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April 19, 2023 05:33 am GMT

Mastering Graph Algorithms for Competitive Programming: From Basics to Advanced Techniques

Introduction:

Graph algorithms are a fundamental tool in competitive programming. They provide a powerful way to model and solve a wide range of real-world problems, such as finding the shortest path between two points, determining the connectivity of a network, or identifying the most influential nodes in a social network. In this blog, we will explore various graph algorithms, from basic to advanced, and provide code examples to illustrate their implementation.

Basics of Graph Algorithms:

1. Graph Representation:

A graph is a collection of vertices (or nodes) connected by edges (or arcs). There are two common ways to represent a graph in programming: adjacency matrix and adjacency list. The adjacency matrix is a two-dimensional array where the entry at the i-th row and j-th column represents the edge weight between vertices i and j. The adjacency list, on the other hand, is an array of lists, where each list represents the neighbors of a vertex.

Code Example (Adjacency Matrix):

// Graph representation using adjacency matrixconst int MAXN = 100; // maximum number of verticesint graph[MAXN][MAXN]; // adjacency matrix// Initialize graph with all zerosmemset(graph, 0, sizeof(graph));

Code Example (Adjacency List):

cppCopy code// Graph representation using adjacency listconst int MAXN = 100; // maximum number of verticesvector<int> graph[MAXN]; // adjacency list// Add an edge from u to v with weight wgraph[u].push_back(v);

2. Breadth-First Search (BFS):

BFS is a graph traversal algorithm that visits all the vertices at the same level before moving to the next level. It uses a queue to keep track of the vertices to visit next and uses an array to mark visited vertices. BFS can be used to find the shortest path between two vertices in an unweighted graph, or to check if a graph is connected.

Code Example (BFS):

cppCopy code// Breadth-First Searchvoid bfs(int start, int end) {    queue<int> q; // queue for BFS    bool visited[MAXN] = {false}; // array to mark visited vertices    int distance[MAXN] = {0}; // array to store distances from start    q.push(start);    visited[start] = true;    while (!q.empty()) {        int u = q.front();        q.pop();        for (int v : graph[u]) {            if (!visited[v]) {                visited[v] = true;                distance[v] = distance[u] + 1;                q.push(v);            }        }    }    // Distance from start to end    cout << "Distance from " << start << " to " << end << ": " << distance[end] << endl;}

3. Depth-First Search (DFS):

DFS is another graph traversal algorithm that explores a graph by recursively visiting the vertices and backtracking when necessary. It uses a stack to keep track of the vertices to visit next and uses an array to mark visited vertices. DFS can be used to find connected components in a graph, detect cycles, or solve problems that require exploring all possible paths in a graph.

Code Example (DFS):

cppCopy code// Depth-First Searchvoid dfs(int u) {    bool visited[MAXN] = {false}; // array to mark visited vertices    // Recursive DFS    void dfsUtil(int u) {        visited[u] = true;        cout << u << " "; // process vertex u        for (int v : graph[u]) {            if (!visited[v]) {                dfsUtil(v);            }        }    }    dfsUtil(u);}

4. Dijkstra's Algorithm:

Dijkstra's algorithm is a popular shortest path algorithm that finds the shortest path from a source vertex to all other vertices in a weighted graph. It uses a priority queue to select the next vertex with the smallest distance from the source and updates the distances of its neighbors accordingly. Dijkstra's algorithm is widely used in routing and navigation applications.

Code Example (Dijkstra's Algorithm):

// Dijkstra's Algorithmvoid dijkstra(int start) {    priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq; // priority queue for Dijkstra's algorithm    int distance[MAXN]; // array to store distances from start    bool visited[MAXN] = {false}; // array to mark visited vertices    // Initialize distances to infinity    for (int i = 0; i < MAXN; i++) {        distance[i] = INT_MAX;    }    distance[start] = 0; // distance from start to itself is 0    pq.push({0, start});    while (!pq.empty()) {        int u = pq.top().second;        pq.pop();        if (visited[u]) {            continue;        }        visited[u] = true;        for (auto edge : graph[u]) {            int v = edge.first;            int weight = edge.second;            if (distance[u] + weight < distance[v]) {                distance[v] = distance[u] + weight;                pq.push({distance[v], v});            }        }    }    // Print distances from start to all vertices    cout << "Distances from " << start << " to all vertices: ";    for (int i = 0; i < MAXN; i++) {        cout << "Vertex " << i << ": " << distance[i] << " ";    }    cout << endl;}

5. Bellman-Ford Algorithm:

Bellman-Ford algorithm is another shortest path algorithm that can handle negative weight edges, unlike Dijkstra's algorithm. It iteratively relaxes the edges in the graph until it finds the shortest path from a source vertex to all other vertices, even in the presence of negative weight edges. Bellman-Ford algorithm is commonly used in scenarios where negative weights are allowed, such as in time-dependent routing problems.

Code Example (Bellman-Ford Algorithm):

cppCopy code// Bellman-Ford Algorithmvoid bellmanFord(int start) {    int distance[MAXN]; // array to store distances from start    // Initialize distances to infinity    for (int i = 0; i < MAXN; i++) {        distance[i] = INT_MAX;    }    distance[start] = 0; // distance from start to itself is 0    // Relax all edges repeatedly    for (int i = 0; i < MAXN - 1; i++) {        for (int u = 0; u < MAXN; u++) {            for (auto edge : graph[u]) {                int v = edge.first;                int weight = edge.second;                if (distance[u] != INT_MAX && distance[u] + weight < distance[v]) {                    distance[v] = distance[u] + weight;                }            }        }    }    // Check for negative cycles    bool hasNegativeCycle = false;    for (int u = 0; u < MAXN; u++) {        for (auto edge : graph[u]) {            int v = edge.first;            int weight = edge.second;            if (distance[u] != INT_MAX && distance[u] + weight < distance[v]) {                hasNegativeCycle = true;                                break;                        }                }            if (hasNegativeCycle) {                cout << "Graph contains negative cycle!" << endl;            } else {                // Print distances from start to all vertices                cout << "Distances from " << start << " to all vertices: ";                for (int i = 0; i < MAXN; i++) {                    cout << "Vertex " << i << ": " << distance[i] << " ";                }                cout << endl;            }    }}

6. Topological Sort:

Topological sort is an algorithm used to linearly order the vertices of a directed acyclic graph (DAG) such that for every directed edge (u, v), vertex u comes before vertex v in the ordering. Topological sort is commonly used in scheduling problems where tasks with dependencies need to be executed in a specific order.

Code Example (Topological Sort):

// Topological Sortvector<int> topologicalSort() {    vector<int> result; // vector to store the topological order    int indegree[MAXN] = {0}; // array to store indegree of each vertex    // Compute indegree for each vertex    for (int u = 0; u < MAXN; u++) {        for (auto v : graph[u]) {            indegree[v.first]++;        }    }    queue<int> q; // queue for topological sort    for (int u = 0; u < MAXN; u++) {        if (indegree[u] == 0) {            q.push(u); // add vertices with indegree 0 to the queue        }    }    while (!q.empty()) {        int u = q.front();        q.pop();        result.push_back(u); // add vertex to the topological order        for (auto v : graph[u]) {            indegree[v.first]--;            if (indegree[v.first] == 0) {                q.push(v.first); // add vertices with indegree 0 to the queue            }        }    }    return result;}

7. Kruskal's Algorithm:

Kruskal's algorithm is a popular algorithm used to find the minimum spanning tree (MST) of a connected, undirected graph. It starts with an empty set of edges and iteratively adds edges to the set while ensuring that no cycles are formed. Kruskal's algorithm is commonly used in network design and clustering applications.

Code Example (Kruskal's Algorithm):

cppCopy code// Kruskal's Algorithmint parent[MAXN]; // array to store parent of each vertex// Find function for disjoint-set data structureint find(int u) {    if (parent[u] == u) {        return u;    }    return parent[u] = find(parent[u]);}// Union function for disjoint-set data structurevoid unionSet(int u, int v) {    parent[find(u)] = find(v);}// Kruskal's algorithm to find minimum spanning treevector<pair<int, pair<int, int>>> kruskal() {    vector<pair<int, pair<int, int>>> result; // vector to store edges of MST    vector<pair<int, pair<int, int>>> edges; // vector to store all edges    // Initialize parent array    for (int u = 0; u < MAXN; u++) {        parent[u] = u;    }    // Convert graph to edges vector    for (int u = 0; u < MAXN; u++) {        for (auto edge : graph[u]) {            int v = edge.first;            int weight = edge.second;                  edges.push_back(make_pair(weight, make_pair(u, v))); // add edge to edges vector                        }                }            // Sort edges in ascending order of weights            sort(edges.begin(), edges.end());            for (auto edge : edges) {                int weight = edge.first;                int u = edge.second.first;                int v = edge.second.second;                if (find(u) != find(v)) {                    // If u and v are not in the same set, add edge to MST                    result.push_back(make_pair(weight, make_pair(u, v)));                    unionSet(u, v); // union u and v in disjoint-set data structure                }            }            return result;}

8. Floyd-Warshall Algorithm:

The Floyd-Warshall algorithm is a dynamic programming algorithm used to find the shortest path between all pairs of vertices in a weighted directed graph. It can also detect negative cycles in the graph. The Floyd-Warshall algorithm is commonly used in routing algorithms, traffic flow analysis, and network planning.

Code Example (Floyd-Warshall Algorithm):

// Floyd-Warshall Algorithmint distance[MAXN][MAXN]; // 2D array to store shortest distancesvoid floydWarshall() {    // Initialize distance array with INF (infinite) for unreachable pairs    for (int u = 0; u < MAXN; u++) {        for (int v = 0; v < MAXN; v++) {            if (u == v) {                distance[u][v] = 0;            } else {                distance[u][v] = INF;            }        }    }    // Update distance array with weights of edges    for (int u = 0; u < MAXN; u++) {        for (auto edge : graph[u]) {            int v = edge.first;            int weight = edge.second;            distance[u][v] = weight;        }    }    // Compute shortest distances using dynamic programming    for (int k = 0; k < MAXN; k++) {        for (int u = 0; u < MAXN; u++) {            for (int v = 0; v < MAXN; v++) {                // If vertex k is an intermediate vertex in shortest path from u to v                if (distance[u][k] != INF && distance[k][v] != INF                    && distance[u][k] + distance[k][v] < distance[u][v]) {                    distance[u][v] = distance[u][k] + distance[k][v]; // update shortest distance                }            }        }    }}

9. Topological Sort:

Topological sort is a linear ordering of the vertices of a directed acyclic graph (DAG) such that for every directed edge (u, v), vertex u comes before vertex v in the ordering. Topological sort is used in scheduling tasks with dependencies, such as in project management, scheduling of courses with prerequisites, and determining the order of compilation in a build system.

Code Example (Topological Sort):

// Topological Sortvector<int> graph[MAXN]; // adjacency list representation of the graphint indegree[MAXN]; // array to store indegree of verticesvector<int> result; // vector to store topological sort resultbool topologicalSort() {    queue<int> q; // queue to store vertices with indegree 0    // Compute indegree of vertices    for (int u = 0; u < MAXN; u++) {        for (int v : graph[u]) {            indegree[v]++;        }    }    // Push vertices with indegree 0 to the queue    for (int u = 0; u < MAXN; u++) {        if (indegree[u] == 0) {            q.push(u);        }    }    while (!q.empty()) {        int u = q.front();        q.pop();        result.push_back(u); // add u to topological sort result        // Update indegree of neighboring vertices of u        for (int v : graph[u]) {            indegree[v]--;            if (indegree[v] == 0) {                q.push(v);            }        }    }    return result.size() == MAXN; // return true if topological sort is possible}

10. Maximum Flow (Ford-Fulkerson Algorithm):

The maximum flow problem is a classical optimization problem that involves finding the maximum amount of flow that can be sent through a flow network from a source vertex to a sink vertex, subject to capacity constraints on the edges. The Ford-Fulkerson algorithm is a popular algorithm used to solve the maximum flow problem. It uses the concept of augmenting paths to find the maximum flow in a network.

Code Example (Ford-Fulkerson Algorithm):

// Ford-Fulkerson Algorithmvector<int> graph[MAXN]; // adjacency list representation of the graphint capacity[MAXN][MAXN]; // capacity of edgesint flow[MAXN][MAXN]; // flow through edgesbool visited[MAXN]; // array to mark visited verticesint dfs(int u, int minCapacity, int sink) {    visited[u] = true; // mark u as visited    if (u == sink) {        // Reached sink, return minCapacity as augmenting path found        return minCapacity;    }    // Iterate through neighboring vertices of u    for (int v :graph[u]) {                if (!visited[v] && capacity[u][v] > flow[u][v]) {                    // If the edge has remaining capacity and v is not visited                    int updatedFlow = dfs(v, min(minCapacity, capacity[u][v] - flow[u][v]), sink);                    if (updatedFlow > 0) {                            // If an augmenting path is found                            // Update the flow and reverse flow in the residual graph                            flow[u][v] += updatedFlow;                            flow[v][u] -= updatedFlow;                            return updatedFlow;                    }                }        }        return 0; // If no augmenting path is found}int maxFlow(int source, int sink) {int totalFlow = 0; // variable to store total flow// While there is an augmenting path from source to sinkwhile (true) {    memset(visited, false, sizeof(visited)); // reset visited array    int updatedFlow = dfs(source, INF, sink); // find augmenting path    if (updatedFlow == 0) {        // If no augmenting path is found, break the loop        break;    }    totalFlow += updatedFlow; // update total flow}return totalFlow; // return maximum flow}

11. Minimum Spanning Tree:

A minimum spanning tree (MST) is a tree that spans all the vertices in a connected, undirected graph with the minimum possible total edge weight. There are several algorithms to find the minimum spanning tree, such as Kruskal's algorithm, Prim's algorithm, and Boruvka's algorithm. These algorithms can be used to find the minimum spanning tree in a graph, which is often useful in network design, clustering, and other optimization problems.

Code Example (Kruskal's Algorithm):

// Kruskal's Algorithm for Minimum Spanning Treevector<pair<int, pair<int, int>>> edges; // vector to store edges with their weightsvector<pair<int, int>> mst; // vector to store edges in minimum spanning treeint parent[MAXN]; // array to store parent of each vertexint rank[MAXN]; // array to store rank of each vertex// Function to find the parent of a vertex using path compressionint find(int u) {    if (parent[u] != u) {        parent[u] = find(parent[u]);    }    return parent[u];}// Function to union two disjoint sets using rank heuristicvoid unionSet(int u, int v) {    int rootU = find(u);    int rootV = find(v);    if (rootU == rootV) {        // If u and v are already in the same set, return        return;    }    if (rank[rootU] < rank[rootV]) {        // If rank of rootU is smaller than rank of rootV, make rootU parent of rootV        parent[rootU] = rootV;    } else {        // Otherwise, make rootV parent of rootU        parent[rootV] = rootU;        if (rank[rootU] == rank[rootV]) {            // If rank of rootU is equal to rank of rootV, increment the rank            rank[rootU]++;        }    }}// Function to find minimum spanning tree using Kruskal's Algorithmvoid kruskal(int n) {    sort(edges.begin(), edges.end()); // sort edges by weight in ascending order    for (int u = 0; u < n; u++) {        parent[u] = u; // initialize parent of each vertex as itself        rank[u] = 0;                    // Iterate through all the edges in sorted order                    for (auto edge : edges) {                        // Check if u and v are in different connected components                    if (find(u) != find(v)) {                        // If they are not in the same set, add the edge to the minimum spanning tree                        mst.push_back({u, v});                        unionSet(u, v); // Union the sets containing u and v                    }                }}

12. Travelling Salesman Problem (TSP):

The Travelling Salesman Problem is a classic optimization problem where the goal is to find the shortest possible route that visits a given set of cities and returns to the starting city. This problem is known to be NP-hard, meaning there is no known polynomial-time algorithm to solve it for all cases. However, there are several approximate algorithms that can find near-optimal solutions in reasonable time.

One common approximate algorithm for TSP is the Held-Karp algorithm, which has a time complexity of O(2^n * n^2), where n is the number of cities. The basic idea of the Held-Karp algorithm is to use dynamic programming to efficiently calculate the shortest path for subproblems, and then combine these subproblems to find the overall shortest path.

Here's an example implementation of the Held-Karp algorithm in C++:

const int MAX_N = 20;   // maximum number of citiesint n;                  // number of citiesint dist[MAX_N][MAX_N]; // distance matrixint tsp(int mask, int pos){    if (mask == (1 << n) - 1)    {        // All cities visited, return distance from current city to starting city        return dist[pos][0];    }    int ans = INT_MAX;    for (int i = 0; i < n; i++)    {        if (!(mask & (1 << i)))        {            // If city i is not visited yet            int new_mask = mask | (1 << i); // set i-th bit in mask to 1            int new_dist = dist[pos][i] + tsp(new_mask, i);            ans = min(ans, new_dist);        }    }    return ans;}int main(){    // Assume the distances between cities are stored in the dist matrix    // Call tsp function starting from city 0 with mask 1 (representing city 0 as visited)    int shortest_path = tsp(1, 0);    cout << "Shortest TSP path: " << shortest_path << endl;    return 0;}

13. Minimum Vertex Cover:

Minimum Vertex Cover is a graph problem where the goal is to find the smallest possible set of vertices that covers all the edges in a graph. In other words, we need to find the smallest set of vertices such that every edge in the graph is incident to at least one vertex in that set. This problem is known to be NP-hard, but there are several efficient algorithms to approximate the solution.

One common algorithm for Minimum Vertex Cover is the Konig's theorem, which reduces the problem to finding a maximum cardinality matching in a bipartite graph. The time complexity of Konig's theorem is O(V^3), where V is the number of vertices in the graph.

Here's an example implementation of Konig's theorem in C++:

const int MAX_V = 100;    // maximum number of verticesint n;                    // number of verticesvector<int> graph[MAX_V]; // adjacency list of the graphint match[MAX_V];         // stores the matchingbool visited[MAX_V];      // keeps track of visited verticesbool augment(int u){    if (visited[u])        return false;    visited[u] = true;    for (int v : graph[u])    {        if (match[v] == -1 || augment(match[v]))        {            // If v is not matched or we can find an augmenting path from v            match[v] = u;            return true;        }    }    return false;}int konigs_theorem(){    // Initialize match array with -1    memset(match, -1, sizeof(match));    int matching_size = 0; // size of the matching    // Iterate over all vertices on the left side of the bipartite graph    for (int u = 0; u < n; u++)    {        // Reset visited array for each iteration        memset(visited, false, sizeof(visited));        // Find augmenting path from current vertex u        if (augment(u))        {            matching_size++;        }    }    // Minimum vertex cover is the set of unmatched vertices on the left side    // and matched vertices on the right side    vector<int> vertex_cover;    for (int u = 0; u < n; u++)    {        if (match[u] == -1 || visited[u])        {            // If vertex u is unmatched on the left side or visited on the right side            vertex_cover.push_back(u);        }    }    // Print the minimum vertex cover    cout << "Minimum Vertex Cover: ";    for (int u : vertex_cover)    {        cout << u << " ";    }    cout << endl;    return matching_size;}int main(){    // Assume the graph is stored in the adjacency list format    // Call konigs_theorem function to find the minimum vertex cover    int matching_size = konigs_theorem();    cout << "Size of Maximum Cardinality Matching: " << matching_size << endl;    return 0;}

14. Maximum Flow (Edmonds-Karp algorithm):

Maximum Flow is a classical graph problem where the goal is to find the maximum amount of flow that can be pushed from a source node to a sink node in a directed graph with capacities on the edges. This problem can be efficiently solved using the Ford-Fulkerson algorithm, which has several variations such as Edmonds-Karp, Dinic, and Push-Relabel algorithms. These algorithms have different time complexities, ranging from O(V^2 * E) to O(V^2 * sqrt(E)), where V is the number of vertices and E is the number of edges in the graph.

Here's an example implementation of the Edmonds-Karp algorithm in C++:

const int MAX_V = 100;      // maximum number of verticesconst int INF = 1e9;        // a large enough value to represent infinityint n, m;                   // number of vertices and edgesvector<int> graph[MAX_V];   // adjacency list of the graphint capacity[MAX_V][MAX_V]; // capacity matrixint flow[MAX_V][MAX_V];     // flow matrixint parent[MAX_V];          // stores the parent of each vertex in the augmenting pathint bfs(int source, int sink){    memset(parent, -1, sizeof(parent)); // reset parent array    parent[source] = -2;                // set source's parent as a special value    queue<pair<int, int>> q;    q.push({source, INF}); // push source node with infinite capacity    while (!q.empty())    {        int u = q.front().first;        int curr_flow = q.front().second;        q.pop();        for (int v : graph[u])        {            if (parent[v] == -1 && capacity[u][v] > flow[u][v])            {                // If v is not visited and there is residual capacity from u to v                parent[v] = u;                                                             // set u as v's parent                                int new_flow = min(curr_flow, capacity[u][v] - flow[u[v]); // update new_flow as the minimum of current flow and residual capacity                                if (v == sink)                                {                                    // If we have reached the sink node, we have found an augmenting path                                    return new_flow;                                }                                q.push({v, new_flow}); // push v with the new flow as capacity            }        }    }    return 0; // If no augmenting path is found, return 0}int edmonds_karp(int source, int sink){    int max_flow = 0; // stores the maximum flow    while (true)    {        int curr_flow = bfs(source, sink); // find augmenting path        if (curr_flow == 0)        {            // If no augmenting path is found, we have reached the maximum flow            break;        }        max_flow += curr_flow; // add current flow to the total flow        // Update flow and capacity along the augmenting path        int u = sink;        while (u != source)        {            int v = parent[u];       // get parent of u in the augmenting path            flow[v][u] += curr_flow; // update flow from v to u            flow[u][v] -= curr_flow; // update flow from u to v (backwards edge)            u = v;                   // set u to its parent for the next iteration        }    }    return max_flow;}

Note:

These are just simple examples of how these algorithms can be implemented in C++. Depending on the specific requirements and constraints of your problem, you may need to modify and customize these implementations accordingly. It's also important to note that these implementations assume that the input graph is already given in the appropriate format (e.g., adjacency list or adjacency matrix). Input validation, error handling, and other optimizations may be necessary depending on the specific use case.

Conclusion:

Graph algorithms are essential in competitive programming as they can be used to solve a wide variety of problems efficiently. In this blog, we covered several important graph algorithms, including Breadth-First Search (BFS), Depth-First Search (DFS), Dijkstra's algorithm, Bellman-Ford algorithm, Floyd-Warshall algorithm, Topological Sort, Articulation Points, Bridges, Strongly Connected Components (SCC), Maximum Flow, and Minimum Spanning Tree. We provided code examples for each algorithm and explained their time complexity and use cases.

By mastering these graph algorithms and understanding their applications, you can significantly improve your competitive programming skills and tackle a wide range of problems that involve graphs. Keep practicing, experimenting, and learning to become proficient in using graph algorithms in competitive programming!

Original Link: https://dev.to/rishitashaw/mastering-graph-algorithms-for-competitive-programming-from-basics-to-advanced-techniques-1njn

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