# Routing on triangles, tori and honeycombs

• Jop F. Sibeyn
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1113)

## Abstract

The standard n×n torus consists of two sets of axes: horizontal and vertical ones. For routing h-relations, the bisection bound gives a lower bound of h · n/4. Several algorithms nearly matching this bound have been given.

In this paper we analyze the routing capacity of modified tori: tessellations of the plane with triangles or hexagons and tori with added diagonals. On some of these networks the ratio of routing capacity and degree is higher than for ordinary tori, even though they are as easily constructed. Hence, they may constitute more cost effective alternatives.

For networks with n2 PUs, we get the following results: on a torus of hexagons, node degree 3, h-relations are performed in 0.37 · h · n steps; on a torus of triangles, node degree 6, in 0.13 ·h·n; and on a torus with added diagonals, node degree 8, in h·n/12. The latter result matches the bisection bound for this network. Even faster is the routing on a torus of hexagons with diagonals, node degree 12: 0.053·h·n.

The algorithm is simple, inspired by the algorithm of Valiant and Brebner. The results can easily be extended to sorting, dynamic routing or routing for average-case inputs.

## Classification

Theory of parallel and distributed computation VLSI structures parallel algorithms

## References

1. [1]
Kaufmann, M., S. Rajasekaran, J.F. Sibeyn, ‘Matching the Bisection Bound for Routing and Sorting on the Mesh,’ Proc. 4th Symposium on Parallel Algorithms and Architectures, pp. 31–40, ACM, 1992.Google Scholar
2. [2]
Kaufmann, M., J.F. Sibeyn, ‘Optimal Multi-Packet Routing on the Torus', Proc. 3rd Scandinavian Workshop on Algorithm Theory, LNCS 621, pp. 118–129, Springer-Verlag, 1992.Google Scholar
3. [3]
Kaufmann, M., J.F. Sibeyn, T. Suel, ‘Derandomizing Routing and Sorting Algorithms for Meshes,’ Proc. 5th Symposium on Discrete Algorithms, pp. 669–679, ACMSIAM, 1994.Google Scholar
4. [4]
Kunde, M., ‘Block Gossiping on Grids and Tori: Deterministic Sorting and Routing Match the Bisection Bound,’ Proc. European Symp. on Algorithms, LNCS 726, pp. 272–283, 1993.Google Scholar
5. [5]
Kunde, M., R. Niedermeier, P. Rossmanith, ‘Faster Sorting and Routing on Grids with Diagonals,’ Proc. 11th Symposium on Theoretical Aspects of Computer Science, LNCS 775, pp. 225–236, Springer Verlag, 1994.Google Scholar
6. [6]
Kunde, M., T. Tensi, ‘Multi-Packet Routing on Mesh Connected Processor Arrays,’ Proc. Symposium on Parallel Algorithms and Architectures, pp. 336–343, ACM, 1989.Google Scholar
7. [7]
Rajasekaran, S., M. Raghavachari, ‘Optimal Randomized Algorithms for Multipacket and Cut Through Routing on the Mesh', Journal of Parallel and Distributed Computing, 26(2), pp. 257–260, 1995.Google Scholar
8. [8]
Reif, J., L.G. Valiant, ‘A logarithmic time sort for linear size networks,’ Journal of the ACM, 34(1), pp. 68–76, 1987.
9. [9]
Reischuk, R., ‘Probabilistic Parallel Algorithms for Sorting and Selection,’ S1AM Journal of Computing, 14, pp. 396–411, 1985.
10. [10]
Sibeyn, J.F., ‘Sample Sort on Meshes,’ Proc. Computing Science in the Netherlands, SION, Amsterdam, 1995. Full version in Techn. Rep. MPI-I-95-112, Max-Planck Institut für Informatik, Saarbrücken, Germany, 1995.Google Scholar
11. [11]
Valiant, L.G., ‘A Bridging Model for Parallel Computation,’ Communications of the ACM, 33(8), pp. 103–111, 1990.
12. [12]
Valiant, L.G., G.J. Brebner, ‘Universal Schemes for Parallel Communication,’ Proc. 13th Symposium on Theory of Computing, pp. 263–277, ACM, 1981.Google Scholar