How do I implement Prim's algorithm in Python?

How do I implement Prim's algorithm in Python?

To implement Prim's algorithm in Python, you can use a priority queue to efficiently select the minimum edge. Would you like a code example?

Yes, please provide a code example.

Here's a simple implementation:

import heapq

class Graph:
    def __init__(self, vertices):
        self.V = vertices  # No. of vertices
        self.graph = []  # default dictionary

    def add_edge(self, u, v, w):
        self.graph.append((w, u, v))

    def prim_mst(self):
        self.graph.sort()  # Sort edges based on weight
        parent = [None] * self.V
        key = [float('inf')] * self.V
        min_heap = [(0, 0)]  # (weight, vertex)
        key[0] = 0

        while min_heap:
            weight, u = heapq.heappop(min_heap)
            for edge in self.graph:
                w, v1, v2 = edge
                if v1 == u and key[v2] > w:
                    key[v2] = w
                    parent[v2] = u
                    heapq.heappush(min_heap, (w, v2))
        return parent

This code defines a graph and implements Prim's algorithm to find the minimum spanning tree.

What are the time and space complexities of this implementation?

The time complexity of Prim's algorithm using a priority queue is O(E log V), where E is the number of edges and V is the number of vertices. The space complexity is O(V) for storing the key and parent arrays.