Is Mergesort a stable sorting algorithm?

Question

BlackTom AI · Accepted Answer

When evaluating whether Mergesort is a stable sorting algorithm, we begin by recalling the definition of stability: a stable sort preserves the relative order of equal elements as they appeared in the input.

Option 1: True. Merge sort is traditionally stable. The standard merge procedure combines two sorted sublists by repeatedly selecting the smaller (or equal) head element from the sublists and placing it into the output. When equal elements appear, a stable implementation tends to take the element from the left sublist first, thereby preserving the original relative order of equal elements from the input. This behavior aligns with the formal notion of stability and is why merge sort is commonly taught as a stable sorting method.

Option 2: False. This claim would be incorrect in typical analyses of merge sort, because the stability property is not inherently violated by the divide-and-conquer approach. A naive or specialized variation could potentially break stability if equal elements are not handled carefully during merging, but the standard algorithm and most practical implementations maintain stability. The misconception here might be confusing stability with in-place constraints or with certain edge-case behaviors, yet those do not automatically negate stability when a correct merge step is used.

To connect the ideas: stability depends on how ties are resolved during the merge. If the merge step always prefers elements from the left subarray when equal, stability is preserved. If, however, the merge takes from the right subarray on ties, stability could be violated. Therefore, the correctness of the statement hinges on the merge implementation details rather than the high-level algorithm name alone.

In summary, understanding stability in the context of mergesort requires focusing on the merge rule for equal keys and ensuring that rule preserves input order; nuances in implementation could alter this property, though the standard, properly-implemented merge sort is stable.

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