Bridge Trees

[UtsavKash19]

adding Bridge trees algo

[github-actions]

Thank you for submitting your pull request! 🙌 We'll review it as soon as possible.). If there are any specific instructions or feedback regarding your PR, we'll provide them here. Thanks again for your contribution! 😊

[github-actions]

⚡️ Lighthouse Report for the Deploy Preview of this PR 🚀

🔗 Site: Algo | Live Site

URL 🌐	Performance	Accessibility	Best Practices	SEO	📊
/algo/	🔴 47	🟢 96	🟡 75	🟡 86	📄
/algo/docs	🔴 45	🟢 96	🟡 75	🟢 100	📄
/algo/blog	🟡 67	🟢 92	🟢 96	🟡 86	📄

[ajay-dhangar]

Please carefully read our contribution guidelines. They provide information on our documentation format and syntax.

[UtsavKash19]

I noticed that the PR validation check is failing, while other checks have passed successfully. I will review the logs for the failing check to address any issues. If there are specific concerns or feedback on the implementation, I’d appreciate your insights!

[ajay-dhangar]

My comment concerns the format of your code in the .md file for our docs. Please follow our format for best practices.

[UtsavKash19]

what changes you requested @ajay-dhangar

[ajay-dhangar]

what changes you requested @ajay-dhangar

We need only markdown files, so please write your code in markdown language and save your file with a .md extension. Also, please follow our format and syntax.

[UtsavKash19]

Bridge Tree Construction

Introduction

A bridge in a graph is an edge that, if removed, would disconnect the graph. A bridge tree is a graph where each node represents a component of the original graph that does not contain any bridges, and the edges of the bridge tree represent the bridges of the original graph.

Properties of the Bridge Tree

1. Bridges as Edges: All the bridges of the original graph are represented as edges in the bridge tree.
2. Connected: The bridge tree is connected if the original graph is connected.
3. Acyclic: The bridge tree does not contain any cycles.
4. Nodes Count: The bridge tree will contain ≤ N nodes, where N is the number of nodes in the original graph.
5. Edges Count: The bridge tree will contain ≤ N−1 edges, where N is the number of nodes in the original graph.

Time Complexity

• Bridges Finding: The bridges in the graph can be found in O(N + M), where N is the number of nodes and M is the number of edges.
• DFS Complexity: The time complexity for the DFS function is O(N + M).

Thus, the total time complexity for constructing the bridge tree is O(N + M).

Pseudocode for Constructing the Bridge Tree

DFS Function

void dfs(int node, int component_number) {
    component[node] = component_number; // Mark the component number for the node
    vis[node] = true; // Mark the node as visited
    for each adjacent edge where the edge is not a bridge {
        int next = other endpoint of the edge;
        if (vis[next] == true) {
            continue; // Skip if already visited
        }
        dfs(next, component_number); // Recursively explore
    }
}
void main() {
    // Step 1: Find all the bridges in the graph
    find_bridges(); // This function is assumed to find all the bridges (refer to prerequisites for this)
    
    // Step 2: For each node, call DFS to identify components
    for (int i = 1; i <= n; ++i) {
        if (vis[i] == false) {
            dfs(i, unique_component_number); // Call DFS for unvisited nodes, with a unique component number
            unique_component_number++; // Increment the component number for each DFS call
        }
    }
}