Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 85
2.4.4 Polyphase Sort
We have now discussed the necessary techniques and have acquired the proper background to
investigate and program yet another sorting algorithm whose performance is superior to the balanced sort.
We have seen that balanced merging eliminates the pure copying operations necessary when the
distribution and the merging operations are united into a single phase. The question arises whether or not
the given sequences could be processed even more efficiently. This is indeed the case; the key to this next
improvement lies in abandoning the rigid notion of strict passes, i.e., to use the sequences in a more
sophisticated way than by always having N sources and as many destinations and exchanging sources and
destinations at the end of each distinct pass. Instead, the notion of a pass becomes diffuse. The method
was invented by R.L. Gilstad [2-3] and called Polyphase Sort.
It is first illustrated by an example using three sequences. At any time, items are merged from two
sources into a third sequence variable. Whenever one of the source sequences is exhausted, it immediately
becomes the destination of the merge operations of data from the non-exhausted source and the previous
destination sequence.
As we know that n runs on each input are transformed into n runs on the output, we need to list only the
number of runs present on each sequence (instead of specifying actual keys). In Fig. 2.14 we assume that
initially the two input sequences f0 and f1 contain 13 and 8 runs, respectively. Thus, in the first pass 8 runs
are merged from f0 and f1 to f2, in the second pass the remaining 5 runs are merged from f2 and f0 to f1,
etc. In the end, f0 is the sorted sequence.
f0 f1 f2
13 8
5 0 8
0 5 3
3 2 0
1 0 2
0 1 1
1 0 0
Fig. 2.14. Polyphase mergesort of 21 runs with 3 sequences
A second example shows the Polyphase method with 6 sequences. Let there initially be 16 runs on f0,
15 on f1, 14 on f2, 12 on f3, and 8 on f4. In the first partial pass, 8 runs are merged onto f5. In the end,
f1 contains the sorted set of items (see Fig. 2.15).