Faster Small-Constant-Periodic Merging Networks

We consider the problem of merging two sorted sequences on a comparator network that is used repeatedly, that is, if the output is not sorted, the network is applied again using the output as input. The challenging task is to construct such networks of small depth (called a period in this context). In our previous paper Faster 3-Periodic Merging Network we reduced twice the time of merging on 3periodic networks, i.e. from 12log N to 6log N, compared to the first construction given by Kutyłowski, Loryś and Oesterdikhoff. Note that merging on 2-periodic networks require linear time. In this paper we extend our construction, which is based on Canfield and Williamson (logN)-periodic sorter, and the analysis from that paper to any period p ≥ 4. For p ≥ 4 our p-periodic network merges two sorted sequences of length N/2 in at most 2p p−2 logN + p p−8 p−2 rounds. The previous bound given by Kutyłowski at al. was 2.25p p−2.42 logN. That means, for example, that our 4-periodic merging networks work in time upper-bounded by 4logN and our 6-periodic ones in time upper-bounded by 3log N compared to the corresponding 5.67log N and 3.8logN previous bounds. Our construction is regular and follows the same periodification schema, whereas some additional techniques were used previously to tune the construction for p ≥ 4. Moreover, our networks are also periodic sorters and tests on random permutations show that average sorting time is closed to log2 N.