Abstract
Solomon and Elkin (SIAM J Discret Math 28(3):1173–1198, 2014) constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight \((1+\epsilon )\)-spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after \(O(\log n)\) hops and requires \(O(\varDelta \log n)\) bits of storage per vertex where \(\varDelta \) is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.
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References
Abraham I, Malkhi D (2004) Compact routing on Euclidian metrics. In: Proceedings of the 23rd annual ACM symposium on principles of distributed computing, pp 141–149
Arya SK, Das G, Mount DM, Salowe JS, Smid MHM (1995) Euclidean spanners: short, thin, and lanky. In: Proceedings of the 27th annual ACM symposium on theory of computing
Ashvinkumar V, Gudmundsson J, Levcopoulos C, Nilsson BJ, van Renssen A (2019) Local routing in sparse and lightweight geometric graphs. In: Proceedings of the 30th international symposium on algorithms and computation. LIPIcs, vol 149, pp 30:1–30:13
Bose P, Brodnik A, Carlsson S, Demaine E, Fleischer R, López-Ortiz A, Morin P, Munro J (2002) Online routing in convex subdivisions. Int J Comput Geom Appl 12(4):283–296
Bose P, Fagerberg R, Renssen A, Verdonschot S (2015) Optimal local routing on Delaunay triangulations defined by empty equilateral triangles. SIAM J Comput 44(6):1626–1649
Brankovic M, Gudmundsson J, van Renssen A (2020) Local routing in a tree metric 1-spanner. In: Computing and combinatorics. COCOON 2020. Lecture notes in computer science, vol 12273. Springer, Berlin, pp 174–185
Chan THH, Gupta A, Maggs BM, Zhou S (2016) On hierarchical routing in doubling metrics. ACM Trans Algorithms 12(4):55:1–55:22
Chan T, Li M, Ning L, Solomon S (2015) New doubling spanners: better and simpler. SIAM J Comput 44(1):37–53
Elkin M, Solomon S (2013) Optimal Euclidean spanners: really short, thin and lanky. In: Proceedings of the 45th annual ACM symposium on theory of computing, pp 645–654
Gao J, Guibas LJ, Nguyen A (2006) Deformable spanners and applications. Comput Geom 35(1):2–19
Gottlieb LA, Roditty L (2008) Improved algorithms for fully dynamic geometric spanners and geometric routing. In: Proceedings of the 19th annual ACM-SIAM symposium on discrete algorithms, pp 591–600
Har-Peled S, Mendel M (2006) Fast construction of nets in low-dimensional metrics and their applications. SIAM J Comput 35(5):1148–1184
Santoro N, Khatib R (1985) Labelling and implicit routing in networks. Comput J 28(1):5–8
Solomon S (2014) From hierarchical partitions to hierarchical covers: Optimal fault-tolerant spanners for doubling metrics. In: Proceedings of the 46th annual ACM symposium on theory of computing, pp 363–372
Solomon S, Elkin M (2014) Balancing degree, diameter, and weight in Euclidean spanners. SIAM J Discret Math 28(3):1173–1198
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A preliminary version of this paper appeared in the proceedings of COCOON’20 (Brankovic et al. 2020).
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Brankovic, M., Gudmundsson, J. & Renssen, A.v. Local routing in a tree metric 1-spanner. J Comb Optim 44, 2642–2660 (2022). https://doi.org/10.1007/s10878-021-00784-4
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DOI: https://doi.org/10.1007/s10878-021-00784-4