cplib-cpp
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combinatorial_opt/test/matroid_intersection.aoj_grl_2_b.test.cpp
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- Last update: 2023-07-23 15:18:37+09:00
- Problem: https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=GRL_2_B
Depends on
- (Weighted) matroid intersection （（重みつき）マトロイド交叉）
(combinatorial_opt/matroid_intersection.hpp)
- Graphic matroid （グラフマトロイド）
(combinatorial_opt/matroids/graphic_matroid.hpp)
- Partition matroid （分割マトロイド）
(combinatorial_opt/matroids/partition_matroid.hpp)
- Shortest Path （単一始点最短路）
(graph/shortest_path.hpp)
Code
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=GRL_2_B"
#include "../matroid_intersection.hpp"
#include "../matroids/graphic_matroid.hpp"
#include "../matroids/partition_matroid.hpp"
#include <iostream>
#include <utility>
#include <vector>
using namespace std;
int main() {
int V, E, r;
cin >> V >> E >> r;
vector<vector<int>> partition(V);
vector<int> R(V, 1);
R[r] = 0;
vector<pair<int, int>> edges;
vector<int> weights;
for (int e = 0; e < E; ++e) {
int s, t, w;
cin >> s >> t >> w;
partition[t].push_back(e);
edges.emplace_back(s, t);
weights.emplace_back(w);
}
PartitionMatroid M1(E, partition, R);
GraphicMatroid M2(V, edges);
auto sol = MatroidIntersection(M1, M2, weights);
int ret = 0;
for (int e = 0; e < E; ++e) ret += sol[e] * weights[e];
cout << ret << '\n';
}#line 1 "combinatorial_opt/test/matroid_intersection.aoj_grl_2_b.test.cpp"
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=GRL_2_B"
#line 2 "graph/shortest_path.hpp"
#include <algorithm>
#include <cassert>
#include <deque>
#include <fstream>
#include <functional>
#include <limits>
#include <queue>
#include <string>
#include <tuple>
#include <utility>
#include <vector>
template <typename T, T INF = std::numeric_limits<T>::max() / 2, int INVALID = -1>
struct shortest_path {
int V, E;
bool single_positive_weight;
T wmin, wmax;
std::vector<std::pair<int, T>> tos;
std::vector<int> head;
std::vector<std::tuple<int, int, T>> edges;
void build_() {
if (int(tos.size()) == E and int(head.size()) == V + 1) return;
tos.resize(E);
head.assign(V + 1, 0);
for (const auto &e : edges) ++head[std::get<0>(e) + 1];
for (int i = 0; i < V; ++i) head[i + 1] += head[i];
auto cur = head;
for (const auto &e : edges) {
tos[cur[std::get<0>(e)]++] = std::make_pair(std::get<1>(e), std::get<2>(e));
}
}
shortest_path(int V = 0) : V(V), E(0), single_positive_weight(true), wmin(0), wmax(0) {}
void add_edge(int s, int t, T w) {
assert(0 <= s and s < V);
assert(0 <= t and t < V);
edges.emplace_back(s, t, w);
++E;
if (w > 0 and wmax > 0 and wmax != w) single_positive_weight = false;
wmin = std::min(wmin, w);
wmax = std::max(wmax, w);
}
void add_bi_edge(int u, int v, T w) {
add_edge(u, v, w);
add_edge(v, u, w);
}
std::vector<T> dist;
std::vector<int> prev;
// Dijkstra algorithm
// - Requirement: wmin >= 0
// - Complexity: O(E log E)
using Pque = std::priority_queue<std::pair<T, int>, std::vector<std::pair<T, int>>,
std::greater<std::pair<T, int>>>;
template <class Heap = Pque> void dijkstra(int s, int t = INVALID) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF);
prev.assign(V, INVALID);
dist[s] = 0;
Heap pq;
pq.emplace(0, s);
while (!pq.empty()) {
T d;
int v;
std::tie(d, v) = pq.top();
pq.pop();
if (t == v) return;
if (dist[v] < d) continue;
for (int e = head[v]; e < head[v + 1]; ++e) {
const auto &nx = tos[e];
T dnx = d + nx.second;
if (dist[nx.first] > dnx) {
dist[nx.first] = dnx, prev[nx.first] = v;
pq.emplace(dnx, nx.first);
}
}
}
}
// Dijkstra algorithm
// - Requirement: wmin >= 0
// - Complexity: O(V^2 + E)
void dijkstra_vquad(int s, int t = INVALID) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF);
prev.assign(V, INVALID);
dist[s] = 0;
std::vector<char> fixed(V, false);
while (true) {
int r = INVALID;
T dr = INF;
for (int i = 0; i < V; i++) {
if (!fixed[i] and dist[i] < dr) r = i, dr = dist[i];
}
if (r == INVALID or r == t) break;
fixed[r] = true;
int nxt;
T dx;
for (int e = head[r]; e < head[r + 1]; ++e) {
std::tie(nxt, dx) = tos[e];
if (dist[nxt] > dist[r] + dx) dist[nxt] = dist[r] + dx, prev[nxt] = r;
}
}
}
// Bellman-Ford algorithm
// - Requirement: no negative loop
// - Complexity: O(VE)
bool bellman_ford(int s, int nb_loop) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF), prev.assign(V, INVALID);
dist[s] = 0;
for (int l = 0; l < nb_loop; l++) {
bool upd = false;
for (int v = 0; v < V; v++) {
if (dist[v] == INF) continue;
for (int e = head[v]; e < head[v + 1]; ++e) {
const auto &nx = tos[e];
T dnx = dist[v] + nx.second;
if (dist[nx.first] > dnx) dist[nx.first] = dnx, prev[nx.first] = v, upd = true;
}
}
if (!upd) return true;
}
return false;
}
// Bellman-ford algorithm using deque
// - Requirement: no negative loop
// - Complexity: O(VE)
void spfa(int s) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF);
prev.assign(V, INVALID);
dist[s] = 0;
std::deque<int> q;
std::vector<char> in_queue(V);
q.push_back(s), in_queue[s] = 1;
while (!q.empty()) {
int now = q.front();
q.pop_front(), in_queue[now] = 0;
for (int e = head[now]; e < head[now + 1]; ++e) {
const auto &nx = tos[e];
T dnx = dist[now] + nx.second;
int nxt = nx.first;
if (dist[nxt] > dnx) {
dist[nxt] = dnx;
if (!in_queue[nxt]) {
if (q.size() and dnx < dist[q.front()]) { // Small label first optimization
q.push_front(nxt);
} else {
q.push_back(nxt);
}
prev[nxt] = now, in_queue[nxt] = 1;
}
}
}
}
}
// 01-BFS
// - Requirement: all weights must be 0 or w (positive constant).
// - Complexity: O(V + E)
void zero_one_bfs(int s, int t = INVALID) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF), prev.assign(V, INVALID);
dist[s] = 0;
std::vector<int> q(V * 4);
int ql = V * 2, qr = V * 2;
q[qr++] = s;
while (ql < qr) {
int v = q[ql++];
if (v == t) return;
for (int e = head[v]; e < head[v + 1]; ++e) {
const auto &nx = tos[e];
T dnx = dist[v] + nx.second;
if (dist[nx.first] > dnx) {
dist[nx.first] = dnx, prev[nx.first] = v;
if (nx.second) {
q[qr++] = nx.first;
} else {
q[--ql] = nx.first;
}
}
}
}
}
// Dial's algorithm
// - Requirement: wmin >= 0
// - Complexity: O(wmax * V + E)
void dial(int s, int t = INVALID) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF), prev.assign(V, INVALID);
dist[s] = 0;
std::vector<std::vector<std::pair<int, T>>> q(wmax + 1);
q[0].emplace_back(s, dist[s]);
int ninq = 1;
int cur = 0;
T dcur = 0;
for (; ninq; ++cur, ++dcur) {
if (cur == wmax + 1) cur = 0;
while (!q[cur].empty()) {
int v = q[cur].back().first;
T dnow = q[cur].back().second;
q[cur].pop_back(), --ninq;
if (v == t) return;
if (dist[v] < dnow) continue;
for (int e = head[v]; e < head[v + 1]; ++e) {
const auto &nx = tos[e];
T dnx = dist[v] + nx.second;
if (dist[nx.first] > dnx) {
dist[nx.first] = dnx, prev[nx.first] = v;
int nxtcur = cur + int(nx.second);
if (nxtcur >= int(q.size())) nxtcur -= q.size();
q[nxtcur].emplace_back(nx.first, dnx), ++ninq;
}
}
}
}
}
// Solver for DAG
// - Requirement: graph is DAG
// - Complexity: O(V + E)
bool dag_solver(int s) {
assert(0 <= s and s < V);
build_();
dist.assign(V, INF), prev.assign(V, INVALID);
dist[s] = 0;
std::vector<int> indeg(V, 0);
std::vector<int> q(V * 2);
int ql = 0, qr = 0;
q[qr++] = s;
while (ql < qr) {
int now = q[ql++];
for (int e = head[now]; e < head[now + 1]; ++e) {
const auto &nx = tos[e];
++indeg[nx.first];
if (indeg[nx.first] == 1) q[qr++] = nx.first;
}
}
ql = qr = 0;
q[qr++] = s;
while (ql < qr) {
int now = q[ql++];
for (int e = head[now]; e < head[now + 1]; ++e) {
const auto &nx = tos[e];
--indeg[nx.first];
if (dist[nx.first] > dist[now] + nx.second)
dist[nx.first] = dist[now] + nx.second, prev[nx.first] = now;
if (indeg[nx.first] == 0) q[qr++] = nx.first;
}
}
return *max_element(indeg.begin(), indeg.end()) == 0;
}
// Retrieve a sequence of vertex ids that represents shortest path [s, ..., goal]
// If not reachable to goal, return {}
std::vector<int> retrieve_path(int goal) const {
assert(int(prev.size()) == V);
assert(0 <= goal and goal < V);
if (dist[goal] == INF) return {};
std::vector<int> ret{goal};
while (prev[goal] != INVALID) {
goal = prev[goal];
ret.push_back(goal);
}
std::reverse(ret.begin(), ret.end());
return ret;
}
void solve(int s, int t = INVALID) {
if (wmin >= 0) {
if (single_positive_weight) {
zero_one_bfs(s, t);
} else if (wmax <= 10) {
dial(s, t);
} else {
if ((long long)V * V < (E << 4)) {
dijkstra_vquad(s, t);
} else {
dijkstra(s, t);
}
}
} else {
bellman_ford(s, V);
}
}
// Warshall-Floyd algorithm
// - Requirement: no negative loop
// - Complexity: O(E + V^3)
std::vector<std::vector<T>> floyd_warshall() {
build_();
std::vector<std::vector<T>> dist2d(V, std::vector<T>(V, INF));
for (int i = 0; i < V; i++) {
dist2d[i][i] = 0;
for (const auto &e : edges) {
int s = std::get<0>(e), t = std::get<1>(e);
dist2d[s][t] = std::min(dist2d[s][t], std::get<2>(e));
}
}
for (int k = 0; k < V; k++) {
for (int i = 0; i < V; i++) {
if (dist2d[i][k] == INF) continue;
for (int j = 0; j < V; j++) {
if (dist2d[k][j] == INF) continue;
dist2d[i][j] = std::min(dist2d[i][j], dist2d[i][k] + dist2d[k][j]);
}
}
}
return dist2d;
}
void to_dot(std::string filename = "shortest_path") const {
std::ofstream ss(filename + ".DOT");
ss << "digraph{\n";
build_();
for (int i = 0; i < V; i++) {
for (int e = head[i]; e < head[i + 1]; ++e) {
ss << i << "->" << tos[e].first << "[label=" << tos[e].second << "];\n";
}
}
ss << "}\n";
ss.close();
return;
}
};
#line 5 "combinatorial_opt/matroid_intersection.hpp"
// Find minimum weight augmenting path of matroid intersection.
// m1, m2: matroids
// I: independent set (will be updated if augmenting path is found)
//
// Return `true` iff augmenting path is found.
// Complexity: O(Cn + n^2) (C: circuit query)
template <class Matroid1, class Matroid2, class T = int>
bool matroid_intersection_augment(Matroid1 &m1, Matroid2 &m2, std::vector<bool> &I,
const std::vector<T> &weights = {}) {
const int n = m1.size();
assert(m2.size() == n);
assert((int)I.size() == n);
auto weight = [&](int e) { return weights.empty() ? T() : weights.at(e) * (n + 1); };
const int gs = n, gt = n + 1;
shortest_path<T> sssp(n + 2);
m1.set(I);
m2.set(I);
for (int e = 0; e < n; ++e) {
if (I.at(e)) continue;
auto c1 = m1.circuit(e), c2 = m2.circuit(e);
if (c1.empty()) sssp.add_edge(e, gt, T());
for (int f : c1) {
if (f != e) sssp.add_edge(e, f, -weight(f) + T(1));
}
if (c2.empty()) sssp.add_edge(gs, e, weight(e) + T(1));
for (int f : c2) {
if (f != e) sssp.add_edge(f, e, weight(e) + T(1));
}
}
sssp.solve(gs, gt);
if (auto aug_path = sssp.retrieve_path(gt); aug_path.empty()) {
return false;
} else {
for (auto e : aug_path) {
if (e != gs and e != gt) I.at(e) = !I.at(e);
}
return true;
}
}
// Minimum weight matroid intersection solver
// Algorithm based on http://dopal.cs.uec.ac.jp/okamotoy/lect/2015/matroid/
// Complexity: O(Cn^2 + n^3) (C : circuit query, non-weighted)
template <class Matroid1, class Matroid2, class T = int>
std::vector<bool>
MatroidIntersection(Matroid1 matroid1, Matroid2 matroid2, std::vector<T> weights = {}) {
const int n = matroid1.size();
assert(matroid2.size() == n);
assert(weights.empty() or (int) weights.size() == n);
std::vector<bool> I(n);
if (weights.empty()) {
matroid1.set(I);
matroid2.set(I);
for (int e = 0; e < n; ++e) {
if (matroid1.circuit(e).empty() and matroid2.circuit(e).empty()) {
I.at(e) = true;
matroid1.set(I);
matroid2.set(I);
}
}
}
while (matroid_intersection_augment(matroid1, matroid2, I, weights)) {}
return I;
}
#line 5 "combinatorial_opt/matroids/graphic_matroid.hpp"
// GraphicMatroid: subgraph of undirected graphs, without loops
class GraphicMatroid {
using Vertex = int;
using Element = int;
int M;
int V; // # of vertices of graph
std::vector<std::vector<std::pair<Vertex, Element>>> to;
std::vector<std::pair<Vertex, Vertex>> edges;
std::vector<Element> backtrack;
std::vector<Vertex> vprev;
std::vector<int> depth, root;
public:
GraphicMatroid(int V, const std::vector<std::pair<Vertex, Vertex>> &edges_)
: M(edges_.size()), V(V), to(V), edges(edges_) {
for (int e = 0; e < int(edges_.size()); e++) {
int u = edges_[e].first, v = edges_[e].second;
assert(0 <= u and u < V);
assert(0 <= v and v < V);
if (u != v) {
to[u].emplace_back(v, e);
to[v].emplace_back(u, e);
}
}
}
int size() const { return M; }
std::vector<Vertex> que;
template <class State> void set(State I) {
assert(int(I.size()) == M);
backtrack.assign(V, -1);
vprev.assign(V, -1);
depth.assign(V, -1);
root.assign(V, -1);
que.resize(V);
int qb = 0, qe = 0;
for (Vertex i = 0; i < V; i++) {
if (backtrack[i] >= 0) continue;
que[qb = 0] = i, qe = 1, depth[i] = 0;
while (qb < qe) {
Vertex now = que[qb++];
root[now] = i;
for (auto nxt : to[now]) {
if (depth[nxt.first] < 0 and I[nxt.second]) {
backtrack[nxt.first] = nxt.second;
vprev[nxt.first] = now;
depth[nxt.first] = depth[now] + 1;
que[qe++] = nxt.first;
}
}
}
}
}
std::vector<Element> circuit(const Element e) const {
assert(0 <= e and e < M);
Vertex s = edges[e].first, t = edges[e].second;
if (root[s] != root[t]) return {};
std::vector<Element> ret{e};
auto step = [&](Vertex &i) { ret.push_back(backtrack[i]), i = vprev[i]; };
int ddepth = depth[s] - depth[t];
for (; ddepth > 0; --ddepth) step(s);
for (; ddepth < 0; ++ddepth) step(t);
while (s != t) step(s), step(t);
return ret;
}
};
#line 4 "combinatorial_opt/matroids/partition_matroid.hpp"
// Partition matroid (partitional matroid) : direct sum of uniform matroids
class PartitionMatroid {
using Element = int;
int M;
std::vector<std::vector<Element>> parts;
std::vector<int> belong;
std::vector<int> R;
std::vector<int> cnt;
std::vector<std::vector<Element>> circuits;
public:
// parts: partition of [0, 1, ..., M - 1]
// R: only R[i] elements from parts[i] can be chosen for each i.
PartitionMatroid(int M, const std::vector<std::vector<int>> &parts_, const std::vector<int> &R_)
: M(M), parts(parts_), belong(M, -1), R(R_) {
assert(parts.size() == R.size());
for (int i = 0; i < int(parts.size()); i++) {
for (Element e : parts[i]) belong[e] = i;
}
for (Element e = 0; e < M; e++) {
// assert(belong[e] != -1);
if (belong[e] == -1) {
belong[e] = parts.size();
parts.push_back({e});
R.push_back(1);
}
}
}
int size() const { return M; }
template <class State> void set(const State &I) {
cnt = R;
for (int e = 0; e < M; e++) {
if (I[e]) cnt[belong[e]]--;
}
circuits.assign(cnt.size(), {});
for (int e = 0; e < M; e++) {
if (I[e] and cnt[belong[e]] == 0) circuits[belong[e]].push_back(e);
}
}
std::vector<Element> circuit(const Element e) const {
assert(0 <= e and e < M);
int p = belong[e];
if (cnt[p] == 0) {
auto ret = circuits[p];
ret.push_back(e);
return ret;
}
return {};
}
};
#line 5 "combinatorial_opt/test/matroid_intersection.aoj_grl_2_b.test.cpp"
#include <iostream>
#line 8 "combinatorial_opt/test/matroid_intersection.aoj_grl_2_b.test.cpp"
using namespace std;
int main() {
int V, E, r;
cin >> V >> E >> r;
vector<vector<int>> partition(V);
vector<int> R(V, 1);
R[r] = 0;
vector<pair<int, int>> edges;
vector<int> weights;
for (int e = 0; e < E; ++e) {
int s, t, w;
cin >> s >> t >> w;
partition[t].push_back(e);
edges.emplace_back(s, t);
weights.emplace_back(w);
}
PartitionMatroid M1(E, partition, R);
GraphicMatroid M2(V, edges);
auto sol = MatroidIntersection(M1, M2, weights);
int ret = 0;
for (int e = 0; e < E; ++e) ret += sol[e] * weights[e];
cout << ret << '\n';
}