Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Parameterized and Exact Computation

IPEC 2014: Parameterized and Exact Computation pp 85–96Cite as

The \(k\)-Distinct Language: Parameterized Automata Constructions

  • Conference paper
  • First Online:

  • 533 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8894))

Abstract

In this paper, we pioneer a study of parameterized automata constructions for languages relevant to the design of parameterized algorithms. We focus on the \(k\) -Distinct language \(L_k(\varSigma )\subseteq \varSigma ^k\), defined as the set of words of length \(k\) whose symbols are all distinct. This language is implicitly related to several breakthrough techniques, developed during the last two decades, to design parameterized algorithms for fundamental problems such as \(k\) -Path and \(r\) -Dimensional \(k\) -Matching. Building upon the well-known color coding, divide-and-color and narrow sieves techniques, we obtain the following automata constructions for \(L_k(\varSigma )\). We develop non-deterministic automata (NFAs) of sizes \(4^{k+o(k)}\!\cdot \! n^{O(1)}\) and \((2e)^{k+o(k)}\!\cdot \! n^{O(1)}\), where the latter satisfies a ‘bounded ambiguity’ property relevant to approximate counting, as well as a non-deterministic xor automaton (NXA) of size \(2^k\!\cdot \! n^{O(1)}\), where \(n=|\varSigma |\). We show that our constructions lead to a unified approach for the design of both deterministic and randomized algorithms for parameterized problems, considering also their approximate counting variants. To demonstrate our approach, we consider the \(k\) -Path, \(r\) -Dimensional \(k\) -Matching and Module Motif problems.

The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We consider vertex weights, rather than color similarity scores (see [41]), for the sake of clarity. Our algorithms can be easily modified to handle color similarity scores.

  2. 2.

    We slightly improve the previous randomized algorithm that approximately counts only 3-dimensional \(k\)-matchings, and deterministic algorithm for Module Motif.

  3. 3.

    We note that determining the minimal NFA size, even for a finite regular language, is computationally hard [20].

  4. 4.

    If while reading a word we reach a state where we cannot progress by reading the next symbol, this run is rejected and the state we are at is not added to \(M(w)\).

  5. 5.

    We assume NXAs do not contain directed cycles of only \(\epsilon \)-transitions, as their XOR-language is not properly defined in this case.

  6. 6.

    Usually the Hankel matrix is defined as an infinite matrix, and we are defining a submatrix of it; for a finite language \(L\subseteq [n]^k\), that the rank of this submatrix is clearly the same as the rank of the entire matrix, and we are only concerned with its rank.

References

  1. Abasi, H., Bshouty, N.: A simple algorithm for undirected hamiltonicity. Electron. Colloquium Comput. Complex. (ECCC) 20, 12 (2013)

    Google Scholar 

  2. Alon, N., Gutner, S.: Balanced hashing, color coding and approximate counting. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 1–16. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Alon, N., Yuster, R., Zwick, U.: Color coding. J. Assoc. Comput. Mach. 42(4), 844–856 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ben-Basat, R.: M.Sc. thesis. Technical reports and theses, Technion (2015)

    Google Scholar 

  5. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Narrow sieves for parameterized paths and packings. CoRR abs/1007.1161 (2010)

  6. Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, J., Feng, Q., Liu, Y., Lu, S., Wang, J.: Improved deterministic algorithms for weighted matching and packing problems. Theor. Comput. Sci. 412(23), 2503–2512 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, J., Friesen, D., Jia, W., Kanj, I.: Using nondeterminism to design effcient deterministic algorithms. Algorithmica 40(2), 83–97 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, J., Kneis, J., Lu, S., Molle, D., Richter, S., Rossmanith, P., Sze, S.H., Zhang, F.: Randomized divide-and-conquer: improved path, matching, and packing algorithms. SIAM J. Comput. 38(6), 2526–2547 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, J., Liu, Y., Lu, S., Sze, S.H., Zhang, F.: Iterative expansion and color coding: an improved algorithm for 3D-matching. ACM Trans. Algorithms 8(1), 6 (2012)

    Article  MathSciNet  Google Scholar 

  11. Chen, S., Chen, Z.: Faster deterministic algorithms for packing, matching and \(t\)-dominating set problems. CoRR abs/1306.360 (2013)

  12. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  Google Scholar 

  13. Downey, R.G., Fellows, M.R., Koblitz, M.: Techniques for exponential parameterized reductions in vertex set problems (Unpublished, reported in [12], Sect. 8.3)

    Google Scholar 

  14. Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F.A., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. Algorithmica 52(2), 167–176 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fomin, F., Lokshtanov, D., Saurabh, S.: Efficient computation of representative sets with applications in parameterized and exact agorithms. In: SODA, pp. 142–151 (2014)

    Google Scholar 

  16. Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Representative sets of product families. In: ESA (2014, to appear)

    Google Scholar 

  17. Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  18. Goyal, P., Misra, N., Panolan, F.: Faster deterministic algorithms for \(r\)-dimensional matching using representative sets. In: FSTTCS, pp. 237–248 (2013)

    Google Scholar 

  19. Goyal, P., Misra, N., Panolan, F., Zehavi, M.: Faster deterministic algorithms for matching and packing problems. Unpublished, results reported in [18, 40] (2014)

    Google Scholar 

  20. Gruber, H., Holzer, M.: Computational complexity of NFA minimization for finite and unary languages. In: LATA, pp. 261–272 (2007)

    Google Scholar 

  21. Hüffner, F., Wernicke, S., Zichner, T.: Algorithm engineering for color-coding with applications to signaling pathway detection. Algorithmica 52(2), 114–132 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kleitman, D.J., Spencer, J.: Families of k-independent sets. Discrete Math. 6(3), 255–262 (1972)

    Article  MathSciNet  Google Scholar 

  23. Koutis, I.: A faster parameterized algorithm for set packing. Inf. Proc. Lett. 94, 7–9 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 575–586. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  25. Koutis, I., Williams, R.: Limits and applications of group algebras for parameterized problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  26. Liu, Y., Chen, J., Wang, J.: On efficient FPT algorithms for weighted matching and packing problems. In: TAMC, pp. 575–586 (2007)

    Google Scholar 

  27. Liu, Y., Chen, J., Wang, J.: A randomized approximation algorithm for parameterized 3-d matching counting problem. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 349–359. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  28. Liu, Y., Lu, S., Chen, J., Sze, S.-H.: Greedy localization and color-coding: improved matching and packing algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 84–95. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  29. M. Naor, L.J.S., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191 (1995)

    Google Scholar 

  30. Mendelzon, A.O., Wood, P.T.: Finding regular simple paths in graph databases. SIAM J. Comput. 24(6), 1235–1258 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Monien, B.: How to find long paths efficiently. Ann. Discrete Math. 25, 239–254 (1985)

    MathSciNet  Google Scholar 

  32. Rizzi, R., Sikora, F.: Some results on more flexible versions of graph motif. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 278–289. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  33. Ryser, H.J.: Combinatorial Mathematics. The Carus Mathematical Monographs. The Mathematical Association of America, Washington (1963)

    MATH  Google Scholar 

  34. Seroussi, G., Bshouty, N.H.: Vector sets for exhaustive testing of logic circuits. IEEE Trans. Inf. Theory 34(3), 513–522 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  35. Shachnai, H., Zehavi, M.: Representative families: a unified tradeoff-based approach. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 786–797. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  36. Vuillemin, J., Gama, N.: Compact normal form for regular languages as xor automata. In: Maneth, S. (ed.) CIAA 2009. LNCS, vol. 5642, pp. 24–33. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  37. Wang, J., Feng, Q.: Improved parameterized algorithms for weighted 3-set packing. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 130–139. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  38. Wang, J., Feng, Q.: An \({\rm {O}}^*(3.523^k)\) parameterized algorithm for 3-set packing. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 82–93. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  39. Williams, R.: Finding paths of length \(k\) in \({O}^*(2^k)\) time. Inf. Proc. Let. 109(6), 315–318 (2009)

    Article  MATH  Google Scholar 

  40. Zehavi, M.: Deterministic parameterized algorithms for matching and packing problems. CoRR abs/1311.0484 (2013)

  41. Zehavi, M.: Parameterized algorithms for module motif. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 825–836. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

Download references

Acknowledgement

We thank Hasan Abasi, Nader Bshouty, Michael Forbes and Amir Shpilka for helpful conversations.

Author information

Authors and Affiliations

  1. Department of Computer Science, Technion, 32000, Haifa, Israel

    Ran Ben-Basat, Ariel Gabizon & Meirav Zehavi

Authors
  1. Ran Ben-Basat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ariel Gabizon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meirav Zehavi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meirav Zehavi .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Marek Cygan

  2. University of Bergen, Bergen, Norway

    Pinar Heggernes

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Ben-Basat, R., Gabizon, A., Zehavi, M. (2014). The \(k\)-Distinct Language: Parameterized Automata Constructions. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-13524-3_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13523-6

  • Online ISBN: 978-3-319-13524-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation