Difference between the shortest and second shortest path in an Unweighted Bidirectional Graph

Introduction

In the area of graph theory, unweighted bidirectional graphs form a fundamental framework for modeling various real-world scenarios. These graphs allow us to explore relationships between different entities, such as road networks or social connections. One key aspect that captures our attention is finding paths between two nodes and determining their respective lengths. In this article, we dive into one interesting facet of this topic – understanding the distinction between the shortest path and second shortest path in an unweighted bidirectional graph.

Shortest and second shortest path

Unweighted bidirectional (or undirected) graphs consist of vertices or nodes connected by edges. These edges hold no specific weight or distance assigned to them but instead only indicate a connection exists without any directional bias.

A path represents a sequence of interconnected vertices where each vertex is directly linked to its adjacent counterparts through edges. Here, we focus on finding paths that connect two distinct vertices explicitly called source and destination nodes with minimal total edge traversals.

Shortest Path

The concept of finding the shortest path has gained significant attention due to its widespread applications across computer science disciplines such as networking algorithms or GPS route planning systems. The shortest path refers to the most direct sequence of vertices from a given source node to its corresponding destination with minimal edge traversals.

In other words, it depicts journeys that cover fewer connecting edges compared to alternative routes present within an unweighted bidirectional graph. Various algorithms exist for computing these minimal paths; one notable example is Dijkstra's algorithm which ensures efficiency even when dealing with large-scale graphs effectively.

Second Shortest Path

The crucial distinguishing aspect lies in the number of edges traversed. While both paths aim to connect the same source and destination nodes, they differ regarding edge utilization. The second shortest path guarantees that it still represents a valid connection between two nodes while intentionally excluding any overlap with the shortest path. This exclusion means that at least one edge used in the primary shortest path will be avoided in this alternative route.

While this might seem counterintuitive at first, understanding why these distinctions exist can help us explore more efficient and diverse routing possibilities within unweighted bidirectional graphs.

Example

We have an undirected graph with n=4 nodes and m=4 edges. The edge connections can be described as follows −

arr[0] = {1, 2}
arr[1] = {2, 3}
arr[2] = {3, 4}
arr[3] = {1, 4}

Our goal is to find both the length of the shortest path from node "1" to node "4" and calculate its difference from the length of the second-shortest path. In the below code, the shortest path for the above graph is,

1 -> 2-> 3 -> 4

The second shortest path is,

1 -> 4

The length of shortest path is 3 and the second shortest path is 2, so the difference between them is 1.

Approach 1: C++ Program to find the difference between the shortest and second shortest path

To solve this problem optimally, we'll utilize Dijkstra's algorithm modified to retrieve not only one but two minimum distances.

Algorithm

• Step 1 − Initialize necessary data structures including adjacency matrix or linked list representation.

• Step 2 − Create a priority queue or min-heap data structure based on distance values (minimum at top).

• Step 3 − Start by setting all distances as infinite except for source vertex 's', which should be zero.

• Step 4 − Insert the parameters as distance[s] and ‘s’ into priority_queue or heap.

• Step 5 − While priority_queue / heap is not empty −

  • Extract top vertex 'u' from priority_queue / heap

  • For each neighbor vertex v adjacent to u −

    • Update distance[v] = distance[u] + 1

    • Insert (distance[v], v) into priority_queue / heap

• Step 6 − After the entire graph is processed, distances array will store shortest paths from source vertex 's' to all other vertices.

Example

//Including the required header files
#include <iostream>
#include <vector>
#include <queue>
#include <limits>

using namespace std;

const int INF = numeric_limits<int>::max();
//Declaring the function with three arguments
int shortestDifference(int numNodes, int numEdges, vector<pair<int,int>>& edges) {
   vector<vector<int>> adjacency(numNodes+1);
   for(auto& edge : edges) {
      adjacency[edge.first].push_back(edge.second);
      adjacency[edge.second].push_back(edge.first); // since it is an undirected graph
   }

   priority_queue<pair<int,int>, vector<pair<int,int>>, greater<pair<int,int>>> pq;

   vector<int> dist(numNodes+1, INF);

   const int source = 1; // Start node

   dist[source] = 0;

   pq.push({dist[source],source});

   while(!pq.empty()) {
      int currentDistance=pq.top().first;
      int currentNode=pq.top().second;

      pq.pop();

      if(currentDistance>dist[currentNode])
      continue;

      for(auto neighbor:adjacency[currentNode]) {
         if(dist[neighbor]>currentDistance+1) {
            dist[neighbor]=currentDistance+1;
            pq.push({dist[neighbor],neighbor});
         }
      }
   }

   return abs(dist[numNodes]-2*dist[4]);
}

// Testing the code with the required input 

int main() {
   vector<pair<int,int>> edges={{0,0}, {1,2}, {2,3}, {3,4}, {1,4}};
   int numNodes = 4;
   int numEdges = 5;
   cout << shortestDifference(numNodes, numEdges, edges) << endl;
}

Output

1

Conclusion

The former establishes an optimal connection by minimizing edge traversals without consideration for alternatives. On the other hand, the latter embraces diversity by offering an alternate route while avoiding overlapping edges with its shorter counterpart. Mastering these concepts allows us to build powerful algorithms applicable across various domains such as network optimization or navigation systems, enabling us to unlock new potentials within graph theory research.