What is the optimal time complexity of solving Prim?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Prim?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Prim?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Prim?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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DSA SectionEasy

Prim

Learn how to solve the 'Prim' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function prim_mst(graph) that takes an undirected, connected, weighted graph represented as an adjacency list of dictionaries (where graph[u][v] is the weight of edge (u, v)) and returns the sum of weights of the Minimum Spanning Tree (MST) using Prim's algorithm.

Constraints

•1 <= V <= 500
•0 <= E <= 1000

Examples

Example 1

Input

graph = {0: {1: 2, 3: 6}, 1: {0: 2, 2: 3, 3: 8, 4: 5}, 2: {1: 3, 4: 7}, 3: {0: 6, 1: 8, 4: 9}, 4: {1: 5, 2: 7, 3: 9}}

Output

Explanation

MST edges selected are: (0,1) wt 2, (1,2) wt 3, (1,4) wt 5, (0,3) wt 6. Total weight = 16.

Need a Hint?

Represent graph node connections as an adjacency list/matrix, then use standard BFS or DFS graph traversal.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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