Document Open Access Logo

Low-Stretch Spanning Trees of Graphs with Bounded Width

Authors Glencora Borradaile, Erin Wolf Chambers, David Eppstein, William Maxwell, Amir Nayyeri

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2020.15.pdf
  • Filesize: 0.75 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Treewidth
  • low-stretch spanning tree
  • fundamental cycle basis

Metrics

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

Abstract

We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph G with a linear arrangement of bandwidth b can be embedded into a distribution T of spanning trees such that the expected stretch of each edge of G is O(b²). Our proof implies a linear time algorithm for sampling from T. Therefore, we have a linear time algorithm that finds a spanning tree of G with average stretch O(b²) with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of G with average stretch O(b³). For graphs of cutwidth c, we construct a spanning tree with stretch O(c²) in linear time. Finally, when G has treewidth k we provide a dynamic programming algorithm computing a minimum stretch spanning tree of G that runs in polynomial time with respect to the number of vertices of G.

Cite As Get BibTex

Glencora Borradaile, Erin Wolf Chambers, David Eppstein, William Maxwell, and Amir Nayyeri. Low-Stretch Spanning Trees of Graphs with Bounded Width. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SWAT.2020.15

Author Details

Glencora Borradaile
  • Oregon State University, Corvallis, OR, USA
Erin Wolf Chambers
  • Saint Louis University, MO, USA
David Eppstein
  • University of California, Irvine, CA, USA
William Maxwell
  • Oregon State University, Corvallis, OR, USA
Amir Nayyeri
  • Oregon State University, Corvallis, OR, USA

Funding

This material is based upon work supported by the National Science Foundation under Grant Nos. NSF CCF-1614562, DBI-1759807, CCF-1907612, CCF-1616248, CCF-1617951, CCF-1618301, and CCF-1816442.

References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS '08, pages 781-790, Washington, DC, USA, 2008. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2008.62.
  2. Ittai Abraham and Ofer Neiman. Using petal-decompositions to build a low stretch spanning tree. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC '12, pages 395-406, New York, NY, USA, 2012. ACM. Google Scholar
  3. Noga Alon, Richard M. Karp, David Peleg, and Douglas B. West. A graph-theoretic game and its application to the k-server problem. SIAM J. Comput., 24(1):78-100, 1995. URL: https://doi.org/10.1137/S0097539792224474.
  4. Stefan Arnborg, Derek G. Corneil, and Andrzej Proskurowski. Complexity of Finding Embeddings in a k-tree. SIAM. J. on Algebraic and Discrete Methods, 8(2):277-284, 1987. Google Scholar
  5. Yair Bartal, Douglas E. Carroll, Adam Meyerson, and Ofer Neiman. Bandwidth and low dimensional embedding. Theor. Comput. Sci., 500:44-56, 2013. URL: https://doi.org/10.1016/j.tcs.2013.05.038.
  6. Hans L. Bodlaender, Paul Bonsma, and Daniel Lokshtanov. The fine details of fast dynamic programming over tree decompositions. In Gregory Gutin and Stefan Szeider, editors, Parameterized and Exact Computation, pages 41-53, Cham, 2013. Springer International Publishing. Google Scholar
  7. Bela Bollobas. Modern Graph Theory. Springer, 1998. Google Scholar
  8. Douglas E. Carroll, Ashish Goel, and Adam Meyerson. Embedding bounded bandwidth graphs into 𝓁₁. In Proceedings of the 33rd International Conference on Automata, Languages and Programming - Volume Part I, ICALP'06, pages 27-37, Berlin, Heidelberg, 2006. Springer-Verlag. URL: https://doi.org/10.1007/11786986_4.
  9. A. C. Cassell, J. C. de C. Henderson, and A. Kaveh. Cycle bases for the flexibility analysis of structures. International Journal for Numerical Methods in Engineering, 8(3):521-528, 1974. URL: https://doi.org/10.1002/nme.1620080308.
  10. F. R. K. Chung. On the cutwidth and topological bandwidth of a tree. SIAM Journal of Algebraic Discrete Methods, 6:268-277, 1985. Google Scholar
  11. Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. Lower-stretch spanning trees. SIAM J. Comput., 38(2):608-628, May 2008. Google Scholar
  12. Yuval Emek. k-outerplanar graphs, planar duality, and low stretch spanning trees. Algorithmica, 61(1):141-160, September 2011. Google Scholar
  13. Michael L. Fredman and Dan E. Willard. Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. Syst. Sci., 48(3):533-551, June 1994. Google Scholar
  14. Thomas M. J. Fruchterman and Edward M. Reingold. Graph drawing by force-directed placement. Softw. Pract. Exper., 21(11):1129-1164, November 1991. Google Scholar
  15. F. Gavril. Some NP-complete problems on graphs. 11th Conference on Information Sciences and Systems, pages 91-95, 1977. Google Scholar
  16. Petra M. Gleiss. Short Cycle. PhD thesis, Universitat Wien, 2001. Google Scholar
  17. Anupam Gupta, Ilan Newman, Yuri Rabinovich, and Alistair Sinclair. Cuts, trees and 𝓁₁-embeddings of graphs. Combinatorica, 24(2):233-269, April 2004. Google Scholar
  18. Dov Harel. A linear time algorithm for the lowest common ancestors problem. 21st Annual Symposium on Foundations of Computer Science, 1980. Google Scholar
  19. Tom Kloks. Treewidth: Computations and Approximations. Springer-Verlag, 1994. Google Scholar
  20. Ekkehard Köhler, Rolf H. Möhring, and Gregor Wünsch. Minimizing total delay in fixed-time controlled traffic networks. In OR, pages 192-199, 2004. Google Scholar
  21. I. Koutis, G. L. Miller, and R. Peng. Approaching optimality for solving sdd linear systems. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 235-244, 2010. Google Scholar
  22. Ioannis Koutis, Gary L. Miller, and Richard Peng. A nearly-mlog n time solver for sdd linear systems. In Proceedings of the 2011 IEEE 52Nd Annual Symposium on Foundations of Computer Science, FOCS '11, pages 590-598, Washington, DC, USA, 2011. IEEE Computer Society. Google Scholar
  23. James R. Lee and Anastasios Sidiropoulos. Pathwidth, trees, and random embeddings. Combinatorica, 33(3):349-374, June 2013. URL: https://doi.org/10.1007/s00493-013-2685-8.
  24. Christian Liebchen. Periodic timetable optimization in public transport. In Karl-Heinz Waldmann and Ulrike M. Stocker, editors, Operations Research Proceedings 2006, pages 29-36, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. Google Scholar
  25. Ch. H. Papadimitriou. The NP-completeness of the bandwidth minimization problem. Computing, 16(3):263-270, 1976. Google Scholar
  26. David Peleg and Eilon Reshef. Deterministic polylog approximation for minimum communication spanning trees. In ICALP, volume 1443 of Lecture Notes in Computer Science, pages 670-681. Springer, 1998. Google Scholar
  27. Alexander Reich. Cycle Bases of Graphs and Spanning Trees with Many Leaves. PhD thesis, BTU Cottbus, 2014. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact