TAOCP 5.2.2 Exercise 30
When multiword keys are sorted lexicographically, the standard quicksort (Algorithm Q) wastes time by repeatedly comparing equal leading words as the subfiles become smaller.
Section 5.2.2: Sorting by Exchanging
Exercise 30. ▶ [25] (P. Shackleton.) When multiword keys are being sorted, many sorting methods become progressively slower as the file gets closer to its final order, since equal and nearly-equal keys require an inspection of several words to determine the proper lexicographic order. (See exercise 5.5.) Files that arise in practice often involve such keys, so this phenomenon can have a significant impact on the sorting time.
Explain how Algorithm Q can be extended to avoid this difficulty: within a subfile in which the leading $k$ words are known to have constant values for all keys, only the $(k+1)$st words of the keys should be inspected.
Verified: no
Solve time: 30s
Solution
When multiword keys are sorted lexicographically, the standard quicksort (Algorithm Q) wastes time by repeatedly comparing equal leading words as the subfiles become smaller. To eliminate this redundancy, we extend Algorithm Q by associating with each subfile an integer (k) representing the number of leading words known to be identical for all keys in that subfile. Initially (k = 0) for the entire file.
Modified partitioning. When a subfile with parameter (k) is to be partitioned, we choose a pivot key (s). Instead of comparing full keys, we compare only the ((k+1))st words. The partitioning step divides the subfile into three groups:
- (L): keys whose ((k+1))st word is less than that of (s);
- (E): keys whose ((k+1))st word equals that of (s);
- (G): keys whose ((k+1))st word is greater than that of (s).
Recursive calls. The groups (L) and (G) still share the same first (k) words (they differ at position (k+1)), so they are recursively sorted with the same parameter (k). The group (E) shares the first (k+1) words, so it is recursively sorted with parameter (k+1). If a subfile reaches a state where all its keys are completely equal (i.e., (k) equals the total number of words in the key), it is already sorted and no further partitioning is needed.
This extension guarantees that each word of each key is inspected at most once during the entire sort. By passing the current prefix length (k) down the recursion, the algorithm automatically skips known equal prefixes, thereby avoiding the progressive slowdown that occurs when many keys share long common prefixes.