Abstract
The sizes of permutation networks for special sets of permutations are investigated. The study of the planar realization and the search for small but hard sets of permutations are also included. Several asymptotically optimal estimations for distinct subsets of the set of all permutations are established here.
The two main results are:
-
(i)
an asymptotically optimal permutation network of size 6·N·log log N for shifts of power 2.
-
(ii)
an asymptotically optimal planar permutation network of size Θ(N 2·(loglog N/log N)2) for shifts of power 2.
A consequence of our results is a construction of a 4-degree network which can simulate each communication step of any hypercube algorithm using edges from at most a constant number of different dimensions in one step in O(loglog N) communication steps. A new sorting network as well as an essential improvement of gossiping in vertex-disjoint path mode in bounded-degree networks follow.
This work was partially supported by grants Mo 285/9-1 and Me 872/6-1 (Leibniz Award) of the German Research Association (DFG), and by the ESPRIT Basic Research Action No. 7141 (ALCOM II).
Supported by SAV Grant 2/1138/94 and by EC Cooperation Action IC 1000 ALTEC
Supported by Polish Government grants KBN 2 1197 91 01, KBN 8 S503 002 07, KBN 2 P301 034 07 & Volkswagen Stiftung (Joint project of Wroclaw University and Heinz-Nixdorf-Institut, University of Paderborn).
This author was supported by the Ministerium für Wissenschaft und Forschung des Landes Nordrhein-Westfalen.
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Hromkovič, J., Loryś, K., Kanarek, P., Klasing, R., Unger, W., Wagener, H. (1995). On the sizes of permutation networks and consequences for efficient simulation of hypercube algorithms on bounded-degree networks. In: Mayr, E.W., Puech, C. (eds) STACS 95. STACS 1995. Lecture Notes in Computer Science, vol 900. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59042-0_78
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