How do I implement Prim's algorithm?

How do I implement Prim's algorithm?

Prim's algorithm is used to find the minimum spanning tree of a graph. Would you like a step-by-step guide?

Yes, please provide the steps.

1. Start with a graph and select an arbitrary vertex as the starting point.
2. Create a set to keep track of visited vertices.
3. While there are unvisited vertices, find the edge with the smallest weight that connects a visited vertex to an unvisited vertex.
4. Add the selected edge and vertex to the minimum spanning tree and mark the vertex as visited.
5. Repeat until all vertices are visited.

Can you show me a code example?

Sure! Here's a simple implementation in Python:

import heapq

def prim(graph, start):
    mst = []
    visited = set([start])
    edges = [(cost, start, to) for to, cost in graph[start].items()]
    heapq.heapify(edges)

    while edges:
        cost, frm, to = heapq.heappop(edges)
        if to not in visited:
            visited.add(to)
            mst.append((frm, to, cost))
            for to_next, cost in graph[to].items():
                if to_next not in visited:
                    heapq.heappush(edges, (cost, to, to_next))
    return mst

This will give you the minimum spanning tree.

What is the time complexity of Prim's algorithm?

The time complexity of Prim's algorithm is O(E log V) when using a priority queue, where E is the number of edges and V is the number of vertices.