1. **Problem Statement:** Determine if the sorting algorithm A described is stable. 2. **Recall the definition of stability in sorting:** A sorting algorithm is stable if it preserves the relative order of equal elements from the input in the output. 3. **Description of algorithm A:** - Input array of length $n$ (where $n$ is a power of 3). - Divide into three subarrays each of length $\frac{n}{3}$. - Sort first subarray with Merge Sort (which is stable). - Sort second subarray with Insertion Sort (which is stable). - Sort third subarray recursively with algorithm A. - Merge the three sorted subarrays using a standard merge procedure. 4. **Key points:** - Merge Sort and Insertion Sort are stable. - The recursive call to A on the third subarray is assumed to be stable if A is stable (inductive assumption). - The final step merges three sorted subarrays. 5. **Is the final merge stable?** - Standard merge procedure for two arrays is stable. - Here, merging three sorted subarrays is done by merging all three together. - If the merge procedure merges three arrays by repeatedly merging two arrays at a time using a stable merge, stability is preserved. - However, if the merge merges all three arrays simultaneously, the stability depends on the implementation. 6. **Crucial observation:** - The three subarrays come from different parts of the original array. - Elements in different subarrays are distinct in position. - Stability concerns only elements that are equal. - Since the subarrays are sorted independently, equal elements that were originally in different subarrays will be merged. - The merge procedure must preserve the order of equal elements from the subarrays. 7. **Conclusion:** - Since each subarray is sorted stably, and the merge procedure is a standard stable merge extended to three arrays, the relative order of equal elements is preserved. - Therefore, algorithm A is stable. **Final answer:** True