# List ranking on interconnection networks

## Abstract

The list-ranking problem is considered for parallel computers which communicate through an interconnection network. Each PU holds *k* nodes of a set of singly linked lists. An easy randomized algorithm gives a considerable improvement over earlier ones.

For a large class of networks, the algorithm takes only twice the number of steps required by a *k-k* routing. The only conditions are that: (1) *k=ω*(*k*^{*}), where k^{*} is so large that the time consumption of *k*^{*} -*k*^{*} routing is determined by the bisection bound, and (2) the routing time slightly increases with the number of PUs in the network.

For special networks we can prove stronger results. Particularly, for n×...×n meshes, the list ranking problem is solved in (1/2+*o*(1)) · *k* · *n* steps, if k=ω(1). For hypercubes with *N* PUs, assuming all-port communication, the algorithm requires only (2+*o*(1)) · *k* steps, if k=ω(log^{2}*N)*.

We show that list ranking requires at least the time required for *k-k* routing. So, the results are within a factor two from optimal. For meshes we even match the lower bound up to lower-order terms.

## Keywords

parallel algorithms bdinterconnection networks list ranking randomization## Preview

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## References

- [1]Anderson, R.J., G.L. Miller, ‘Deterministic Parallel List Ranking,’
*Algorithmica*6 pp. 859–868, 1991.CrossRefGoogle Scholar - [2]Atallah, M.J., S.E. Hambrusch, ‘Solving Tree Problems on a Mesh-Connected Processor Array,’
*Information and Control*, 69, pp. 168–187, 1986.CrossRefGoogle Scholar - [3]Chang, Y., J. Simon, ‘Continuous Routing and Batch Routing on the Hypercube,’
*Proc. 18th Symp. on Theory of Computing*, pp. 272–281, ACM, 1986.Google Scholar - [4]Cole, R., U. Vishkin, ‘Deterministic Coin Tossing and Accelerated Cascades: Micro and Macro Techniques for Designing Parallel Algorithms,’
*Proc. 18th Symp. on Theory of Computing*, pp. 206–219, ACM, 1986.Google Scholar - [5]Gibbons, A.M., Y. N. Srikant, ‘A Class of Problems Efficiently Solvable on Mesh-Connected Computers Including Dynamic Expression Evaluation,’
*Information Processing Letters*, 32, pp. 305–311, 1989.CrossRefGoogle Scholar - [6]Hagerup, T., C. Rüb, ‘A Guided Tour of Chernoff Bounds,’
*Information Processing Letters*, 33, 305–308, 1990.Google Scholar - [7]Hwang, K.,
*Advanced Computer Architecture; Parallelism, Scalability, Programmability*, McGraw-Hill, Inc., 1993.Google Scholar - [8]JáJá, J.,
*An Introduction to Parallel Algorithms*, Addison-Wesley Publishing Company, Inc., 1992.Google Scholar - [9]Kaufmann, M., J.F. Sibeyn, T. Suel, ‘Derandomizing Routing and Sorting Algorithms for Meshes,’
*Proc. 5th Symp. on Discrete Algorithms*, pp. 669–679, ACM-SIAM, 1994.Google Scholar - [10]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, Springer-Verlag, 1993.Google Scholar - [11]McDiarmid, C., ‘On the Method of Bounded Differences,’ in
*Surveys in Combinatorics*, J. Siemons, editor, 1989 London Mathematical Society Lecture Note Series 141, pp. 148–188, Cambridge University Press, 1989.Google Scholar - [12]Reid-Miller, M., ‘List Ranking and List Scan on the Cray C-90,’
*Proc. 6th Symp. on Parallel Algorithms and Architectures*, pp. 104–113, ACM, 1994.Google Scholar - [13]Reif, J., L.G. Valiant, ‘A logarithmic time sort for linear size networks,’
*Journal of the ACM*, 34(1), pp. 68–76, 1987.CrossRefGoogle Scholar - [14]Reischuk, R., ‘Probabilistic Parallel Algorithms for Sorting and Selection,’
*SIAM Journal of Computing*, 14, pp. 396–411, 1985.CrossRefGoogle Scholar - [15]Ryu, K.W., J. JáJá, ‘Efficient Algorithms for List Ranking and for Solving Graph Problems on the Hypercube,’
*IEEE Transactions on Parallel and Distributed Systems*,Vol. 1, No. 1, pp. 83–90, 1990.Google Scholar - [16]Sibeyn, J.F., ‘List Ranking on Interconnection Networks,’
*Techn. Rep. 11/1995, SFB 124-D6*, Universität Saarbrücken, Saarbrücken, Germany, 1995. Preliminary version in*Proc. Computing Science in the Netherlands*, pp. 271–280, SION, Amsterdam, 1994. Submitted to*Acta Informatica*.Google Scholar - [17]Valiant, L.G., ‘A Bridging Model for Parallel Computation,’
*Communications of the ACM*, 33(8), pp. 103–111, 1990.CrossRefGoogle Scholar