Compare Merge Sort with Factorial and Randomized Inefficient Sorts

Explain a comparison-sorting algorithm with better than quadratic worst-case time, then contrast it with sorting procedures whose cost grows faster than quadratic. ### Part 1 — An Efficient Sort Use merge sort to explain division, merging, the base case, time complexity, and auxiliary space. Discuss stability and how an adaptive merge-based implementation can exploit existing order. #### What This Part Should Cover A correct merge invariant, O(n log n) reasoning, and clear separation between an algorithm and a runtime's implementation choice. ### Part 2 — Inefficient Sorts Analyze checking permutations for a sorted order and repeatedly shuffling until the input is sorted. Distinguish finite worst-case bounds from expected time, and assess whether there is a meaningful absolute worst sorting algorithm. #### What This Part Should Cover Factorial growth, the assumptions behind random-shuffle analysis, and why arbitrary extra work prevents a unique worst algorithm. ### Constraints Assume n distinct comparable elements for the permutation and random-shuffle discussion, constant-cost comparisons, and independent uniform shuffles when randomness is used. Do not generalize one JavaScript engine's sorting implementation to all virtual machines. ### Clarifying Questions - Are we comparing worst-case time, expected time, memory, or practical runtime? - Must the output be stable or produced in place? ```hint Separate the cost of one attempt from the number of attempts A shuffle-based procedure must both create a candidate order and determine whether that order is sorted. ``` ### What a Strong Answer Covers - Correct merge-sort mechanics and complexity. - Factorial enumeration and randomized stopping-time analysis. - Qualified runtime implementation claims and no unsupported absolute worst-case label. ### Follow-up Questions - Why should equal elements be taken from the left run first for a stable merge? - What changes in the shuffle probability when some values are duplicates?