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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M. Ajtai, J. Komlós, E. Szemerédi, An O(n log n) sorting network. Proc. 15th ACM Symposium on Theory of Computing, pp. 1–9, 1983.
V. Beneš, “Permutation groups, complexes, and rearrangeable multistage Connecting networks”, Bell System Technical Journal, vol. 43, pp. 1619–1640, 1964.
V. Beneš, “Mathematical Theory of Connecting Networks and Telephone Traffic”, Academic Press, New York, NY, 1965.
R. Cypher, C.G. Plaxton: Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers. Journal of Computer and System Sciences, No. 47, 1993, pp. 501–548.
M. Cutler, Y. Shiloach: Permutation layout. Networks, vol. 8 (1978) 253–278.
O. Gabber, Z. Galil: Explicit Construction of Linear-Sized Superconcentrators. Journal of Computer and System Sciences, No. 22, 1981, pp. 407–420.
J. Hromkovič, R. Klasing, E.A. Stöhr, “Gossiping in Vertex-Disjoint Paths Mode in Interconnection Networks”, Proc. of the 19th Int. Workshop on Graph-Theoretic Concepts in Computer Science (WG'93), Springer LNCS 790, pp. 288–300.
J. Hromkovič, R. Klasing, E.A. Stöhr, H. Wagener, “Gossiping in Vertex-Disjoint Paths Mode in d-Dimensional Grids and Planar Graphs”, Proc. of the First Annual European Symposium on Algorithms (ESA'93), Springer LNCS 726, pp. 200–211.
J. Hromkovič, R. Klasing, W. Unger, H. Wagener, “Optimal Algorithms for Broadcast and Gossip in the Edge-Disjoint Path Modes”, Proc. of the 4th Scandinavian Workshop on Algorithm Theory (SWAT'94), Springer LNCS 824, pp. 219–230.
R. Klasing, “The Relationship Between Gossiping in Vertex-Disjoint Paths Mode and Bisection Width”, Proc. of the 19th International Symposium on Mathematical Foundations of Computer Science (MFCS'94), Springer LNCS 841, pp. 473–483.
M. Klawe, T. Leighton: A tight lower bound on the size of planar permutation networks. SIAM J. Disc. Math., Vol. 5, No. 4, pp. 558–563, November 1992
M. Kutyłowski, K. Loryś, B. Oesterdiekhoff, R. Wanka: Fast and feasible periodic sorting networks of constant depth; Proc. 35th Annual Symposium on Foundations of Computer Science (FOCS '94), to appear.
F.T. Leighton: Introduction to parallel algorithms and architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann Publisher (1992).
R.J. Lipton, R.E. Tarjan, “A separator theorem for planar graphs”, SIAM J. Appl. Math. 36(2), 1979, pp. 177–189.
N. Pippenger: Superconcentrators. SIAM Journal of Computing, No. 6, 1977, pp. 298–304.
N. Pippenger, L. Valiant: Shifting graphs and their applications. Journal of the ACM, Vol. 23, No. 3, July 1976, pp. 423–432.
Ullman, J.D., Computational Aspects of VLSI Computer Science Press, Rockville, MD, 1984, 495 p.
A. Waksman, “A permutation network”, Journal of the ACM, vol. 15, no. 1, pp. 159–163, January 1968.
R. Werchner, personal communication.
D.B. Wilson: Embedding leveled hypercube algorithms into hypercubes. Proc. 4th ACM Symposium on Parallel Algorithms and Architectures (SPAA '92), pp. 264–270.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/3-540-59042-0_78
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59042-2
Online ISBN: 978-3-540-49175-0
eBook Packages: Springer Book Archive