Explain a Sorting Algorithm Faster Than O(n^2) and Name One Slower Than O(n^2)

This question comes from a rapid-fire technical screen: the interviewer asks many short questions in a row, and the goal is to answer as many as possible, each one clearly and correctly. This question has two parts. ### Clarifying Questions - Are the elements only compared with each other (comparison sorting), or may the algorithm use properties of the keys, such as a small integer range? - Do the time bounds refer to the worst case, or is an expected (average) bound acceptable? ### Part 1 — A sort faster than $O(n^2)$ Explain in detail a sorting algorithm that runs faster than $O(n^2)$ time: how it works step by step, why it achieves its running time, how much extra memory it uses, and whether it is stable. ```hint Count the work per level Many fast sorts split the input repeatedly. Count how much work one level of splitting does in total, and how many levels there are. ``` #### What This Part Should Cover - A correct, step-by-step description of the chosen algorithm - A derivation of its running time, not just the final bound - Extra space, stability, and how the worst case differs from the typical case ### Part 2 — A sort slower than $O(n^2)$ Now give a sorting algorithm that is slower than $O(n^2)$, and state its running time. ```hint Correct, not sensible The algorithm only has to produce sorted output, not be practical. Think about procedures that search among arrangements of the input, or recursive schemes whose subproblems overlap heavily. ``` #### Clarifying Questions for this Part - Must the algorithm be deterministic and always terminate, or is a randomized algorithm with an expected bound acceptable? #### What This Part Should Cover - A procedure that always produces sorted output, or does so with probability 1 if it is randomized - A precise running time that separates the expected case from the worst case for a randomized procedure - An argument for why the bound exceeds quadratic ### What a Strong Answer Covers - Short, correct answers delivered quickly, because the format rewards breadth - Every complexity claim backed by a one-line justification - Awareness of the gap between asymptotic bounds and practical speed (constant factors, memory traffic, recursion overhead) ### Follow-up Questions - Why must every comparison sort make $\Omega(n \log n)$ comparisons in the worst case? - When would you choose counting sort or radix sort instead, and what do they cost in memory? - Why do standard libraries usually ship hybrid sorts rather than a textbook merge sort or quicksort?