#include <bits/stdc++.h>
using namespace std;
ifstream fin("ctc.in");
ofstream fout("ctc.out");
class Graph {
private:
//Variables
int vertices;
int edges;
bool oriented;
vector<int> *adjacency_list;
//To compute:
vector<int> distances;
vector<list<int>> biconnected_components;
vector<vector<int>> strongly_connected_components;
vector<int> topological;
vector<pair<int,int>> bridges;
private: //Functions
void BFS(int starting_vertex); //Breadth-first search
void DFS(int vertex, int *visited); //Depth-first search
void BCC(int vertex, vector<int> &parent, stack<int> &vertices_stack, vector<int> &discovery_time,
vector<int> &lowest_reachable, int &timer); //Biconnected Components
void SCC(int vertex, vector<int> &discovery_time, vector<int> &lowest_reachable, stack<int> &vertices_stack,
vector<bool> &on_stack, int &timer); //Strongly connected components
void CCN(int vertex,vector<int>& discovery_time, vector<int>& lowest_reachable,vector<bool>& visited, vector<int>& parent, int& timer); //Critical Connections
void TOPOLOGICAL_SORT(int vertex, vector<bool> &visited);
vector<int> BFSMD(int starting_vertex);
public:
Graph(int vertices = 0, int edges = 0, bool oriented = false);
~Graph();
void infoarena_graph();
void show_my_graph();
void solve_distances(int starting_vertex);
void solve_connected_components();
void solve_biconnected();
void solve_strongly_connected();
void solve_topological();
void solve_critical_connections();
void solve_starting_ending_distance(int starting_vertex, int ending_vertex);
};
//Havel Hakimi problem
bool solve_havel_hakimi(vector<int> degrees);
//function to print a vector
void printv(vector<int> xs){
for(int i : xs) cout<<i<<' ';
cout<<'\n';
}
int main() {
int n,m;
fin>>n>>m;
Graph g(n,m,true);
g.infoarena_graph();
g.solve_strongly_connected();
}
#pragma region Initialization
Graph::Graph(int vertices, int edges, bool oriented) : vertices(vertices), edges(edges), oriented(oriented) {
adjacency_list = new vector<int>[vertices + 1];
}
Graph::~Graph() {
delete[] adjacency_list;
}
void Graph::infoarena_graph() {
int x, y;
if (oriented) {
for (int i = 1; i <= edges; i++) {
fin >> x >> y;
adjacency_list[x].push_back(y);
}
} else {
for (int i = 1; i <= edges; i++) {
fin >> x >> y;
adjacency_list[x].push_back(y);
adjacency_list[y].push_back(x);
}
}
}
void Graph::show_my_graph() {
for (int i = 1; i < vertices + 1; i++) {
cout << i << "=>";
for (int j : adjacency_list[i]) {
cout << j << ' ';
}
cout << '\n';
}
}
#pragma endregion
//Algorithm implementations
#pragma region Algorithms
void Graph::BFS(int starting_vertex) {
int *visited = (int *) calloc(vertices + 1, sizeof(int));
queue<int> que;
que.push(starting_vertex);
distances[starting_vertex] = 1;
visited[starting_vertex] = 1;
while (!que.empty()) {
int current_node = que.front();
que.pop();
for (auto neighbor : adjacency_list[current_node]) {
if (!visited[neighbor]) {
que.push(neighbor);
visited[neighbor] = 1;
distances[neighbor] = distances[current_node] + 1;
}
}
}
free(visited);
}
void Graph::DFS(int vertex, int *visited) {
visited[vertex] = 1;
for (auto neighbor : adjacency_list[vertex])
if (!visited[neighbor])
DFS(neighbor, visited);
}
vector<int> Graph::BFSMD(int starting_vertex){
int *visited = (int *) calloc(vertices + 1, sizeof(int));
queue<int> que;
que.push(starting_vertex);
vector<int> distances(vertices+1,0);
distances[starting_vertex] = 0;
visited[starting_vertex] = 1;
while (!que.empty()) {
int current_node = que.front();
que.pop();
for (auto neighbor : adjacency_list[current_node]) {
if (!visited[neighbor]) {
que.push(neighbor);
visited[neighbor] = 1;
distances[neighbor] = distances[current_node] + 1;
}
}
}
free(visited);
return distances;
}
//Used Tarjan Algorithm for biconnected components, strongly connected components, and bridges (critical connections) implementation
//The following implementations are similar, because you need to keep track of the some things that are happening during the dfs
void Graph::BCC(int vertex, vector<int> &parent, stack<int> &vertices_stack, vector<int> &discovery_time,
vector<int> &lowest_reachable, int &timer) {
// consider lowest reachable value as being a better path from a node to another
// for example if we want to reach a certain point and we have 2 neighbors with different discovery time we will chose
// the one with the less value because we want to reach faster that certain point
// increment the discovery time of the vertex you are visiting
// this is the only information you posses at the moment
discovery_time[vertex] = lowest_reachable[vertex] = ++timer;
for (int neighbor : adjacency_list[vertex]) {
if (neighbor != parent[vertex]) {
//now for each neighbor you are checking in adjacency list you are pushing on stack the vertex you are currently visiting
//assuming it is an articulation point
vertices_stack.push(vertex);
//if the neighbor you are checking has not been visited yet you will visit him next via DFS
if (parent[neighbor] == -1) {
parent[neighbor] = vertex; //set the parent of neighbor to be te current node
BCC(neighbor, parent, vertices_stack, discovery_time, lowest_reachable, timer); //DFS
//After you reach a point when DFS cant visit unvisited nodes you get back and update the values of your vertex lowest_reachable point
//with the values of the neighbor you currently visited and set it to the min value between both of them
lowest_reachable[vertex] = min(lowest_reachable[neighbor],
lowest_reachable[vertex]);
//You do this operation until you are in a vertex which has its discovery time less than or equal to the lowest reachable
//value of the neighbor you currently visited
//if you manage to find such a vertex, that means it is an articulation point, and all the vertices
//you pushed into the stack by now are part of a biconnected component
if (discovery_time[vertex] <= lowest_reachable[neighbor]) {
list<int> component;
vector<bool> pushed(vertices + 1, false);
component.push_back(vertex);
pushed[vertex] = true;
int temp = vertices_stack.top();
while (temp != vertex) {
if (!pushed[temp]) {
component.push_back(temp);
pushed[temp] = true;
}
vertices_stack.pop();
temp = vertices_stack.top();
}
biconnected_components.push_back(component);
}
}
//if the neighbor you are visiting has been actually visited, we need to check if it's discovery time is less then our lowest reachable value
//if so, that means you have a path to that node that is better than the path you are on right now, so when you
//get back by recursion it will tell the other vertices that he found a better path and will set all other vertices' lowest_reachable value to the minimal one
//so it will form a biconnected component
else {
lowest_reachable[vertex] = min(discovery_time[neighbor],
lowest_reachable[vertex]);
}
}
}
}
void Graph::SCC(int vertex, vector<int> &discovery_time, vector<int> &lowest_reachable, stack<int> &vertices_stack,
vector<bool> &has_component, int &timer) {
//set the discovery time and lowest_reachable value at the time you discover this node
discovery_time[vertex] = lowest_reachable[vertex] = ++timer;
//save your node intro the stack assuming it could be an articulation point
vertices_stack.push(vertex);
for (auto neighbor : adjacency_list[vertex]) {
//for every neighbor if it has't been visited recursively do dfs and update the lowest_reachable vertex;
if (discovery_time[neighbor] == -1) {
SCC(neighbor, discovery_time, lowest_reachable, vertices_stack, has_component, timer);
lowest_reachable[vertex] = min(lowest_reachable[vertex], lowest_reachable[neighbor]);
}
//if the vertex is already on the stack, just update the lowest_reachable value
else if (!has_component[neighbor]) {
lowest_reachable[vertex] = min(lowest_reachable[vertex], discovery_time[neighbor]);
}
}
//A strongly connected component will have all its vertices' lowest_reachabqle values equal
if (lowest_reachable[vertex] == discovery_time[vertex]) {
vector<int> component;
int temp;
do {
temp = vertices_stack.top();
vertices_stack.pop();
has_component[temp] = true;
component.push_back(temp);
} while (temp != vertex);
strongly_connected_components.push_back(component);
}
}
//similar to articulation points for an edge to be a bridge the condition is that the vertex X should have
//the lowest_reachable value greater than his parent's discovery time
// ( NO PATH TO X or one of its ancestors )
//Explanations similar to the previous algorithms.
void Graph::CCN(int vertex, vector<int> &discovery_time, vector<int> &lowest_reachable,vector<bool>& visited,vector<int>& parent, int &timer) {
discovery_time[vertex] = lowest_reachable[vertex] = ++timer;
visited[vertex] = true;
for(int neighbor : adjacency_list[vertex]){
if(!visited[neighbor]){
parent[neighbor] = vertex;
CCN(neighbor,discovery_time,lowest_reachable,visited,parent,timer);
lowest_reachable[vertex] = min(lowest_reachable[vertex],lowest_reachable[neighbor]);
if(discovery_time[vertex] < lowest_reachable[neighbor])
bridges.push_back({vertex,neighbor});
}
else if(parent[vertex] != neighbor){
lowest_reachable[vertex] = min(lowest_reachable[vertex],discovery_time[vertex]);
}
}
}
void Graph::TOPOLOGICAL_SORT(int vertex, vector<bool> &visited) {
visited[vertex] = true;
for (int neighbor : adjacency_list[vertex]) {
if (!visited[neighbor])
TOPOLOGICAL_SORT(neighbor, visited);
}
topological.push_back(vertex);
}
#pragma endregion
//infoarena solutions
#pragma region Solutions
//Minimal distances BFS problem
void Graph::solve_distances(int starting_vertex) {
distances.resize(vertices+1,0);
BFS(starting_vertex);
for (int i = 1; i < vertices + 1; i++)
fout << distances[i] - 1 << ' ';
}
//Connected compontents DFS problem
void Graph::solve_connected_components() {
int counter = 0;
int *visited = (int *) calloc(vertices + 1, sizeof(int));
for (int i = 1; i < vertices + 1; i++)
if (!visited[i]) {
DFS(i, visited);
counter++;
}
fout << counter;
free(visited);
}
void Graph::solve_biconnected() {
//initialize the stuff you will work with
stack<int> vertices_stack;
vector<int> parent(vertices + 1, -1);
vector<int> discovery_time(vertices + 1, -1);
vector<int> lowest_reachable(vertices + 1, -1);
//global timer for discovery time and lowest reachable value
int timer = 0;
parent[1] = 1;
BCC(1, parent, vertices_stack, discovery_time, lowest_reachable, timer);
list<int>::iterator it;
fout << biconnected_components.size() << '\n';
for (auto components : biconnected_components) {
for (it = components.begin(); it != components.end(); it++)
fout << *it << ' ';
fout << '\n';
}
}
void Graph::solve_strongly_connected() {
vector<int> discovery_time(vertices + 1, -1);
vector<int> lowest_reachable(vertices + 1, -1);
vector<bool> has_component(vertices + 1, false);
stack<int> vertices_stack;
int timer = 0;
for (int i = 1; i < vertices + 1; i++) {
if (discovery_time[i] == -1)
SCC(i, discovery_time, lowest_reachable, vertices_stack, has_component, timer);
}
fout << strongly_connected_components.size() << '\n';
for (auto component : strongly_connected_components) {
for (int i : component)
fout << i << ' ';
fout << '\n';
}
}
void Graph::solve_topological() {
vector<bool> visited(vertices + 1, false);
for (int i = 1; i < vertices + 1; i++) {
if (!visited[i])
TOPOLOGICAL_SORT(i, visited);
}
for (int i = topological.size() - 1; i >= 0; i--) {
fout << topological[i] << ' ';
}
}
void Graph::solve_critical_connections(){
vector<bool> visited(vertices+1,false);
vector<int> discovery_time(vertices + 1, -1);
vector<int> lowest_reachable(vertices + 1, -1);
vector<int> parent(vertices + 1,-1);
int timer = 0;
parent[1] = 1;
for(int i = 1;i<=vertices+1;i++){
if(!visited[i])
CCN(i,discovery_time,lowest_reachable,visited,parent,timer);
}
for(auto br : bridges){
cout<<br.first<<"--"<<br.second;
cout<<'\n';
}
}
void Graph::solve_starting_ending_distance(int starting_vertex, int ending_vertex){
vector<int> start_min_dist = BFSMD(starting_vertex);
vector<int> end_min_dist = BFSMD(ending_vertex);
vector<int> frequency(vertices+1,0);
int min_dist = start_min_dist[ending_vertex];
for(int i = 1;i<=vertices;i++){
if(start_min_dist[i] + end_min_dist[i] == min_dist)
frequency[start_min_dist[i]]++;
}
vector<int> min_dist_vertices;
for(int i = 0;i<=vertices;i++){
if(frequency[start_min_dist[i]] == 1 && start_min_dist[i] + end_min_dist[i] == min_dist)
min_dist_vertices.push_back(i);
}
fout<<min_dist_vertices.size()<<'\n';
for(auto i : min_dist_vertices){
fout<<i<<' ';
}
}
//Havel Hakimi - given a vector of vertices' degree
//say if there is a graph which coresponds to the given data
bool solve_havel_hakimi(vector<int> degrees){
sort(degrees.begin(),degrees.end(),greater<int>());
while(!degrees.empty()){
printv(degrees);
int current = degrees[0];
if(current > degrees.size())
return false;
else if (current == 0)
return true;
for(int i = 0;i<=current;i++)
if(degrees[i] - 1 < 0)
return false;
else
degrees[i]--;
degrees.erase(degrees.begin());
sort(degrees.begin(),degrees.end(),greater<int>());
}
return true;
}
#pragma endregion