Implement DFS and tree algorithms

Q: Implement DFS and tree algorithms

This question evaluates graph traversal and tree-algorithm competencies, specifically depth-first search for connected components and binary tree operations such as level-order traversal and height-balance checking, including complexity analysis and scalability considerations for very large sparse graphs.

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Question

Part 1 (DFS on graph): Given an undirected graph with n nodes labeled 0..n-1 and an edge list, implement a function that returns all connected components using depth-first search. Each component should be returned as a list of node labels in ascending order; the list of components should be sorted by each component’s smallest label. Discuss time/space complexity and how you would handle very large sparse graphs.

Part 2 (Binary tree): Given the root of a binary tree, implement (a) level-order traversal that returns values grouped by level, and (b) a function that determines whether the tree is height-balanced (for every node, the height difference between left and right subtrees is at most 1). Provide complexity and key test cases.

Implement DFS and tree algorithms

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