Abstract
In this paper we show some important randomization techniques for the parallel processing of discrete problems. In particular, we present several parallel randomized algorithms frequently used for sorting, packet routing, shortest paths problems, matching problems, depth first search, minimum cost spanning trees, and maximal independent set problems. We also discuss the connection between randomization and approximation, showing how randomization yields approximate solutions and we illustrate this connection by means of network flow problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Ajtai, J. Komlós and E. Szemerédi, An O(n log n) sorting network, Proc. 15th Annual ACM STOC, 1983, pp. 1–9.
N. Alon and J. H. Spencer, The Probabilistic Method, (WileyInterscience Publication, 1992 ).
A. Aggarwal, R. J. Anderson, A random NC algorithm for depth first search, Proc. 19th Annual ACM STOC, 1987, pp. 325–334.
R. J. Anderson, A parallel algorithm for the maximal path problem, Combinatorica 7 (3), 1987, pp. 400–415.
L. Babai, F. L. Levin, and M. Szegedy, Checking computation in poly-logarithmic time, Proc. 23rd Annual ACM STOC, 1991, pp. 21–28.
K. E. Batcher, Sorting networks and their applications, Proc. Spring Joint Computer Conference 32, (AFIPS Press, 1968 ), pp. 307–314.
P. Beame and J. Hastad, Optimal bounds for decision problems on the CRCW PRAM, Proc. 19th Annual ACM STOC, 1987, pp. 83–93.
G. E. Blelloch, C. E. Leiserson, B. M. Maggs, C. G. Plaxton, S. J. Smith, and M. Zagha, A comparison of sorting algorithms for the connection machine CM-2, Proc. 3rd Annual ACM Symposium on Parallel Algorithms and Architectures, 1991.
B. Bollobas, Random Graphs, (Academic Press, 1985 ).
A. Clementi, L. Kucera, and J. Rolim, A note on parallel randomized algorithms for searching problems, to appear in DIMACS Series in Discrete Mathematics and Theoretical Computer Sciences, (American Mathematical Society, 1994 ).
E. Cohen, Polylog-time and near-linear work approximation scheme for undirected shortest paths, Proc. 26th Annual ACM STOC, 1994, pp. 16–26.
R. Cole, Parallel merge sort, SIAM J. Comp, 17 (4), 1988, pp. 770–785.
R. Cole, P. Klein, and R. Tarjan, Finding minimum spanning forests in logarithmic time and linear work using random sampling, Proc. Eighth Annual Symposium on Parallel Algorithms and Architectures, 1996, pp. 243–250.
D. Coppersmith, P. Raghavan, and M. Tompa, Parallel graph algorithms that are efficient on average, Proc. 28th Annual IEEE FOCS, 1987, pp. 260–269.
L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comp 5, 1976, pp. 618–623.
R. E. Cypher and C. G. Plaxton, Deterministic sorting in nearly logarithmic time on the hypercube and related computers, Proc. 22nd Annual ACM STOC, 1990, pp. 193–203.
P. Erdos and A. Renyi, On random graphs I, Publ. Math. Debrecen 6, 1959, pp. 290–297.
L. R. Ford and D. R. Fulkerson, Flows in Networks, (Princeton University Press, 1962 ).
W. D. Frazer and A. C. McKellar, Samplesort: a sampling approach to minimal storage tree sorting, Journal of the ACM, 17 (3), 1970, pp. 496–507.
Z. Galil, and V. Pan, Improved processor bounds for algebraic and combinatorial problems in RNC, Proc. 26th Annual IEEE FOCS, 1985, pp. 490–495.
R. K. Ghosh and G. P. Bhattacharjee, A parallel search algorithm for directed acyclic graphs, BIT 24, 1984, pp. 134–150.
R. Greenlaw, Polynomial completeness and parallel computation, in J. H. Reif (ed.) Synthesis of Parallel Algorithms, (Morgan-Kaufmann Publishers, 1993 ).
J. Gil, Y. Matias, and U. Vishkin, Towards a theory of nearly constant time parallel algorithms, Proc. 32nd Annual IEEE FOCS, 1991, pp. 698–710.
W. L. Hightower, J. F. Prins, and J. H. Reif, Implementation of randomized sorting on large parallel machines, Proc. 4th Annual ACM Symposium on Parallel Algorithms and Architectures, 1992, pp. 158–167.
C. A. R. Hoare, Quicksort, Computer Journal 5, 1962, pp. 10–15.
E. Horowitz, S. Sahni, and S. Rajasekaran, Computer Algorithms, (W. H. Freeman Press, 1998 ).
J. Já Já, An Introduction to Parallel Algorithms, (Addison-Wesley Publishers, 1992 ).
C. Kaklamanis and D. Krizanc, Optimal sorting on mesh-connected processor arrays, Proc. 4th Annual ACM Symposium on Parallel Algorithms and Architectures, 1992, pp. 50–59.
D. R. Karger, P. N. Klein, and R. E. Tarjan, A randomized linear-time algorithm to find minimum spanning trees, Journal of the ACM 42 (2), 1995, pp. 321–328.
R. M. Karp and A. Wigderson, A fast parallel algorithm for the maximal independent set problem, Journal of the ACM 32, 1985, pp. 762–773.
R. M. Karp, E. Upfal, and A. Wigderson, Constructing a maximum matching is in random NC. Combinatorica 6(1), 1986, pp. 35–48. A preliminary version also appeared in Proc. 17th Annual ACM STOC,1985.
R. M. Karp and V. Ramachandran, Parallel algorithms for shared-memory machines, in J. van Leeuwen (ed.) Handbook of Theoretical Computer Science, ( Elsevier Science, 1990 ), Vol. A, Chapter 17.
R. Karp, An introduction to randomized algorithms, Discr. Appl. Math 34, 1991, pp. 165–201.
M. Kaufmann, S. Torsten, and J. Sibeyn, Derandomizing algorithms for routing and sorting on meshes, Proc. 5th Annual ACM SIAM Symposium on Discrete Algorithms, 1994, pp. 669–679.
D. Kavvadias, G. E. Pantziou, P. G. Spirakis, and C. D. Zaroliagis, Hammock-on-ears decomposition: a technique for the efficient parallel solution of shortest paths and other problems, Proc. 19th MFCS, LNCS 841, 1994, pp. 462–472.
P. N. Klein and S. Sairam, A parallel randomized approximation scheme for shortest paths, Proc. 24th Annual ACM STOC, 1992, pp. 750–758.
P. N. Klein and S. Sairam, A linear-processor polylog-time algorithm for shortest paths in planar graphs, Proc. 34th Annual IEEE FOCS, 1994, pp. 259–270.
L. Kučera, Expected behavior of graph coloring algorithms, Proc. Fundamentals in Computation Theory, LNCS 56, 1984, pp. 447–451.
M. Kunde, Block gossiping on grids and tori: sorting and routing match the bisection bound deterministically, Proc. European Symposium on Algorithms, 1993, pp. 272–283.
T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays-Trees-Hypercube, (Morgan-Kaufmann Publishers, 1992 ).
T. Leighton, Tight bounds on the complexity of parallel sorting, IEEE Transactions on Computers C34 (4), 1985, pp. 344–354.
L. Lovasz, On determinants, matchings and random algorithm, in L. Budach (ed.) Fundamentals of Computing Theory, ( Berlin, AkademiaVerlag, 1979 ).
M. Luby, A simple parallel algorithm for the maximal independent set problem, SIAM J. Comp. 15, 1986, pp. 1036–1053. (Also in Proc. 17th Annual ACM STOC).
Y. Ma, S. Sen, and D. Scherson, The distance bound for sorting on mesh connected processor arrays is tight, Proc. 26th Annual IEEE FOCS, 1986, pp. 255–263.
S. Micali and V. V. Vazirani, An \(O(\sqrt {\left| V \right|} \left| E \right|)\)algorithm for finding maximum matching in general graphs, Proc. 21st Annual IEEE FOCS, 1980, pp. 17–27.
K. Mulmuley, U. V. Vazirani, and V. V. Vazirani, Matching is as easy as matrix inversion, Combinatorica 7, 1987, pp. 105–113. (Also in Proc. 19th Annual ACM STOC, 1987, pp. 345–354.)
D. Nassimi and S. Sahni, Parallel permutation and sorting algorithms and a new generalized connection network, Journal of the ACM 29 (3), 1982, pp. 642–667.
M. Nigam and S Sahni, Sorting n2 numbers on n x n meshes, Proc. International Parallel Processing Symposium, 1993, pp. 73–78.
S. Nikoletseas, K. Palem, P. Spirakis, and M. Yung, Short vertex disjoint paths and multiconnectivity in random graphs: reliable networks for computing, Proc. 21st ICALP, LNCS, 1994, pp. 508–519.
V. Pan, Fast and efficient algorithms for the exact inversion of integer matrices, Proc. Fifth Annual Symposium on Foundations of Software Technology and Theoretical Computer Science, 1985, pp. 504–521.
G. Pantziou, P. Spirakis, and C. Zaroliagis, Coloring random graphs efficiently in parallel through adaptive techniques, CTI TR-90.10.25, Comp. Techn. Institute, Patras. Also presented in the ALCOM Workshop on Graphs Algorithms, Data Structures and Computational Geometry, Berlin, October, 1990.
P. M. Pardalos, S. Rajasekaran, editors, Advances in Randomized Parallel Computing, (Kluwer Academic Press, 1998 ).
F. Preparata, New parallel sorting schemes, IEEE Transactions on Computers C27 (7), 1978, pp. 669–673.
P. Raghavan and C. D. Thompson, Provably good routing in graphs: regular arrays, Proc. 17th Annual ACM STOC, 1985, pp. 79–87.
S. Rajasekaran, A simple parallel sorting algorithm, Technical Report 26, Dept. of CISE, University of Florida, 1997.
S. Rajasekaran, Basic algorithms on parallel optical models of computing, in P. M. Pardalos (ed.) Parallel Processing of Discrete Problems, (Springer-Verlag, 1998 ).
S. Rajasekaran, k — k routing, k — k sorting, and cut through routing on the mesh, Journal of Algorithms 19, 1995, pp. 361–382.
S. Rajasekaran, Sorting and selection on interconnection networks, DI-MACS Series in Discrete Mathematics and Theoretical Computer Science 21, 1995, pp. 275–296.
S. Rajasekaran and J. H. Reif, Optimal and sub-logarithmic time randomized parallel sorting algorithms, SIAM J. Comp, 18 (3), 1989, pp. 594–607.
S. Rajasekaran and Th. Tsantilas, Optimal routing algorithms for mesh connected processor arrays, Algorithmica 8, 1992, pp. 21–38.
A. G. Ranade, How to emulate shared memory, Proc. 28th Annual IEEE FOCS, 1987, pp. 185–192.
E. Reghbati and D. Corniel, Parallel computations in graph theory, SIAM J. Comp 7, 1978, pp. 230–237.
J. H. Reif, Depth first search is inherently sequential, Information processing Letters 20, 1985, pp. 229–234.
J. H. Reif and L. G. Valiant, A logarithmic time sort for linear size networks, Journal of the ACM 34 (1), 1987, pp. 60–76.
R. Reischuk, Probabilistic parallel algorithms for sorting and selection, SIAM J. Comp 14 (2), 1985, pp. 396–409.
C. Schnorr and A. Shamir, An optimal sorting algorithm for mesh-connected computers, Proc. 18th Annual ACM STOC, 1986, pp. 255–263.
J.T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, Journal of the ACM 27 (4), 1980, pp. 701–717.
M. Serna and P. G. Spirakis, Tight RNC approximations to max flow, Proc. 8th Annual STAGS, LNCS 480, 1991, pp. 118–126.
J. R. Smith, Parallel algorithms for depth first searches I: planar graphs, SIAM J. Comp 15 (3), 1986, pp. 814–830.
T. M. Stricker, Supporting the hypercube programming model on mesh architectures (A fast sorter for iWarp tori), Proc. 4th Annual ACM Symposium on Parallel Algorithms and Architectures, 1992, pp. 148–157.
C. D. Thompson and H. T. Kung, Sorting on a mesh connected parallel computer, Communications of the ACM 20 (4), 1977, pp. 263–271.
W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22, 1947, pp. 107–111.
J. Ullmann and M. Yannakakis, High probability parallel transitive closure algorithms, SIAM J. Comp 20, 1991, pp. 100–125.
E. Urland, Experimental tests of efficient shortest paths heuristics for random graphs on the CM-2, Technical Report 71, University of Geneva, August, 1994.
L. G. Valiant, A scheme for fast parallel communication, SIAM J. Comp 11, 1982, pp. 350–361.
L. G. Valiant and G. J. Brebner, Universal schemes for parallel communication, Proc. 13th Annual ACM STOC, 1981, pp. 263–277.
J. van Leeuwen, Graph Algorithms, in J. van Leeuwen (ed.) Handbook of Theoretical Computer Science, ( Elsevier Science, 1990 ), Vol. A, 10.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Kluwer Academic Publishers
About this chapter
Cite this chapter
Rajasekaran, S., Rolim, J.D.P. (1998). Randomized Parallel Algorithms for Combinatorial Optimization. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_32
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0303-9_32
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7987-4
Online ISBN: 978-1-4613-0303-9
eBook Packages: Springer Book Archive