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graph/dag_path_cover.hpp
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- Last update: 2023-11-07 22:29:27+09:00
- Include:
#include "graph/dag_path_cover.hpp"
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Code
#include "graph/base.hpp"
#include "flow/maxflow.hpp"
#include "ds/unionfind/unionfind.hpp"
// 各頂点の色をかえす。各色はひとつのパス上にあるようにする
template <typename DAG>
vc<int> dag_path_cover(DAG& G) {
static_assert(DAG::is_directed);
for (auto&& e: G.edges) assert(e.frm < e.to);
int N = G.N;
int source = 2 * N, sink = 2 * N + 1;
MaxFlow<int> F(2 * N + 2, source, sink);
FOR(v, N) {
F.add(source, 2 * v + 1, 1);
F.add(2 * v + 0, sink, 1);
F.add(2 * v + 0, 2 * v + 1, infty<int>);
}
for (auto&& e: G.edges) F.add(2 * e.frm + 1, 2 * e.to + 0, infty<int>);
F.flow();
auto paths = F.path_decomposition();
UnionFind uf(N);
for (auto& P: paths) {
int a = P[1], b = P[len(P) - 2];
uf.merge(a / 2, b / 2);
}
vc<int> ANS(N, -1);
int p = 0;
FOR(v, N) if (uf[v] == v) ANS[v] = p++;
FOR(v, N) if (uf[v] != v) ANS[v] = ANS[uf[v]];
return ANS;
};#line 2 "graph/base.hpp"
template <typename T>
struct Edge {
int frm, to;
T cost;
int id;
};
template <typename T = int, bool directed = false>
struct Graph {
static constexpr bool is_directed = directed;
int N, M;
using cost_type = T;
using edge_type = Edge<T>;
vector<edge_type> edges;
vector<int> indptr;
vector<edge_type> csr_edges;
vc<int> vc_deg, vc_indeg, vc_outdeg;
bool prepared;
class OutgoingEdges {
public:
OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}
const edge_type* begin() const {
if (l == r) { return 0; }
return &G->csr_edges[l];
}
const edge_type* end() const {
if (l == r) { return 0; }
return &G->csr_edges[r];
}
private:
const Graph* G;
int l, r;
};
bool is_prepared() { return prepared; }
Graph() : N(0), M(0), prepared(0) {}
Graph(int N) : N(N), M(0), prepared(0) {}
void build(int n) {
N = n, M = 0;
prepared = 0;
edges.clear();
indptr.clear();
csr_edges.clear();
vc_deg.clear();
vc_indeg.clear();
vc_outdeg.clear();
}
void add(int frm, int to, T cost = 1, int i = -1) {
assert(!prepared);
assert(0 <= frm && 0 <= to && to < N);
if (i == -1) i = M;
auto e = edge_type({frm, to, cost, i});
edges.eb(e);
++M;
}
#ifdef FASTIO
// wt, off
void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }
void read_graph(int M, bool wt = false, int off = 1) {
for (int m = 0; m < M; ++m) {
INT(a, b);
a -= off, b -= off;
if (!wt) {
add(a, b);
} else {
T c;
read(c);
add(a, b, c);
}
}
build();
}
#endif
void build() {
assert(!prepared);
prepared = true;
indptr.assign(N + 1, 0);
for (auto&& e: edges) {
indptr[e.frm + 1]++;
if (!directed) indptr[e.to + 1]++;
}
for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
auto counter = indptr;
csr_edges.resize(indptr.back() + 1);
for (auto&& e: edges) {
csr_edges[counter[e.frm]++] = e;
if (!directed)
csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
}
}
OutgoingEdges operator[](int v) const {
assert(prepared);
return {this, indptr[v], indptr[v + 1]};
}
vc<int> deg_array() {
if (vc_deg.empty()) calc_deg();
return vc_deg;
}
pair<vc<int>, vc<int>> deg_array_inout() {
if (vc_indeg.empty()) calc_deg_inout();
return {vc_indeg, vc_outdeg};
}
int deg(int v) {
if (vc_deg.empty()) calc_deg();
return vc_deg[v];
}
int in_deg(int v) {
if (vc_indeg.empty()) calc_deg_inout();
return vc_indeg[v];
}
int out_deg(int v) {
if (vc_outdeg.empty()) calc_deg_inout();
return vc_outdeg[v];
}
#ifdef FASTIO
void debug() {
print("Graph");
if (!prepared) {
print("frm to cost id");
for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
} else {
print("indptr", indptr);
print("frm to cost id");
FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
}
}
#endif
vc<int> new_idx;
vc<bool> used_e;
// G における頂点 V[i] が、新しいグラフで i になるようにする
// {G, es}
Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
if (len(new_idx) != N) new_idx.assign(N, -1);
if (len(used_e) != M) used_e.assign(M, 0);
int n = len(V);
FOR(i, n) new_idx[V[i]] = i;
Graph<T, directed> G(n);
vc<int> history;
FOR(i, n) {
for (auto&& e: (*this)[V[i]]) {
if (used_e[e.id]) continue;
int a = e.frm, b = e.to;
if (new_idx[a] != -1 && new_idx[b] != -1) {
history.eb(e.id);
used_e[e.id] = 1;
int eid = (keep_eid ? e.id : -1);
G.add(new_idx[a], new_idx[b], e.cost, eid);
}
}
}
FOR(i, n) new_idx[V[i]] = -1;
for (auto&& eid: history) used_e[eid] = 0;
G.build();
return G;
}
private:
void calc_deg() {
assert(vc_deg.empty());
vc_deg.resize(N);
for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
}
void calc_deg_inout() {
assert(vc_indeg.empty());
vc_indeg.resize(N);
vc_outdeg.resize(N);
for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
}
};
#line 1 "flow/maxflow.hpp"
// incremental に辺を追加してよい
template <typename Cap>
struct MaxFlow {
struct Edge {
int to, rev;
Cap cap;
Cap flow = 0;
};
const int N, source, sink;
vvc<Edge> edges;
vc<int> prog, level;
vc<int> que;
bool calculated;
Cap flow_ans;
MaxFlow(int N, int source, int sink)
: N(N),
source(source),
sink(sink),
edges(N),
calculated(0),
flow_ans(0) {}
void add(int frm, int to, Cap cap, Cap rev_cap = 0) {
calculated = 0;
assert(0 <= frm && frm < N);
assert(0 <= to && to < N);
assert(frm != to);
assert(Cap(0) <= cap);
if (frm == to) return;
int a = len(edges[frm]);
int b = len(edges[to]);
edges[frm].eb(Edge{to, b, cap, 0});
edges[to].eb(Edge{frm, a, rev_cap, 0});
}
// frm, to, flow
vc<tuple<int, int, Cap>> get_flow_edges() {
vc<tuple<int, int, Cap>> res;
FOR(frm, N) {
for (auto&& e: edges[frm]) {
if (e.flow <= 0) continue;
res.eb(frm, e.to, e.flow);
}
}
return res;
}
// 差分ではなくこれまでの総量
Cap flow() {
if (calculated) return flow_ans;
calculated = true;
while (set_level()) {
prog.assign(N, 0);
while (1) {
Cap x = flow_dfs(source, infty<Cap>);
if (x == 0) break;
flow_ans += x;
chmin(flow_ans, infty<Cap>);
if (flow_ans == infty<Cap>) return flow_ans;
}
}
return flow_ans;
}
// 最小カットの値および、カットを表す 01 列を返す
pair<Cap, vc<int>> cut() {
flow();
vc<int> res(N);
FOR(v, N) res[v] = (level[v] >= 0 ? 0 : 1);
return {flow_ans, res};
}
// O(F(N+M)) くらい使って経路復元
// simple path になる
vvc<int> path_decomposition() {
flow();
auto edges = get_flow_edges();
vvc<int> TO(N);
for (auto&& [frm, to, flow]: edges) { FOR(flow) TO[frm].eb(to); }
vvc<int> res;
vc<int> vis(N);
FOR(flow_ans) {
vc<int> path = {source};
vis[source] = 1;
while (path.back() != sink) {
int to = POP(TO[path.back()]);
while (vis[to]) { vis[POP(path)] = 0; }
path.eb(to), vis[to] = 1;
}
for (auto&& v: path) vis[v] = 0;
res.eb(path);
}
return res;
}
void debug() {
print("source", source);
print("sink", sink);
print("edges (frm, to, cap, flow)");
FOR(v, N) {
for (auto& e: edges[v]) {
if (e.cap == 0 && e.flow == 0) continue;
print(v, e.to, e.cap, e.flow);
}
}
}
private:
bool set_level() {
que.resize(N);
level.assign(N, -1);
level[source] = 0;
int l = 0, r = 0;
que[r++] = source;
while (l < r) {
int v = que[l++];
for (auto&& e: edges[v]) {
if (e.cap > 0 && level[e.to] == -1) {
level[e.to] = level[v] + 1;
if (e.to == sink) return true;
que[r++] = e.to;
}
}
}
return false;
}
Cap flow_dfs(int v, Cap lim) {
if (v == sink) return lim;
Cap res = 0;
for (int& i = prog[v]; i < len(edges[v]); ++i) {
auto& e = edges[v][i];
if (e.cap > 0 && level[e.to] == level[v] + 1) {
Cap a = flow_dfs(e.to, min(lim, e.cap));
if (a > 0) {
e.cap -= a, e.flow += a;
edges[e.to][e.rev].cap += a, edges[e.to][e.rev].flow -= a;
res += a;
lim -= a;
if (lim == 0) break;
}
}
}
return res;
}
};
#line 2 "ds/unionfind/unionfind.hpp"
struct UnionFind {
int n, n_comp;
vc<int> dat; // par or (-size)
UnionFind(int n = 0) { build(n); }
void build(int m) {
n = m, n_comp = m;
dat.assign(n, -1);
}
void reset() { build(n); }
int operator[](int x) {
while (dat[x] >= 0) {
int pp = dat[dat[x]];
if (pp < 0) { return dat[x]; }
x = dat[x] = pp;
}
return x;
}
ll size(int x) {
x = (*this)[x];
return -dat[x];
}
bool merge(int x, int y) {
x = (*this)[x], y = (*this)[y];
if (x == y) return false;
if (-dat[x] < -dat[y]) swap(x, y);
dat[x] += dat[y], dat[y] = x, n_comp--;
return true;
}
};
#line 4 "graph/dag_path_cover.hpp"
// 各頂点の色をかえす。各色はひとつのパス上にあるようにする
template <typename DAG>
vc<int> dag_path_cover(DAG& G) {
static_assert(DAG::is_directed);
for (auto&& e: G.edges) assert(e.frm < e.to);
int N = G.N;
int source = 2 * N, sink = 2 * N + 1;
MaxFlow<int> F(2 * N + 2, source, sink);
FOR(v, N) {
F.add(source, 2 * v + 1, 1);
F.add(2 * v + 0, sink, 1);
F.add(2 * v + 0, 2 * v + 1, infty<int>);
}
for (auto&& e: G.edges) F.add(2 * e.frm + 1, 2 * e.to + 0, infty<int>);
F.flow();
auto paths = F.path_decomposition();
UnionFind uf(N);
for (auto& P: paths) {
int a = P[1], b = P[len(P) - 2];
uf.merge(a / 2, b / 2);
}
vc<int> ANS(N, -1);
int p = 0;
FOR(v, N) if (uf[v] == v) ANS[v] = p++;
FOR(v, N) if (uf[v] != v) ANS[v] = ANS[uf[v]];
return ANS;
};