Second Best Minimum Spanning Tree

Prerequisites – Graph, Spanning tree, Disjoint Set (Union – Find).

A minimum spanning tree (MST) T, for a given graph G, spans over all vertices of a given graph and has minimum weight sum of all edges, out of all the possible spanning trees. 

Second best MST, T’, is a spanning tree with the second minimum weight sum of all edges, out of all spanning trees of graph G.

T and T’ differ by only one edge replacement. So, we should find an edge e_new which is not in T and replace it with an edge in T (say e_old) such that T’ = T union {e_new} – {e_old} is a spanning tree and weight difference of (e_new – e_old) is minimum (e_new, e_old are edges in the graph G).

Using Kruskal’s Algorithm –

• Use Kruskal’s algorithm to find MST T of graph G. Remove a single edge from it and replace it with another to obtain T’.
• Sort the edges in O(ElogE) time (E-no.of edges) and find MST using Kruskal’s algorithm in O(E) time (No.of edges in MST = V-1 where V = no.of vertices in the graph G).
• For each edge in MST, temporarily exclude it from the edge list (so that we cannot choose it).
• Then, try to find MST in O(E) using the remaining edges. (no need to sort again)
• Repeat the above for all edges in MST and take the best one. (with 2nd minimum weight sum). Thus, we obtained  T’ – second-best MST.
• Overall Time Complexity – O(ElogE + E +VE) = O(VE)

Below is the implementation of the above approach:

C++

// C++ implementation to find the
// second best MST

#include <bits/stdc++.h>
using namespace std;

// used to implement union-find algorithm
int parent[100005];

// to keep track of edges in MST
vector<int> present;

// to keep track of number of edges
// in spanning trees other than the MST
int edg;

// a structure to represent a
// weighted edge in graph
struct edge {
    int src, dest, weight;
} edges[100005];
// array edges is of type edge.

// Compare two edges according
// to their weights.
// Used in sort() for sorting
// an array of edges
bool cmp(edge x, edge y)
{
    return x.weight < y.weight;
}

// initialising the array -
// each vertex is its own parent
// initially
void initialise(int n)
{
    // 1-indexed
    for (int i = 1; i <= n; i++)
        parent[i] = i;
}

// Implementing the union-find algorithm
int find(int x)
{
    if (parent[x] == x)
        return x;
    return parent[x] = find(parent[x]);
}

// Function to find the union
// for the Minimum spanning Tree
int union1(int i, int sum)
{
    int x, y;
    x = find(edges[i].src);
    y = find(edges[i].dest);
    if (x != y) {

        // parent of x = y (LCA) -
        // both are edge connected
        parent[x] = y;

        // keeping track of edges in MST
        present.push_back(i);

        // finding sum of weights
        // of edges in MST
        sum += edges[i].weight;
    }
    return sum;
}

// Function to find the second
// best minimum spanning Tree
int union2(int i, int sum)
{
    int x, y;
    x = find(edges[i].src);
    y = find(edges[i].dest);
    if (x != y) {
        // parent of x = y (LCA) -
        // both are edge connected
        parent[x] = y;

        // sum of weights of edges
        // in spanning tree
        sum += edges[i].weight;
        edg++;
    }
    return sum;
}

// Driver Code
int main()
{
    // V-> Number of vertices,
    // E-> Number of edges
    int V, E;
    V = 5;
    E = 8;

    // initialising the array to
    // be used for union-find
    initialise(V);

    // src, dest and weights can
    // also be taken from user as
    // input the following vectors
    // represent - source[0],
    // destination[0] are connected
    // by an edge with
    // weight[0]

    vector<int> source = { 1, 3, 2, 3,
                           2, 5, 1, 3 };
    vector<int> destination = { 3, 4, 4,
                                2, 5, 4, 2, 5 };
    vector<int> weights = { 75, 51, 19,
                            95, 42, 31, 9, 66 };
    for (int i = 0; i < E; i++) {
        edges[i].src = source[i];
        edges[i].dest = destination[i];
        edges[i].weight = weights[i];
    }

    // sorting the array of edges
    // based on edge weights
    sort(edges, edges + E, cmp);

    int sum = 0;
    for (int i = 0; i < E; i++) {
        sum = union1(i, sum);
    }

    // printing the cost of MST
    cout << "MST: " << sum << "\n";

    // initialising cost of second best MST
    int sec_best_mst = INT_MAX;

    // setting the sum to zero again.
    sum = 0;
    int j;
    for (j = 0; j < present.size(); j++) {
        initialise(V);
        edg = 0;
        for (int i = 0; i < E; i++) {

            // excluding one edge of
            // MST at a time
            // and forming spanning tree
            // with remaining
            // edges
            if (i == present[j])
                continue;
            sum = union2(i, sum);
        }
        // checking if number of edges = V-1 or not
        // since number of edges in a spanning tree of
        // graph with V vertices is (V-1)
        if (edg != V - 1) {
            sum = 0;
            continue;
        }

        // storing the minimum sum
        // in sec_best_mst
        if (sec_best_mst > sum)
            sec_best_mst = sum;
        sum = 0;
    }

    // printing the cost of second best MST
    cout << "Second Best MST: "
         << sec_best_mst << "\n";
    return 0;
}

Output

MST: 110
Second Best MST: 121

Time Complexity – O(VE) where V – number of vertices in the input graph, E – number of edges in the input graph.