Document Open Access Logo

Polynomial Min/Max-weighted Reachability is in Unambiguous Log-space

Authors Anant Dhayal, Jayalal Sarma, Saurabh Sawlani



PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2014.597.pdf
  • Filesize: 476 kB
  • 13 pages

Document Identifiers

Author Details

Anant Dhayal
Jayalal Sarma
Saurabh Sawlani

Cite AsGet BibTex

Anant Dhayal, Jayalal Sarma, and Saurabh Sawlani. Polynomial Min/Max-weighted Reachability is in Unambiguous Log-space. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 597-609, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)
https://doi.org/10.4230/LIPIcs.FSTTCS.2014.597

Abstract

For a graph G(V,E) and a vertex s in V, a weighting scheme (w : E -> N) is called a min-unique (resp. max-unique) weighting scheme, if for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by n^c for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this paper, we propose an unambiguous non-deterministic log-space (UL) algorithm for the problem of testing reachability in layered directed acyclic graphs (DAGs) augmented with a min-poly weighting scheme. This improves the result due to Reinhardt and Allender [Reinhardt/Allender, SIAM J. Comp., 2000] where a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique is a triple inductive counting, which generalizes the techniques of [Immermann, Siam J. Comp.,1988; Szelepcsényi, Acta Inf.,1988] and [Reinhardt/Allender, SIAM J. Comp., 2000], combined with a hashing technique due to [Fredman et al.,J. ACM, 1984] (also used in [Garvin et al., Comp. Compl.,2014]). We combine this with a complementary unambiguous verification method, to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-poly property of the graph. Using our techniques, we generalize the double inductive counting method in [Limaye et al., CATS, 2009] where UL algorithms were given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for DAGs.
Keywords
  • Reachability Problem
  • Space Complexity
  • Unambiguous Algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Eric Allender. Reachability problems: An update. In Proc. of CiE 2007, pages 25-27, 2007. Google Scholar
  2. Eric Allender, David A. Mix Barrington, Tanmoy Chakraborty, Samir Datta, and Sambuddha Roy. Planar and grid graph reachability problems. Theor. Comp. Sys., 45(4):675-723, July 2009. Google Scholar
  3. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, New York, NY, USA, 1st edition, 2009. Google Scholar
  4. Chris Bourke, Raghunath Tewari, and N. V. Vinodchandran. Directed planar reachability is in unambiguous log-space. ACM Trans. Comput. Theory, 1(1):4:1-4:17, February 2009. Google Scholar
  5. Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a sparse table with 0(1) worst case access time. J. ACM, 31(3):538-544, June 1984. Google Scholar
  6. Brady Garvin, Derrick Stolee, Raghunath Tewari, and N.V. Vinodchandran. ReachFewL = ReachUL. computational complexity, 23(1):85-98, 2014. Google Scholar
  7. Neil Immerman. Nondeterministic space is closed under complementation. SIAM J. Comput., 17(5):935-938, October 1988. Google Scholar
  8. Nutan Limaye, Meena Mahajan, and Prajakta Nimbhorkar. Longest paths in planar dags in unambiguous logspace. In Proc. of CATS 2009, pages 101-108, 2009. Google Scholar
  9. Aduri Pavan, Raghunath Tewari, and N. V. Vinodchandran. On the power of unambiguity in log-space. Computational Complexity, 21(4):643-670, 2012. Google Scholar
  10. Omer Reingold. Undirected st-connectivity in log-space. In Proceedings of STOC 2005, pages 376-385, 2005. Google Scholar
  11. Klaus Reinhardt and Eric Allender. Making nondeterminism unambiguous. SIAM J. Comput., 29(4):1118-1131, 2000. Google Scholar
  12. R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta Inf., 26(3):279-284, November 1988. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail