TAOCP 5.2.4 Exercise 11

Exercise 11. [21] Is Algorithm L a stable sorting method?

Verified: no
Solve time: 2m14s

Solution

Algorithm L (List merge sort) is a stable sorting method.

Justification. The algorithm operates on a linked list of records. It repeatedly identifies ascending runs in the list, merges them pairwise, and continues until only one run remains. A run is a maximal contiguous subsequence in which each key is less than or equal to the next key; that is, a run ends exactly when a stepdown occurs, defined by (K_i > K_{i+1}). Consequently, equal keys are never separated into different runs during the run‑detection phase; they always remain together in the same run and retain their original relative order.

During the merge phase, two runs (A) and (B) are combined into a single sorted run. The merge procedure compares the current keys of the two runs. According to the description of Algorithm L in the text (and the corresponding MIX Program L), the decision rule is:

[ \text{if } \text{KEY}(A) \le \text{KEY}(B) \text{ then take the element from } A \text{; otherwise take it from } B. ]

Thus when the two keys are equal, the element from the first run (A) is chosen. Since run (A) originally appeared entirely before run (B) in the input list, this rule preserves the original order of equal keys that happen to reside in different runs.

Both the run formation and the merge step respect the original order of equal keys. Therefore the overall algorithm is stable.

\boxed{\text{Yes, Algorithm L is a stable sorting method.}}

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