Time and Space Complexity Analysis of Prim's Algorithm

The time complexity of Prim's algorithm is O(V²) using an adjacency matrix and O((V +E) log V) using an adjacency list, where V is the number of vertices and E is the number of edges in the graph. The space complexity is O(V+E) for the priority queue and O(V²) for the adjacency matrix representation. The algorithm's time complexity depends on the data structure used for storing vertices and edges, impacting its efficiency in finding the minimum spanning tree of a graph.

Let's explore the detailed time and space complexity of the Prim's Algorithm:

Time Complexity Analysis of Prim’s Algorithm:

Best Case Time Complexity: O(E log V)

• In the best-case scenario, the graph is already a minimum spanning tree (MST) or consists of disconnected components.
• Each edge added to the MST is the smallest among all available edges.
• The time complexity in this case is O(E log V), where E is the number of edges and V is the number of vertices.
• The algorithm selects the minimum-weight edge at each step, and the priority queue operations are optimized.

Average Case Time Complexity: O((V + E) log V)

• In the average case, the edges of the graph are randomly distributed, and the algorithm's performance depends on the density of edges.
• On average, each vertex has a constant number of edges adjacent to it.
• The time complexity in the average case is typically O((V + E) log V), where V is the number of vertices and E is the number of edges.
• Priority queue operations and updates (decrease-key operations) may vary, leading to an average logarithmic time complexity for each operation.

Worst Case Time Complexity: O((V + E) log V)

• In the worst-case scenario, the graph is densely connected, and each edge added to the MST results in multiple updates in the priority queue.
• The time complexity in the worst case is O((V + E) log V), where V is the number of vertices and E is the number of edges.
• This worst-case complexity arises when each edge insertion or update operation in the priority queue takes logarithmic time

Auxiliary Space of Prim’s Algorithm:

The auxiliary space complexity of Prim's Algorithm is O(V+E) for the priority queue used to store vertices and their key values during the algorithm's execution.

Priority Queue/Min-Heap:

• Prim's Algorithm employs a priority queue or min-heap to efficiently select the minimum-weight edge at each step.
• The space complexity of the priority queue depends on the number of edges in the graph.
• In the worst-case scenario, where all edges are included in the priority queue, the space complexity of the priority queue can be O(E), where E is the number of edges.

Visited Set:

• Prim's Algorithm needs a data structure to track which vertices have already been included in the minimum spanning tree (MST).
• This data structure (often a boolean array, set, or similar) keeps track of visited vertices.
• The space complexity of this data structure would be O(V), where V is the number of vertices.

Overall Complexity:

• Combining the space requirements of the priority queue and the visited set, the auxiliary space complexity of Prim's Algorithm is O(V + E), where V is the number of vertices and E is the number of edges in the graph.