article

A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows

Authors:

Andrzej Lingas,

Mia PerssonAuthors Info & Claims

Algorithmica, Volume 72, Issue 2

Pages 607 - 619

https://doi.org/10.1007/s00453-013-9865-1

Published: 01 June 2015 Publication History

Abstract

We present a new approach to the minimum-cost integral flow problem for small values of the flow. It reduces the problem to the tests of simple multivariate polynomials over a finite field of characteristic two for non-identity with zero. In effect, we show that a minimum-cost flow of value k in a network with n vertices, a sink and a source, integral edge capacities and positive integral edge costs polynomially bounded in n can be found by a randomized PRAM, with errors of exponentially small probability in n, running in O(klog(kn)+log2(kn)) time and using 2k(kn)O(1) processors. Thus, in particular, for the minimum-cost flow of value O(logn), we obtain an RNC2 algorithm, improving upon time complexity of earlier NC and RNC algorithms.

References

[1]

Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, New York (1993)

[2]

Anderson, R.J., Setubal, J.C.: A parallel implementation of the push-relabel algorithm for the maximum flow problem. J. Parallel Distrib. Comput. 29(1), 17---26 (1995)

Digital Library

[3]

Björklund, A.: Determinant sums for undirected Hamiltonicity. In: Proceedings of the 51th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 173---182 (2010)

Digital Library

[4]

Björklund, A., Husfeldt, T., Taslaman, N.: Shortest Cycle Through Specified Elements. In: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1747---1753 (2012)

[5]

DeMillo, R.A., Lipton, R.J.: A probabilistic remark on algebraic program testing. Inf. Process. Lett. 7, 193---195 (1978)

[6]

Edmonds, J.: Systems of distinct representatives and linear algebra. J. Res. Natl. Bur. Stand. 71B(4), 241---245 (1967)

[7]

Even, S.: Graph Algorithms. Computer Science (1979)

[8]

Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248---264 (1972)

Digital Library

[9]

Ford, L.R. Jr., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8, 399---404 (1956)

[10]

Fulkerson, D.R.: An out-of-Kilter method for minimal cost flow problems. SIAM J. Appl. Math. 9, 18---27 (1961)

[11]

Galil, Z., Pan, V.: Improved processor bounds for combinatorial problems in RNC. Combinatorica 8(2), 189---200 (1988)

[12]

Goldberg, A.: Parallel algorithms for network flow problems. In: Reif, J.H. (ed.) Synthesis of Parallel Algorithms, Morgan Kaufmann, San Mateo (1993)

[13]

Goldschlager, L.M., Shaw, R.A., Staples, J.: The maximum flow problem is log space complete for P. Theor. Comput. Sci. 21(1), 105---111 (1982)

[14]

JáJá, J.: An Introduction to Parallel Algorithms. Addison-Wesley, Reading (1992)

[15]

Karp, R., Upfal, E., Wigderson, A.: Constructing a perfect matching is in random NC. Combinatorica 6(1), 35---48 (1986)

Digital Library

[16]

Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Proceedings of the 35th Annual International Colloquium on Automata, Languages and Programming (ICALP). Lecture Notes in Computer Science, vol. 5125, pp. 575---586 (2008)

[17]

Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1967)

[18]

Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as Easy as Matrix Inversion. Combinatorica 7(1), 105---113 (1987)

Digital Library

[19]

Orlin, J.B., Stein, C.: Parallel Algorithms for the Assignment and Minimum-Cost Flow Problems. Oper. Res. Lett. 14, 181---186 (1993)

Digital Library

[20]

Reif, J.H. (ed.): Synthesis of Parallel Algorithms. Morgan Kaufmann, San Mateo (1993)

[21]

Vazirani, V.V.: Parallel graph matching. In: Reif, J.H. (ed.) Synthesis of Parallel Algorithms. Morgan-Kaufmann, San Mateo (1993)

[22]

Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701---717 (1980)

Digital Library

[23]

Vassilevska Williams, V.: Multiplying matrices faster than coppersmith-winograd. In: Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), pp. 887---898 (2012)

[24]

Williams, R.: Finding paths of length k in O'(2k). Inf. Process. Lett. 109, 301---338 (2009)

Digital Library

Cited By

Jiang JChen JWang C(2020)Multi-granularity hybrid parallel network simplex algorithm for minimum-cost flow problemsThe Journal of Supercomputing10.1007/s11227-020-03227-976:12(9800-9826)Online publication date: 4-Mar-2020
https://dl.acm.org/doi/10.1007/s11227-020-03227-9
van Melkebeek DPrakriya G(2017)Derandomizing isolation in space-bounded settingsProceedings of the 32nd Computational Complexity Conference10.5555/3135595.3135600(1-32)Online publication date: 9-Jul-2017
https://dl.acm.org/doi/10.5555/3135595.3135600

Index Terms

A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms

Index terms have been assigned to the content through auto-classification.

Recommendations

A fast parallel algorithm for minimum-cost small integral flows
Euro-Par'12: Proceedings of the 18th international conference on Parallel Processing

We present a new approach to the minimum-cost integral flow problem for small values of the flow. It reduces the problem to the tests of simple multi-variable polynomials over a finite field of characteristic two for non-identity with zero. In effect, ...
Doubly Logarithmic Communication Algorithms for Optical-Communication Parallel Computers

In this paper, we consider the problem of interprocessor communication on parallel computers that have optical communication networks. We consider the completely connected optical-communication parallel computer (OCPC), which has a completely connected ...
Fast parallel algorithms for minimum and related problems with small integer inputs
IPPS '95: Proceedings of the 9th International Symposium on Parallel Processing

The fundamental problem of minimum computation has several well studied generalizations such as prefix minima, range minima, and all nearest smaller values computations. Recent papers introduced parallel algorithms for these problems when the n input ...

Comments

Information & Contributors

Information

Published In

cover image Algorithmica

Algorithmica Volume 72, Issue 2

June 2015

327 pages

ISSN:0178-4617

Issue’s Table of Contents

Copyright © Copyright © 2015 Springer Science+Business Media New York.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 June 2015

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 11 Aug 2024

Other Metrics

View Author Metrics

Citations

Cited By

Jiang JChen JWang C(2020)Multi-granularity hybrid parallel network simplex algorithm for minimum-cost flow problemsThe Journal of Supercomputing10.1007/s11227-020-03227-976:12(9800-9826)Online publication date: 4-Mar-2020
https://dl.acm.org/doi/10.1007/s11227-020-03227-9
van Melkebeek DPrakriya G(2017)Derandomizing isolation in space-bounded settingsProceedings of the 32nd Computational Complexity Conference10.5555/3135595.3135600(1-32)Online publication date: 9-Jul-2017
https://dl.acm.org/doi/10.5555/3135595.3135600

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents