#ifndef LAB1_GRAPH_H
#define LAB1_GRAPH_H
#include <bits/stdc++.h>
using namespace std;
ifstream fin("dfs.in");
ofstream fout("dfs.out");
typedef pair<int, int> pii;
typedef pair<pii, int> ppi;
typedef pair<int, pii> pip;
const int inf = 1 << 30;
class DisjointSet {
private:
int m_n;
vector<int> m_root;
vector<int> m_size;
public:
// Constructors
DisjointSet();
explicit DisjointSet(int n);
// Union & Find
int find(int x);
void unite(int x, int y);
};
DisjointSet::DisjointSet() : m_n(0), m_root(), m_size() {}
DisjointSet::DisjointSet(int n) : m_n(n) {
m_root.resize(m_n + 1);
m_size.resize(m_n + 1);
for (int i = 1; i <= m_n; ++i) {
m_root[i] = i;
m_size[i] = 1;
}
}
int DisjointSet::find(int x) {
int rx = x, aux;
while (rx != m_root[rx]) {
rx = m_root[rx];
}
while (x != m_root[x]) {
aux = m_root[x];
m_root[x] = rx;
x = aux;
}
return rx;
}
void DisjointSet::unite(int x, int y) {
int rx = find(x);
int ry = find(y);
if (m_size[rx] > m_size[ry]) {
m_root[ry] = rx;
m_size[rx] += m_size[ry];
} else {
m_root[rx] = ry;
m_size[ry] += m_size[rx];
}
}
template<typename T>
class Graph {
public:
// Constructors
Graph();
explicit Graph(int n);
Graph(int n, vector<vector<T>> &ad);
// addEdge Methods
void addEdge(int v, int u);
void addEdgeWeight(int v, int u, int weight);
// Find Connected Components
int components();
// Find Minimum Distance from start to Other Vertices
vector<int> minDist(int start);
// Topological Sort
vector<int> getTopologicalSort();
// Find Strongly Connected Components
vector<vector<int>> stronglyConnectedComponents();
// Find Biconnected Components
vector<vector<int>> biconnectedComponents();
// Check if a graph can be built given the degree sequence
static string canBuildGraph(vector<int> &v);
// Optimal Paths
pair<int, vector<int>> getOptimalVertices(int start, int end);
// Minimum Spanning Tree (Kruskal's Algorithm)
pair<int, vector<pii>> minimumSubtree();
// Dijkstra's Algorithm
vector<int> dijkstra();
// Bellman-Ford Algorithm
pair<int, vector<int>> bellmanFord();
// Maximum Flow in a Network
int getMaxFlow();
// Roy-Floyd Algorithm
vector<vector<int>> royFloyd();
// Diameter of a Tree
int diameter();
// Eulerian Cycle
vector<int> eulerianCycle(int edgeCount);
// Minimum Weight Hamiltonian Circuit
int minHamiltonianCircuit();
// Maximum Bipartite Matching
vector<pii> maxBipartiteMatching(int n);
// Critical Connections in a Network (https://leetcode.com/problems/critical-connections-in-a-network/)
vector<vector<int>> criticalConnections(int n, vector<vector<int>> &connections);
private:
int m_n;
vector<vector<T>> m_ad;
vector<pair<int, T>> m_edges;
// Utility Methods
void DFS(int v, vector<bool> &vis);
void topologicalSort(int v, vector<bool> &vis, vector<int> &stack);
vector<vector<int>> getReversedGraph();
void tDFS(int v, vector<vector<int>> &tAd, vector<bool> &vis, int compCount, vector<vector<int>> &comp);
void bccDFS(int v, int parent, vector<int> &disc, vector<int> &low, vector<int> &stack, vector<vector<int>> &comp);
static bool eligible(vector<int> &v);
void findPaths(vector<vector<int>> &paths, vector<int> &path, vector<vector<int>> &parent, int current);
void BFS(vector<vector<int>> &parents, int start);
bool foundPath(vector<vector<int>> &c, vector<vector<int>> &flow, vector<int> &parent);
static bool comp(pip &X, pip &Y);
void treeDFS(int v, int d, int &dMax, int &vMax, vector<bool> &vis);
int foundMatch(int l, vector<bool> &vis, vector<int> &matchL, vector<int> &matchR);
void critDFS(int v, int parent, vector<int> &disc, vector<int> &low, vector<vector<int>> &cc);
vector<vector<int>> getWeightMatrix();
};
// Constructors
template<typename T>
Graph<T>::Graph() : m_n(0), m_ad(), m_edges() {}
template<typename T>
Graph<T>::Graph(int n) : m_n(n), m_edges() {
m_ad.resize(m_n + 1);
}
template<typename T>
Graph<T>::Graph(int n, vector<vector<T>> &ad) :
m_n(n), m_ad(ad), m_edges() {}
// addEdge Methods
template<typename T>
void Graph<T>::addEdge(int v, int u) {
m_ad[v].push_back(u);
m_edges.emplace_back(v, u);
}
template<typename T>
void Graph<T>::addEdgeWeight(int v, int u, int weight) {
m_ad[v].push_back({u, weight});
m_edges.push_back({v, {u, weight}});
}
// Find Connected Components (using Depth-First Search)
template<typename T>
int Graph<T>::components() {
int compCount = 0;
vector<bool> vis(m_n, false);
for (int i = 1; i <= m_n; ++i) {
if (!vis[i]) {
++compCount;
DFS(i, vis);
}
}
return compCount;
}
// Find Minimum Distance from start to Other Vertices (using Breadth-First Search)
template<typename T>
vector<int> Graph<T>::minDist(int start) {
vector<int> dist(m_n + 1, -1);
queue<int> Q;
int x;
dist[start] = 0;
Q.push(start);
while (!Q.empty()) {
x = Q.front(); Q.pop();
for (auto &y : m_ad[x]) {
if (dist[y] == -1) {
dist[y] = dist[x] + 1;
Q.push(y);
}
}
}
return dist;
}
// Topological Sort
template<typename T>
vector<int> Graph<T>::getTopologicalSort() {
vector<bool> vis(m_n + 1, false);
vector<int> stack;
vector<int> ans;
for (int i = 1; i <= m_n; ++i) {
if (!vis[i]) {
topologicalSort(i, vis, stack);
}
}
for (int i = stack.size() - 1; i >= 0; --i) {
ans.push_back(stack[i]);
}
return ans;
}
// Find Strongly Connected Components (Kosaraju's Algorithm)
template<typename T>
vector<vector<int>> Graph<T>::stronglyConnectedComponents() {
vector<vector<int>> tAd = getReversedGraph();
vector<bool> vis1(m_n + 1, false);
vector<bool> vis2(m_n + 1, false);
vector<int> stack;
int compCount = 0;
vector<vector<int>> comp;
for (int i = 1; i <= m_n; ++i) {
if (!vis1[i]) {
topologicalSort(i, vis1, stack);
}
}
for (int i = stack.size() - 1; i >= 0; --i) {
if (!vis2[stack[i]]) {
++compCount;
comp.emplace_back();
tDFS(stack[i], tAd, vis2, compCount, comp);
}
}
return comp;
}
// Find Biconnected Components
template<typename T>
vector<vector<int>> Graph<T>::biconnectedComponents() {
vector<int> disc(m_n + 1, 0);
vector<int> low(m_n + 1, 0);
vector<int> stack;
vector<vector<int>> comp;
bccDFS(1, 0, disc, low, stack, comp);
return comp;
}
// Check if a graph can be built given the degree sequence (Havel-Hakimi Algorithm)
template<typename T>
string Graph<T>::canBuildGraph(vector<int> &v) {
if (!eligible(v)) {
return "No";
}
sort(v.begin(), v.end(), greater<>());
while (v[0]) {
for (int i = 1; i <= v[0]; ++i) {
if (v[i] == 0) {
return "No";
}
--v[i];
}
v.erase(v.begin());
sort(v.begin(), v.end(), greater<>());
}
return "Yes";
}
// Optimal Paths
template<typename T>
pair<int, vector<int>> Graph<T>::getOptimalVertices(int start, int end) {
vector<vector<int>> paths;
vector<int> path;
vector<vector<int>> parents(m_n + 1);
BFS(parents, start);
findPaths(paths, path, parents, end);
vector<int> check(m_n + 1, 0);
auto pathCount = paths.size();
int count = 0;
for (auto &p : paths) {
for (auto &v : p) {
++check[v];
}
}
for (int i = 1; i <= m_n; ++i) {
if (check[i] == pathCount) {
++count;
}
}
vector<int> sol;
for (int i = 1; i <= m_n; ++i) {
if (check[i] == pathCount) {
sol.push_back(i);
}
}
return {count, sol};
}
// Minimum Spanning Tree (Kruskal's Algorithm)
template<typename T>
pair<int, vector<pii>> Graph<T>::minimumSubtree() {
int totalCost = 0;
DisjointSet S(m_n);
vector<pair<int, int>> sol;
sort(m_edges.begin(), m_edges.end(), comp);
for (auto &edge : m_edges) {
if (sol.size() == m_n - 1) {
break;
}
int x = S.find(edge.first);
int y = S.find(edge.second.first);
if (x != y) {
S.unite(x, y);
totalCost += edge.second.second;
sol.emplace_back(edge.first, edge.second.first);
}
}
return {totalCost, sol};
}
// Dijkstra's Algorithm
template<typename T>
vector<int> Graph<T>::dijkstra() {
vector<int> dist(m_n + 1, inf);
vector<bool> vis(m_n + 1, false);
priority_queue<pii, vector<pii>, greater<>> heap;
dist[1] = 0;
heap.push({0, 1});
while (!heap.empty()) {
int current = heap.top().second;
heap.pop();
if (!vis[current]) {
vis[current] = true;
for (auto &w : m_ad[current]) {
if (dist[current] + w.second < dist[w.first]) {
dist[w.first] = dist[current] + w.second;
heap.push({dist[w.first], w.first});
}
}
}
}
return dist;
}
// Bellman-Ford Algorithm
template<typename T>
pair<int, vector<int>> Graph<T>::bellmanFord() {
vector<int> dist(m_n + 1, inf);
vector<bool> inHeap(m_n + 1, false);
vector<int> count(m_n + 1, 0);
priority_queue<pii, vector<pii>, greater<>> heap;
bool foundCycle = false;
dist[1] = 0;
inHeap[1] = true;
heap.push({0, 1});
while (!heap.empty() && !foundCycle) {
int current = heap.top().second;
heap.pop();
inHeap[current] = false;
for (auto &w : m_ad[current]) {
if (dist[current] + w.second < dist[w.first]) {
if (!inHeap[w.first]) {
++count[w.first];
inHeap[w.first] = true;
heap.push({dist[w.first], w.first});
if (count[w.first] > m_n) {
foundCycle = true;
}
}
dist[w.first] = dist[current] + w.second;
}
}
}
return {foundCycle, dist};
}
// Maximum Flow in a Network (Edmonds-Karp Algorithm)
template<typename T>
int Graph<T>::getMaxFlow() {
int maxFlow = 0;
auto c = getWeightMatrix();
vector<int> parent(m_n + 1);
vector<vector<int>> flow(m_n + 1);
for (int i = 1; i <= m_n; ++i) {
flow[i].resize(m_n + 1, 0);
}
while (foundPath(c, flow, parent)) {
for (auto &p : m_ad[m_n]) {
auto w = p.first;
if (c[w][m_n] == flow[w][m_n] || !parent[w]) {
continue;
}
int minFlow = c[w][m_n] - flow[w][m_n];
for (int v = w; v != 1; v = parent[v]) {
minFlow = min(minFlow, c[parent[v]][v] - flow[parent[v]][v]);
}
if (minFlow == 0) {
continue;
}
flow[w][m_n] += minFlow;
flow[m_n][w] -= minFlow;
for (int v = w; v != 1; v = parent[v]) {
flow[parent[v]][v] += minFlow;
flow[v][parent[v]] -= minFlow;
}
maxFlow += minFlow;
}
}
return maxFlow;
}
// Roy-Floyd Algorithm
template<typename T>
vector<vector<int>> Graph<T>::royFloyd() {
int i, j, k;
for (k = 1; k <= m_n; ++k) {
for (i = 1; i <= m_n; ++i) {
for (j = 1; j <= m_n; ++j) {
m_ad[i][j] = min(m_ad[i][j], m_ad[i][k] + m_ad[k][j]);
}
}
}
return m_ad;
}
// Diameter of a Tree
template<typename T>
int Graph<T>::diameter() {
int dMax = 1, vMax;
vector<bool> vis(m_n + 1, false);
treeDFS(1, 1, dMax, vMax, vis);
vis.assign(m_n + 1, false);
dMax = 1;
treeDFS(vMax, 1, dMax, vMax, vis);
return dMax;
}
// Eulerian Cycle
template<typename T>
vector<int> Graph<T>::eulerianCycle(int edgeCount) {
for (int i = 1; i <= m_n; ++i) {
if (m_ad[i].size() & 1) {
return {-1};
}
}
vector<int> ans;
vector<bool> vis(edgeCount + 1);
stack<int> s;
s.push(1);
while (!s.empty()) {
int current = s.top();
while (!m_ad[current].empty() && vis[m_ad[current].back().second]) {
m_ad[current].pop_back();
}
if (!m_ad[current].empty()) {
pii edge = m_ad[current].back();
vis[edge.second] = true;
s.push(edge.first);
} else {
ans.push_back(current);
s.pop();
}
}
return ans;
}
// Minimum Weight Hamiltonian Circuit (Dynamic Programming)
template<typename T>
int Graph<T>::minHamiltonianCircuit() {
auto weight = getWeightMatrix();
vector<vector<int>> dp(1 << m_n);
for (int i = 0; i < 1 << m_n; ++i) {
dp[i].resize(m_n, inf);
}
dp[1][0] = 0;
for (int i = 1; i < 1 << m_n; ++i) {
for (int j = 0; j < m_n; ++j) {
if (dp[i][j] != inf) {
for (auto &p : m_ad[j]) {
int w = p.first;
if (!(i & (1 << w))) { // nodul w nu face parte din drumul reprezentat de i
dp[i | (1 << w)][w] = min(dp[i | (1 << w)][w], dp[i][j] + weight[j][w]);
}
}
}
}
}
vector<int> endpoints;
int ans = inf;
for (int i = 1; i < m_n; ++i) {
for (auto &w : m_ad[i]) {
if (w.first == 0) {
endpoints.push_back(i);
break;
}
}
}
for (auto &e : endpoints) {
ans = min(ans, dp[(1 << m_n) - 1][e] + weight[e][0]);
}
return ans;
}
// Maximum Bipartite Matching
template<typename T>
vector<pii> Graph<T>::maxBipartiteMatching(int n) {
vector<int> matchL(m_n + 1, -1);
vector<int> matchR(m_n + 1, -1);
vector<bool> vis(m_n + 1);
int found = 1;
while (found) {
found = 0;
fill(vis.begin(), vis.end(), 0);
for (int l = 1; l <= n; ++l) {
if (matchL[l] == -1) {
found |= foundMatch(l, vis, matchL, matchR);
}
}
}
vector<pii> ans;
for (int l = 1; l <= n; ++l) {
if (matchL[l] != -1) {
ans.emplace_back(l, matchL[l]);
}
}
return ans;
}
// Critical Connections in a Network
template<typename T>
vector<vector<int>> Graph<T>::criticalConnections(int n, vector<vector<int>> &connections) {
vector<vector<int>> ad(n);
vector<vector<int>> cc;
for (auto &connection : connections) {
ad[connection[0]].push_back(connection[1]);
ad[connection[1]].push_back(connection[0]);
}
m_n = n;
m_ad = ad;
vector<int> disc(m_n, 0);
vector<int> low(m_n, 0);
DFS(0, -1, disc, low, cc);
return cc;
}
// Utilities
template<typename T>
void Graph<T>::DFS(int v, vector<bool> &vis) {
vis[v] = true;
for (auto &x : m_ad[v]) {
if (!vis[x]) {
DFS(x, vis);
}
}
}
template<typename T>
void Graph<T>::topologicalSort(int v, vector<bool> &vis, vector<int> &stack) {
vis[v] = true;
for (auto &x : m_ad[v]) {
if (!vis[x]) {
topologicalSort(x, vis, stack);
}
}
stack.push_back(v);
}
template<typename T>
vector<vector<int>> Graph<T>::getReversedGraph() {
vector<vector<int>> tAd(m_n + 1);
for (int i = 1; i <= m_n; ++i) {
for (auto &w : m_ad[i]) {
tAd[w].push_back(i);
}
}
return tAd;
}
template<typename T>
void Graph<T>::tDFS(int v, vector<vector<int>> &tAd, vector<bool> &vis, int compCount, vector<vector<int>> &comp) {
vis[v] = true;
comp[compCount - 1].push_back(v);
for (auto &x : tAd[v]) {
if (!vis[x]) {
tDFS(x, tAd, vis, compCount, comp);
}
}
}
template<typename T>
void Graph<T>::bccDFS(int v, int parent, vector<int> &disc, vector<int> &low, vector<int> &stack,
vector<vector<int>> &comp) {
static int time = 0;
int children = 0;
disc[v] = low[v] = ++time;
for (auto &w : m_ad[v]) {
if (w == parent) continue;
if (!disc[w]) {
++children;
stack.push_back(w);
bccDFS(w, v, disc, low, stack, comp);
low[v] = min(low[v], low[w]);
if (low[w] >= disc[v]) {
comp.emplace_back();
stack.push_back(v);
while (!stack.empty() && stack.back() != w) {
comp[comp.size() - 1].push_back(stack.back());
stack.pop_back();
}
if (!stack.empty()) {
comp[comp.size() - 1].push_back(stack.back());
stack.pop_back();
}
}
} else if (w != parent) {
low[v] = min(low[v], disc[w]);
}
}
}
template<typename T>
bool Graph<T>::eligible(vector<int> &v) {
int s = 0;
auto n = v.size();
for (auto &x : v) {
if (x > n - 1) {
return false;
}
s += x;
}
return s % 2 == 0;
}
template<typename T>
void Graph<T>::findPaths(vector<vector<int>> &paths, vector<int> &path, vector<vector<int>> &parent, int current) {
if (current == -1) {
paths.push_back(path);
return;
}
for (auto &p : parent[current]) {
path.push_back(current);
findPaths(paths, path, parent, p);
path.pop_back();
}
}
template<typename T>
void Graph<T>::BFS(vector<vector<int>> &parents, int start) {
vector<int> dist(m_n + 1, inf);
queue<int> Q;
Q.push(start);
parents[start] = {-1};
dist[start] = 0;
while (!Q.empty()) {
int current = Q.front(); Q.pop();
for (auto &x : m_ad[current]) {
if (dist[current] + 1 < dist[x]) {
dist[x] = dist[current] + 1;
Q.push(x);
parents[x].clear();
parents[x].push_back(current);
} else if (dist[x] == dist[current] + 1) {
parents[x].push_back(current);
}
}
}
}
template<typename T>
bool Graph<T>::foundPath(vector<vector<int>> &c, vector<vector<int>> &flow, vector<int> &parent) {
queue<int> Q;
vector<bool> vis(m_n + 1, false);
parent[1] = -1;
Q.push(1);
while (!Q.empty()) {
int current = Q.front(); Q.pop();
vis[current] = true;
if (current == m_n) {
continue;
}
for (auto &w : m_ad[current]) {
if (!vis[w.first] && flow[current][w.first] < c[current][w.first]) {
parent[w.first] = current;
Q.push(w.first);
}
}
}
return vis[m_n];
}
template<typename T>
bool Graph<T>::comp(pip &X, pip &Y) {
return X.second.second < Y.second.second;
}
template<typename T>
void Graph<T>::treeDFS(int v, int d, int &dMax, int &vMax, vector<bool> &vis) {
if (d > dMax) {
dMax = d;
vMax = v;
}
vis[v] = true;
for (auto &w : m_ad[v]) {
if (!vis[w]) {
treeDFS(w, d + 1, dMax, vMax, vis);
}
}
}
template<typename T>
int Graph<T>::foundMatch(int l, vector<bool> &vis, vector<int> &matchL, vector<int> &matchR) {
if (!vis[l]) {
vis[l] = true;
for (auto &r : m_ad[l]) {
if (matchR[r] == -1) {
matchL[l] = r;
matchR[r] = l;
return 1;
}
}
for (auto &r : m_ad[l]) {
if (foundMatch(matchR[r], vis, matchL, matchR)) {
matchL[l] = r;
matchR[r] = l;
return 1;
}
}
}
return 0;
}
template<typename T>
void Graph<T>::critDFS(int v, int parent, vector<int> &disc, vector<int> &low, vector<vector<int>> &cc) {
static int time = 0;
disc[v] = low[v] = ++time;
for (auto &w : m_ad[v]) {
if (w == parent) continue;
if (!disc[w]) {
critDFS(w, v, disc, low, cc);
low[v] = min(low[v], low[w]);
if (low[w] > disc[v]) {
vector<int> e;
e.push_back(v);
e.push_back(w);
cc.push_back(e);
}
} else if (w != parent) {
low[v] = min(low[v], disc[w]);
}
}
}
template<typename T>
vector<vector<int>> Graph<T>::getWeightMatrix() {
vector<vector<int>> weight(m_n + 1);
for (int i = 0; i < m_n; ++i) {
weight[i].resize(m_n + 1, 0);
}
for (int i = 0; i < m_n; ++i) {
for (auto &p : m_ad[i]) {
weight[i][p.first] = p.second;
}
}
return weight;
}
int main() {
int n, m, x, y;
fin >> n >> m;
Graph<int> G(n);
for (int i = 1; i <= m; ++i) {
fin >> x >> y;
G.addEdge(x, y);
G.addEdge(y, x);
}
fout << G.components();
return 0;
}