Array Nesting

Array Nesting

What is this problem about?

The Array Nesting interview question involves an array of N integers where each integer is in the range [0, N-1]. You build a set by starting with an index i and jumping to the index specified by the value at i. You continue this until you hit an element already in your set. The goal is to find the longest such set. This Array Nesting coding problem is a graph problem in disguise.

Why is this asked in interviews?

Google and Apple ask this to test your ability to recognize cycles in a permutation. It evaluates whether you can solve the problem in-place and in linear time. The key is realizing that every element belongs to exactly one cycle, so you don't need to re-visit elements you've already seen.

Algorithmic pattern used

This follows the Array, Depth-First Search interview pattern. Each element in the array can be seen as a node with exactly one outgoing edge. This structure forms a set of disjoint cycles. To optimize, you mark visited elements so you only traverse each cycle once, achieving O(N) time complexity.

Example explanation

Consider nums = [5, 4, 0, 3, 1, 6, 2].

1. Start at index 0: nums[0]=5, then nums[5]=6, then nums[6]=2, then nums[2]=0 (back to start). Cycle: {5, 6, 2, 0}. Length = 4.
2. Start at index 1: nums[1]=4, then nums[4]=1 (back to start). Cycle: {4, 1}. Length = 2.
3. Start at index 3: nums[3]=3 (back to start). Cycle: {3}. Length = 1. The longest cycle is 4.

Common mistakes candidates make

• Redundant Work: Starting a search from every index without keeping track of visited elements, leading to O(N^2) complexity.
• Extra Space: Using a large hash set for visited nodes when you could simply mark the visited indices in the original array (if modification is allowed) or use a boolean array.
• Stack Overflow: Using recursion for DFS on very large arrays instead of an iterative approach.

Interview preparation tip

Recognize that if an array represents a mapping where every value is a valid index and all values are unique, the structure consists entirely of closed cycles. This "permutation cycle" property is a frequent theme in medium-level array problems.