Design Most Recently Used Queue

Design Most Recently Used Queue

What is this problem about?

The Design Most Recently Used Queue interview question asks you to create a specialized queue of $n$ elements (initially $1, 2, \dots, n$). You need to implement a fetch(k) function that retrieves the $k$-th element (1-indexed) in the current queue, removes it, and moves it to the very end. The challenge is to do this efficiently as $n$ and the number of calls to fetch can be large.

Why is this asked in interviews?

Google uses this problem to test a candidate's knowledge of advanced data structures beyond simple arrays and queues. A naive array implementation would take $O(n)$ for every fetch (due to shifting elements), which is too slow. It evaluates your ability to use Divide and Conquer interview patterns or balanced trees to handle dynamic indexing and updates in logarithmic or square-root time.

Algorithmic pattern used

There are two main ways to optimize this:

1. Square Root Decomposition: Divide the $n$ elements into $\sqrt{n}$ blocks. Each fetch involves finding the block containing the $k$-th element, which is $O(\sqrt{n})$, and shifting elements within that block and between blocks, also $O(\sqrt{n})$.
2. Binary Indexed Tree (BIT) or Segment Tree: Use a structure to keep track of "active" positions in a larger array. This allows finding the $k$-th existing element in $O(\log^2 n)$ or $O(\log n)$ time.

Example explanation

Initial Queue (n=5): [1, 2, 3, 4, 5]

1. fetch(3):
   • Find the 3rd element: '3'.
   • Move it to the end: [1, 2, 4, 5, 3].
2. fetch(2):
   • Find the 2nd element in the new queue: '2'.
   • Move it to the end: [1, 4, 5, 3, 2].

Common mistakes candidates make

• Brute Force: Using a standard array or list and calling remove(k) and append(), which leads to $O(n)$ complexity and TLE (Time Limit Exceeded).
• Indexing Confusion: Forgetting that the queue is 1-indexed in the problem description but 0-indexed in most programming languages.
• Complexity Analysis: Failing to realize that a standard LinkedList also takes $O(n)$ to find the $k$-th node, even if the insertion is $O(1)$.

Interview preparation tip

This is a classic "dynamic order statistic" problem. If you encounter a problem that requires finding the "$k$-th" element while the data structure is frequently modified, think of Square Root Decomposition or a Segment Tree. These are high-signal patterns for senior-level interviews.