Reverse Delete Algorithm for Minimum Spanning Tree

3.6

Reverse Delete algorithm is closely related to Kruskal’s algorithm. In Kruskal’s algorithm what we do is : Sort edges by increasing order of their weights. After sorting, we one by one pick edges in increasing order. We include current picked edge if by including this in spanning tree not form any cycle until there are V-1 edges in spanning tree, where V = number of vertices.

In Reverse Delete algorithm, we sort all edges in decreasing order of their weights. After sorting, we one by one pick edges in decreasing order. We include current picked edge if excluding current edge causes disconnection in current graph. The main idea is delete edge if its deletion does not lead to disconnection of graph.

The Algorithm

1) Sort all edges of graph in non-increasing order of
   edge weights.

2) Initialize MST as original graph and remove extra
   edges using step 3.

3) Pick highest weight edge from remaining edges and 
   check if deleting the edge disconnects the graph  
   or not.
       If disconnects, then we don't delete the edge.
       Else we delete the edge and continue. 

Illustration:
Let us understand with the following example:

If we delete highest weight edge of weight 14, graph doesn’t become disconnected, so we remove it.
reversedelete2

Next we delete 11 as deleting it doesn’t disconnect the graph.
reversedelete3

Next we delete 10 as deleting it doesn’t disconnect the graph.
reversedelete4

Next is 9. We cannot delete 9 as deleting it causes disconnection.
reversedelete5

We continue this way and following edges remain in final MST.

Edges in MST
(3, 4) 
(0, 7) 
(2, 3) 
(2, 5) 
(0, 1) 
(5, 6) 
(2, 8) 
(6, 7) 


Note :
In case of same weight edges, we can pick any edge of the same weight edges.

Below is C++ implementation of above steps.

// C++ program to find Minimum Spanning Tree
// of a graph using Reverse Delete Algorithm
#include<bits/stdc++.h>
using namespace std;

// Creating shortcut for an integer pair
typedef  pair<int, int> iPair;

// Graph class represents a directed graph
// using adjacency list representation
class Graph
{
    int V;    // No. of vertices
    list<int> *adj;
    vector< pair<int, iPair> > edges;
    void DFS(int v, bool visited[]);

public:
    Graph(int V);   // Constructor

    // function to add an edge to graph
    void addEdge(int u, int v, int w);

    // Returns true if graph is connected
    bool isConnected();

    void reverseDeleteMST();
};

Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
}

void Graph::addEdge(int u, int v, int w)
{
    adj[u].push_back(v); // Add w to v’s list.
    adj[v].push_back(u); // Add w to v’s list.
    edges.push_back({w, {u, v}});
}

void Graph::DFS(int v, bool visited[])
{
    // Mark the current node as visited and print it
    visited[v] = true;

    // Recur for all the vertices adjacent to
    // this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
        if (!visited[*i])
            DFS(*i, visited);
}

// Returns true if given graph is connected, else false
bool Graph::isConnected()
{
    bool visited[V];
    memset(visited, false, sizeof(visited));

    // Find all reachable vertices from first vertex
    DFS(0, visited);

    // If set of reachable vertices includes all,
    // return true.
    for (int i=1; i<V; i++)
        if (visited[i] == false)
            return false;

    return true;
}

// This function assumes that edge (u, v)
// exists in graph or not,
void Graph::reverseDeleteMST()
{
    // Sort edges in increasing order on basis of cost
    sort(edges.begin(), edges.end());

    int mst_wt = 0;  // Initialize weight of MST

    cout << "Edges in MST\n";

    // Iterate through all sorted edges in
    // decreasing order of weights
    for (int i=edges.size()-1; i>=0; i--)
    {
        int u = edges[i].second.first;
        int v = edges[i].second.second;

        // Remove edge from undirected graph
        adj[u].remove(v);
        adj[v].remove(u);

        // Adding the edge back if removing it
        // causes disconnection. In this case this 
        // edge becomes part of MST.
        if (isConnected() == false)
        {
            adj[u].push_back(v);
            adj[v].push_back(u);

            // This edge is part of MST
            cout << "(" << u << ", " << v << ") \n";
            mst_wt += edges[i].first;
        }
    }

    cout << "Total weight of MST is " << mst_wt;
}

// Driver code
int main()
{
    // create the graph given in above fugure
    int V = 9;
    Graph g(V);

    //  making above shown graph
    g.addEdge(0, 1, 4);
    g.addEdge(0, 7, 8);
    g.addEdge(1, 2, 8);
    g.addEdge(1, 7, 11);
    g.addEdge(2, 3, 7);
    g.addEdge(2, 8, 2);
    g.addEdge(2, 5, 4);
    g.addEdge(3, 4, 9);
    g.addEdge(3, 5, 14);
    g.addEdge(4, 5, 10);
    g.addEdge(5, 6, 2);
    g.addEdge(6, 7, 1);
    g.addEdge(6, 8, 6);
    g.addEdge(7, 8, 7);

    g.reverseDeleteMST();
    return 0;
}

Output :

Edges in MST
(3, 4) 
(0, 7) 
(2, 3) 
(2, 5) 
(0, 1) 
(5, 6) 
(2, 8) 
(6, 7) 
Total weight of MST is 37

Notes :

  1. The above implementation is a simple/naive implementation of Reverse Delete algorithm and can be optimized to O(E log V (log log V)3) [Source : Wiki]. But this optimized time complexity is still less than Prim and Kruskal Algorithms for MST.
  2. The above implementation modifies the original graph. We can create a copy of the graph if original graph must be retained.

References:
https://en.wikipedia.org/wiki/Reverse-delete_algorithm

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