Solve the following instance of single source shortest paths problem with vertex 'a' as the source usingsuitable method.

To solve the single-source shortest paths problem with vertex 'a' as the source, we can use Dijkstra's algorithm if all edge weights are non-negative. Here’s how we can approach this problem:

1. ### Break Down the Problem

1. Identify the graph structure including all vertices and edges.
2. Initialize distances from the source vertex 'a' to all other vertices.
3. Implement Dijkstra's algorithm to find the shortest paths.

2. ### Relevant Concepts

• Dijkstra's Algorithm: Used to find the shortest path in a graph with non-negative edge weights.
• Distance Initialization: Distance to the source vertex is zero, and all other vertices are set to infinity.

3. ### Analysis and Detail

Step 1: Graph Representation

• Represent the graph using an adjacency list or matrix.
• Let $V$ be the set of vertices and $E$ be the set of edges.

Step 2: Initialize Distances

• Create a distance array $dist[]$ where:
  • $dist[a] = 0$ (distance to itself)
  • $dist[v] = \infty$ for all $v \neq a$

Step 3: Dijkstra's Algorithm Execution

• Use a priority queue to always expand the least costly node.
• For each neighbor of the current node, update the distances if a shorter path is found.

4. ### Verify and Summarize

• Once Dijkstra's algorithm completes, verify distances for correctness.
• Summarize the shortest path from vertex 'a' to all other vertices.

Final Answer

The result will be an array containing the shortest distances from vertex 'a' to all other vertices in the graph.

If details of the graph (vertices and edges) are provided, specific calculations can be performed to give actual distance values.