What is the optimal time complexity of solving Graph Valid Tree?

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 Graph Valid Tree?

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 Graph Valid Tree?

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 Graph Valid Tree?

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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Graph Valid Tree

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

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Problem Statement

Medium

You have a graph of n nodes labeled from 0 to n - 1. You are given an integer n and a list of edges where edges[i] = [ai, bi] indicates that there is an undirected edge between nodes ai and bi in the graph.

Return True if the edges of the given graph make up a valid tree, and False otherwise.

Write a function validTree(n: int, edges: List[List[int]]) -> bool.

Constraints

•1 <= n <= 2000
•0 <= len(edges) <= 5000
•edges[i].length == 2
•0 <= ai, bi < n

Examples

Example 1

Input

n = 5, edges = [[0,1],[0,2],[0,3],[1,4]]

Output

True

Explanation

The graph is fully connected and contains no cycles, so it is a valid tree.

Example 2

Input

n = 5, edges = [[0,1],[1,2],[2,3],[1,3],[1,4]]

Output

False

Explanation

The graph contains a cycle 1-2-3-1, so it is not a valid tree.

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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