edge-coloring.hpp"
#line 2 "structure/union-find/union-find.hpp"
struct UnionFind {
vector< int > data;
UnionFind() = default;
explicit UnionFind(size_t sz) : data(sz, -1) {}
bool unite(int x, int y) {
x = find(x), y = find(y);
if(x == y) return false;
if(data[x] > data[y]) swap(x, y);
data[x] += data[y];
data[y] = x;
return true;
}
int find(int k) {
if(data[k] < 0) return (k);
return data[k] = find(data[k]);
}
int size(int k) {
return -data[find(k)];
}
bool same(int x, int y) {
return find(x) == find(y);
}
vector< vector< int > > groups() {
int n = (int) data.size();
vector< vector< int > > ret(n);
for(int i = 0; i < n; i++) {
ret[find(i)].emplace_back(i);
}
ret.erase(remove_if(begin(ret), end(ret), [&](const vector< int > &v) {
return v.empty();
}), end(ret));
return ret;
}
};
#line 1 "graph/flow/bipartite-flow.hpp"
/**
* @brief Bipartite Flow(二部グラフのフロー)
*
*/
struct BipartiteFlow {
size_t n, m, time_stamp;
vector< vector< int > > g, rg;
vector< int > match_l, match_r, dist, used, alive;
bool matched;
public:
explicit BipartiteFlow(size_t n, size_t m) :
n(n), m(m), time_stamp(0), g(n), rg(m), match_l(n, -1), match_r(m, -1), used(n), alive(n, 1), matched(false) {}
void add_edge(int u, int v) {
g[u].push_back(v);
rg[v].emplace_back(u);
}
vector< pair< int, int > > max_matching() {
matched = true;
for(;;) {
build_augment_path();
++time_stamp;
int flow = 0;
for(int i = 0; i < (int)n; i++) {
if(match_l[i] == -1) flow += find_min_dist_augment_path(i);
}
if(flow == 0) break;
}
vector< pair< int, int > > ret;
for(int i = 0; i < (int)n; i++) {
if(match_l[i] >= 0) ret.emplace_back(i, match_l[i]);
}
return ret;
}
/* http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=3198 */
void erase_edge(int a, int b) {
if(match_l[a] == b) {
match_l[a] = -1;
match_r[b] = -1;
}
g[a].erase(find(begin(g[a]), end(g[a]), b));
rg[b].erase(find(begin(rg[b]), end(rg[b]), a));
}
/* http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=0334 */
vector< pair< int, int > > lex_max_matching() {
if(!matched) max_matching();
for(auto &vs : g) sort(begin(vs), end(vs));
vector< pair< int, int > > es;
for(int i = 0; i < (int)n; i++) {
if(match_l[i] == -1 || alive[i] == 0) {
continue;
}
match_r[match_l[i]] = -1;
match_l[i] = -1;
++time_stamp;
find_augment_path(i);
alive[i] = 0;
es.emplace_back(i, match_l[i]);
}
return es;
}
vector< int > min_vertex_cover() {
auto visited = find_residual_path();
vector< int > ret;
for(int i = 0; i < (int)(n + m); i++) {
if(visited[i] ^ (i < (int)n)) {
ret.emplace_back(i);
}
}
return ret;
}
/* https://atcoder.jp/contests/utpc2013/tasks/utpc2013_11 */
vector< int > lex_min_vertex_cover(const vector< int > &ord) {
assert(ord.size() == n + m);
auto res = build_risidual_graph();
vector< vector< int > > r_res(n + m + 2);
for(int i = 0; i < (int)(n + m + 2); i++) {
for(auto &j : res[i]) r_res[j].emplace_back(i);
}
queue< int > que;
vector< int > visited(n + m + 2, -1);
auto expand_left = [&](int t) {
if(visited[t] != -1) return;
que.emplace(t);
visited[t] = 1;
while(!que.empty()) {
int idx = que.front();
que.pop();
for(auto &to : r_res[idx]) {
if(visited[to] != -1) continue;
visited[to] = 1;
que.emplace(to);
}
}
};
auto expand_right = [&](int t) {
if(visited[t] != -1) return;
que.emplace(t);
visited[t] = 0;
while(!que.empty()) {
int idx = que.front();
que.pop();
for(auto &to : res[idx]) {
if(visited[to] != -1) continue;
visited[to] = 0;
que.emplace(to);
}
}
};
expand_right(n + m);
expand_left(n + m + 1);
vector< int > ret;
for(auto &t : ord) {
if(t < (int)n) {
expand_left(t);
if(visited[t] & 1) ret.emplace_back(t);
} else {
expand_right(t);
if(~visited[t] & 1) ret.emplace_back(t);
}
}
return ret;
}
vector< int > max_independent_set() {
auto visited = find_residual_path();
vector< int > ret;
for(int i = 0; i < (int)(n + m); i++) {
if(visited[i] ^ (i >= (int)n)) {
ret.emplace_back(i);
}
}
return ret;
}
vector< pair< int, int > > min_edge_cover() {
auto es = max_matching();
for(int i = 0; i < (int)n; i++) {
if(match_l[i] >= 0) {
continue;
}
if(g[i].empty()) {
return {};
}
es.emplace_back(i, g[i][0]);
}
for(int i = 0; i < (int)m; i++) {
if(match_r[i] >= 0) {
continue;
}
if(rg[i].empty()) {
return {};
}
es.emplace_back(rg[i][0], i);
}
return es;
}
// left: [0,n), right: [n,n+m), S: n+m, T: n+m+1
vector< vector< int > > build_risidual_graph() {
if(!matched) max_matching();
const size_t S = n + m;
const size_t T = n + m + 1;
vector< vector< int > > ris(n + m + 2);
for(int i = 0; i < (int)n; i++) {
if(match_l[i] == -1) ris[S].emplace_back(i);
else ris[i].emplace_back(S);
}
for(int i = 0; i < (int)m; i++) {
if(match_r[i] == -1) ris[i + n].emplace_back(T);
else ris[T].emplace_back(i + n);
}
for(int i = 0; i < (int)n; i++) {
for(auto &j : g[i]) {
if(match_l[i] == j) ris[j + n].emplace_back(i);
else ris[i].emplace_back(j + n);
}
}
return ris;
}
private:
vector< int > find_residual_path() {
auto res = build_risidual_graph();
queue< int > que;
vector< int > visited(n + m + 2);
que.emplace(n + m);
visited[n + m] = true;
while(!que.empty()) {
int idx = que.front();
que.pop();
for(auto &to : res[idx]) {
if(visited[to]) continue;
visited[to] = true;
que.emplace(to);
}
}
return visited;
}
void build_augment_path() {
queue< int > que;
dist.assign(g.size(), -1);
for(int i = 0; i < (int)n; i++) {
if(match_l[i] == -1) {
que.emplace(i);
dist[i] = 0;
}
}
while(!que.empty()) {
int a = que.front();
que.pop();
for(auto &b : g[a]) {
int c = match_r[b];
if(c >= 0 && dist[c] == -1) {
dist[c] = dist[a] + 1;
que.emplace(c);
}
}
}
}
bool find_min_dist_augment_path(int a) {
used[a] = time_stamp;
for(auto &b : g[a]) {
int c = match_r[b];
if(c < 0 || (used[c] != (int)time_stamp && dist[c] == dist[a] + 1 && find_min_dist_augment_path(c))) {
match_r[b] = a;
match_l[a] = b;
return true;
}
}
return false;
}
bool find_augment_path(int a) {
used[a] = time_stamp;
for(auto &b : g[a]) {
int c = match_r[b];
if(c < 0 || (alive[c] == 1 && used[c] != (int)time_stamp && find_augment_path(c))) {
match_r[b] = a;
match_l[a] = b;
return true;
}
}
return false;
}
};
#line 2 "graph/others/eulerian-trail.hpp"
#line 4 "graph/others/eulerian-trail.hpp"
/**
* @brief Eulerian Trail(オイラー路)
*
*/
template< bool directed >
struct EulerianTrail {
vector< vector< pair< int, int > > > g;
vector< pair< int, int > > es;
int M;
vector< int > used_vertex, used_edge, deg;
explicit EulerianTrail(int V) : g(V), M(0), used_vertex(V), deg(V) {}
void add_edge(int a, int b) {
es.emplace_back(a, b);
g[a].emplace_back(b, M);
if(directed) {
deg[a]++;
deg[b]--;
} else {
g[b].emplace_back(a, M);
deg[a]++;
deg[b]++;
}
M++;
}
pair< int, int > get_edge(int idx) const {
return es[idx];
}
vector< vector< int > > enumerate_eulerian_trail() {
if(directed) {
for(auto &p : deg) if(p != 0) return {};
} else {
for(auto &p : deg) if(p & 1) return {};
}
used_edge.assign(M, 0);
vector< vector< int > > ret;
for(int i = 0; i < (int) g.size(); i++) {
if(g[i].empty() || used_vertex[i]) continue;
ret.emplace_back(go(i));
}
return ret;
}
vector< vector< int > > enumerate_semi_eulerian_trail() {
UnionFind uf(g.size());
for(auto &p : es) uf.unite(p.first, p.second);
vector< vector< int > > group(g.size());
for(int i = 0; i < (int) g.size(); i++) group[uf.find(i)].emplace_back(i);
vector< vector< int > > ret;
used_edge.assign(M, 0);
for(auto &vs : group) {
if(vs.empty()) continue;
int latte = -1, malta = -1;
if(directed) {
for(auto &p : vs) {
if(abs(deg[p]) > 1) {
return {};
} else if(deg[p] == 1) {
if(latte >= 0) return {};
latte = p;
}
}
} else {
for(auto &p : vs) {
if(deg[p] & 1) {
if(latte == -1) latte = p;
else if(malta == -1) malta = p;
else return {};
}
}
}
ret.emplace_back(go(latte == -1 ? vs.front() : latte));
if(ret.back().empty()) ret.pop_back();
}
return ret;
}
vector< int > go(int s) {
stack< pair< int, int > > st;
vector< int > ord;
st.emplace(s, -1);
while(!st.empty()) {
int idx = st.top().first;
used_vertex[idx] = true;
if(g[idx].empty()) {
ord.emplace_back(st.top().second);
st.pop();
} else {
auto e = g[idx].back();
g[idx].pop_back();
if(used_edge[e.second]) continue;
used_edge[e.second] = true;
st.emplace(e);
}
}
ord.pop_back();
reverse(ord.begin(), ord.end());
return ord;
}
};
#line 6 "graph/others/bipartite-graph-edge-coloring.hpp"
/**
* @brief Bipartite Graph Edge Coloring(二部グラフの辺彩色)
*
* @see https://ei1333.hateblo.jp/entry/2020/08/25/015955
*/
struct BipariteGraphEdgeColoring {
private:
vector< vector< int > > ans;
vector< int > A, B;
int L, R;
struct RegularGraph {
int k{}, n{};
vector< int > A, B;
};
RegularGraph g;
static UnionFind contract(valarray< int > °, int k) {
using pi = pair< int, int >;
priority_queue< pi, vector< pi >, greater<> > que;
for(int i = 0; i < (int) deg.size(); i++) {
que.emplace(deg[i], i);
}
UnionFind uf(deg.size());
while(que.size() > 1) {
auto p = que.top();
que.pop();
auto q = que.top();
que.pop();
if(p.first + q.first > k) continue;
p.first += q.first;
uf.unite(p.second, q.second);
que.emplace(p);
}
return uf;
}
RegularGraph build_k_regular_graph() {
valarray< int > deg[2];
deg[0] = valarray< int >(L);
deg[1] = valarray< int >(R);
for(auto &p : A) deg[0][p]++;
for(auto &p : B) deg[1][p]++;
int k = max(deg[0].max(), deg[1].max());
/* step 1 */
UnionFind uf[2];
uf[0] = contract(deg[0], k);
uf[1] = contract(deg[1], k);
vector< int > id[2];
int ptr[] = {0, 0};
id[0] = vector< int >(L);
id[1] = vector< int >(R);
for(int i = 0; i < L; i++) if(uf[0].find(i) == i) id[0][i] = ptr[0]++;
for(int i = 0; i < R; i++) if(uf[1].find(i) == i) id[1][i] = ptr[1]++;
/* step 2 */
int N = max(ptr[0], ptr[1]);
deg[0] = valarray< int >(N);
deg[1] = valarray< int >(N);
/* step 3 */
vector< int > C, D;
C.reserve(N * k);
D.reserve(N * k);
for(int i = 0; i < (int) A.size(); i++) {
int u = id[0][uf[0].find(A[i])];
int v = id[1][uf[1].find(B[i])];
C.emplace_back(u);
D.emplace_back(v);
deg[0][u]++;
deg[1][v]++;
}
int j = 0;
for(int i = 0; i < N; i++) {
while(deg[0][i] < k) {
while(deg[1][j] == k) ++j;
C.emplace_back(i);
D.emplace_back(j);
++deg[0][i];
++deg[1][j];
}
}
return {k, N, C, D};
}
void rec(const vector< int > &ord, int k) {
if(k == 0) {
return;
} else if(k == 1) {
ans.emplace_back(ord);
return;
} else if((k & 1) == 0) {
EulerianTrail< false > et(g.n + g.n);
for(auto &p : ord) et.add_edge(g.A[p], g.B[p] + g.n);
auto paths = et.enumerate_eulerian_trail();
vector< int > path;
for(auto &ps : paths) {
for(auto &e : ps) path.emplace_back(ord[e]);
}
vector< int > beet[2];
for(int i = 0; i < (int) path.size(); i++) {
beet[i & 1].emplace_back(path[i]);
}
rec(beet[0], k / 2);
rec(beet[1], k / 2);
} else {
BipartiteFlow flow(g.n, g.n);
for(auto &i : ord) flow.add_edge(g.A[i], g.B[i]);
flow.max_matching();
vector< int > beet;
ans.emplace_back();
for(auto &i : ord) {
if(flow.match_l[g.A[i]] == g.B[i]) {
flow.match_l[g.A[i]] = -1;
ans.back().emplace_back(i);
} else {
beet.emplace_back(i);
}
}
rec(beet, k - 1);
}
}
public:
explicit BipariteGraphEdgeColoring() : L(0), R(0) {}
void add_edge(int a, int b) {
A.emplace_back(a);
B.emplace_back(b);
L = max(L, a + 1);
R = max(R, b + 1);
}
vector< vector< int > > build() {
g = build_k_regular_graph();
vector< int > ord(g.A.size());
iota(ord.begin(), ord.end(), 0);
rec(ord, g.k);
vector< vector< int > > res;
for(int i = 0; i < (int) ans.size(); i++) {
res.emplace_back();
for(auto &j : ans[i]) if(j < (int)A.size()) res.back().emplace_back(j);
}
return res;
}
};