Array Nesting

Problem Description

You are given an integer array nums of length n where nums is a permutation of the numbers in the range [0, n - 1]. You should build a set s[k] = {nums[k], nums[nums[k]], nums[nums[nums[k]]], ... } subjected to the following rule:

• The first element in s[k] starts with the selection of the element nums[k] of index k.
• The next element in s[k] should be nums[nums[k]], and then nums[nums[nums[k]]], and so on.
• We stop adding right before a duplicate element occurs in s[k].

Return the longest length of a set s[k].

Key Insights

• Each index in the array leads to a unique value, creating a "cycle" due to the properties of permutations.
• The problem can be visualized as finding cycles in a directed graph where nodes are indices and edges point to their respective values.
• A depth-first search (DFS) approach can be used to traverse through the cycles and count their lengths.
• Keeping track of visited indices will help in avoiding counting the same cycle multiple times.

Space and Time Complexity

Time Complexity: O(n) - Each element is visited once. Space Complexity: O(n) - To track visited nodes.

Solution

To solve the problem, we can use an iterative or recursive depth-first search (DFS) approach. We will maintain a list to track which indices have been visited. For each index k, we will follow the sequence of indices until we revisit an already visited index, thus forming a cycle. The length of each cycle will be compared to find the maximum length.

Code Solutions