Abstract
We consider a 2-edge connected, non-negatively weighted graph G, with n nodes and m edges, and a single-source shortest paths tree (SPT) of G rooted at an arbitrary node. If an edge of the SPT is temporarily removed, a widely recognized approach to reconnect the nodes disconnected from the root consists of joining the two resulting subtrees by means of a single non-tree edge, called a swap edge. This allows to reduce consistently the set-up and computational costs which are incurred if we instead rebuild a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge, and here we restrict our attention to arguably the two most significant measures: the minimization of either the maximum or the average distance between the root and the disconnected nodes. For the former criteria, we present an O(m logα(m,n)) time algorithm to find a best swap edge for every edge of the SPT, thus improving onto the previous O(m logn) time algorithm (B. Gfeller, ESA’08). Concerning the latter criteria, we provide an O(m + n logn) time algorithm for the special but important case where G is unweighted, which compares favorably with the \(O\big(m+n \, \alpha(n,n)\log^2n\big)\) time bound that one would get by using the fastest algorithm known for the weighted case – once this is suitably adapted to the unweighted case.
This work was partially supported by the Research Grant PRIN 2010 “ARS TechnoMedia”, funded by the Italian Ministry of Education, University, and Research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ackermann, W.: Zum hilbertschen aufbau der reellen zahlen. Mathematical Annals 99, 118–133 (1928)
Bilò, D., Gualà, L., Proietti, G.: Finding best swap edges minimizing the routing cost of a spanning tree. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 138–149. Springer, Heidelberg (2010)
Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proc. of the 43rd Symp. on Foundations of Computer Science (FOCS 2002), pp. 617–626. IEEE Comp. Soc. (2002)
Di Salvo, A., Proietti, G.: Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor. Theor. Comp. Science 383(1), 23–33 (2007)
Gualà, L., Proietti, G.: Exact and approximate truthful mechanisms for the shortest-paths tree problem. Algorithmica 49(3), 171–191 (2007)
Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing 13(2), 338–355 (1984)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM 34(3), 596–615 (1987)
Gfeller, B.: Faster swap edge computation in minimum diameter spanning trees. Algorithmica 62(1-2), 169–191 (2012)
Italiano, G.F., Ramaswami, R.: Maintaining spanning trees of small diameter. Algorithmica 22(3), 275–304 (1998)
Ito, H., Iwama, K., Okabe, Y., Yoshihiro, T.: Polynomial-time computable backup tables for shortest-path routing. In: Proc. of the 10th Int. Coll. on Structural Information and Communication Complexity (SIROCCO 2003). Proceedings in Informatics, Carleton Scientific, vol. 17, pp. 163–177 (2003)
Nardelli, E., Proietti, G., Widmayer, P.: How to swap a failing edge of a single source shortest paths tree. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 144–153. Springer, Heidelberg (1999)
Nardelli, E., Proietti, G., Widmayer, P.: Finding all the best swaps of a minimum diameter spanning tree under transient edge failures. Journal of Graph Algorithms and Applications 5(5), 39–57 (2001)
Nardelli, E., Proietti, G., Widmayer, P.: Swapping a failing edge of a single source shortest paths tree is good and fast. Algorithmica 36(4), 361–374 (2003)
Pettie, S.: Sensitivity analysis of minimum spanning trees in sub-inverse-Ackermann time. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 964–973. Springer, Heidelberg (2005)
Proietti, G.: Dynamic maintenance versus swapping: an experimental study on shortest paths trees. In: Näher, S., Wagner, D. (eds.) WAE 2000. LNCS, vol. 1982, pp. 207–217. Springer, Heidelberg (2001)
Wu, B.Y., Hsiao, C.-Y., Chao, K.-M.: The swap edges of a multiple-sources routing tree. Algorithmica 50(3), 299–311 (2008)
Tarjan, R.E.: Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Process. Lett. 14(1), 30–33 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bilò, D., Gualà, L., Proietti, G. (2013). A Faster Computation of All the Best Swap Edges of a Shortest Paths Tree. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-40450-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40449-8
Online ISBN: 978-3-642-40450-4
eBook Packages: Computer ScienceComputer Science (R0)