Abstract
Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). extended the classical +2 (respectively, +4) spanners for unweighted graphs of size \(O(n^{3/2})\) (resp., \(O(n^{7/5})\)) to the weighted setting, where the additive error is \(+2W\) (resp., \(+4W\)). This means that for every pair u, v, the additive stretch is at most \(+2W_{u,v}\), where \(W_{u,v}\) is the maximal edge weight on the shortest \(u-v\) path (weights are normalized so that the minimum edge weight is 1). In addition, as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). showed a randomized algorithm yielding a \(+8W_{max}\) spanner of size \(O(n^{4/3})\), here \(W_{max}\) is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a \(+(6+\varepsilon )W\) spanner for weighted graphs with size \(O(n^{4/3})\) (for any constant \(\varepsilon >0\)), thus nearly matching the classical +6 spanner of size \(O(n^{4/3})\) for unweighted graphs. Furthermore, we show a \(+(2+\varepsilon )W\) subsetwise spanner of size \(O(n\cdot \sqrt{\vert S\vert })\), improving the \(+4W_{max}\) result of as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). (that had the same size). We also show a simple randomized algorithm for a \(+4W\) emulator of size \({\tilde{O}}(n^{4/3})\). In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. It was proved by Abboud A, Bodwin G (J ACM 64(4):28–12820 2017) that such spanners must suffer polynomially large stretches. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size \(+{\tilde{O}}(\sqrt{n}\cdot W)\) spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of Dor D et al. (SIAM J Comput 29:1740–1759, 2000) for unweighted graphs, we devise an efficient randomized algorithm producing a \(+2W\) spanner for weighted graphs of size \({\tilde{O}}(n^{3/2})\) in \({\tilde{O}}(n^2)\) time.
Notes
The notation \({\tilde{O}}(\cdot )\) hides polylogarithmic factors.
In their paper the spanner is claimed to be \(+4W_{\max }\) but a tighter analysis shows it is actually a \(+4W\).
For arbitrary \(\varepsilon > 0\), the size of our spanner is \(O(n^{4/3}/\varepsilon )\).
\(1 - x \le e^{-x}\)
By increasing the leading constant from 2 to c, we can reduce the failure probability to at most \(O(n^{1-c})\).
References
Ahmed, A.R., Bodwin, G., Sahneh, F.D., Kobourov, S.G., Spence, R.: Weighted additive spanners. In: Adler, I., Müller, H. (eds.) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK, June 24-26, 2020, Revised Selected Papers. Lecture Notes in Computer Science, vol. 12301, pp. 401–413 (2020). https://doi.org/10.1007/978-3-030-60440-0_32
Abboud, A., Bodwin, G.: The 4/3 additive spanner exponent is tight. J. ACM 64(4), 28–12820 (2017). https://doi.org/10.1145/3088511
Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29, 1740–1759 (2000)
Peleg, D., Schäffer, A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)
Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discret. Comput. Geom. 9, 81–100 (1993). https://doi.org/10.1007/BF02189308
Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28(4), 1167–1181 (1999). https://doi.org/10.1137/S0097539796303421
Elkin, M., Peleg, D.: (1+epsilon, beta)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004). https://doi.org/10.1137/S0097539701393384
Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: New constructions of (alpha, beta)-spanners and purely additive spanners. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pp. 672–681 (2005). http://dl.acm.org/citation.cfm?id=1070432.1070526
Knudsen, M.B.T.: Additive spanners: A simple construction. In: Ravi, R., Gørtz, I.L. (eds.) Algorithm Theory - SWAT 2014 - 14th Scandinavian Symposium and Workshops, Copenhagen, Denmark, July 2-4, 2014. Proceedings. Lecture Notes in Computer Science, vol. 8503, pp. 277–281 (2014). https://doi.org/10.1007/978-3-319-08404-6_24
Chechik, S.: New additive spanners. In: Khanna, S. (ed.) Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pp. 498–512 (2013). https://doi.org/10.1137/1.9781611973105.36
Bodwin, G.: Some general structure for extremal sparsification problems. CoRR arXiv:abs/2001.07741 (2020)
Pettie, S.: Low distortion spanners. ACM Transactions on Algorithms 6(1), 1–22 (2009)
Bodwin, G., Williams, V.V.: Very sparse additive spanners and emulators. In: Roughgarden, T. (ed.) Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pp. 377–382 (2015). https://doi.org/10.1145/2688073.2688103
Bodwin, G., Williams, V.V.: Better distance preservers and additive spanners. In: Krauthgamer, R. (ed.) Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pp. 855–872 (2016). https://doi.org/10.1137/1.9781611974331.ch61
Elkin, M.: Computing almost shortest paths. In: Proc. 20th ACM Symp. on Principles of Distributed Computing, pp. 53–62 (2001)
Elkin, M., Zhang, J.: Efficient algorithms for constructing \((1+\varepsilon,\beta )\)-spanners in the distributed and streaming models. Distributed Computing 18, 375–385 (2006)
Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: Proc. of Symp. on Discr. Algorithms, pp. 802–809 (2006)
Abboud, A., Bodwin, G., Pettie, S.: A hierarchy of lower bounds for sublinear additive spanners. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. SODA ’17, pp. 568–576. Society for Industrial and Applied Mathematics, USA (2017)
Elkin, M., Neiman, O.: Hopsets with constant hopbound, and applications to approximate shortest paths. SIAM J. Comput. 48(4), 1436–1480 (2019). https://doi.org/10.1137/18M1166791
Peleg, D., Upfal, E.: A tradeoff between size and efficiency for routing tables. J. of the ACM 36, 510–530 (1989)
Thorup, M., Zwick, U.: Compact routing schemes. In: Proceedings of the Thirteenth Annual ACM Symposium on Parallel Algorithms and Architectures. SPAA ’01, pp. 1–10. ACM, New York, NY, USA (2001). https://doi.org/10.1145/378580.378581
Elkin, M., Neiman, O.: On efficient distributed construction of near optimal routing schemes. Distributed Comput. 31(2), 119–137 (2018). https://doi.org/10.1007/s00446-017-0304-4
Elkin, M., Gitlitz, Y., Neiman, O.: Almost shortest paths with near-additive error in weighted graphs. CoRR arXiv:abs/1907.11422 (2019)
Ahmed, A.R., Bodwin, G., Hamm, K., Kobourov, S.G., Spence, R.: On additive spanners in weighted graphs with local error. In: Kowalik, L., Pilipczuk, M., Rzazewski, P. (eds.) Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Warsaw, Poland, June 23-25, 2021, Revised Selected Papers. Lecture Notes in Computer Science, vol. 12911, pp. 361–373 (2021). https://doi.org/10.1007/978-3-030-86838-3_28
Bollobás, B., Coppersmith, D., Elkin, M.: Sparse distance preservers and additive spanners. SIAM J. Discret. Math. 19(4), 1029–1055 (2005). https://doi.org/10.1137/S0895480103431046
Acknowledgements
A preliminary version of this paper was published in DISC 2021. Michael Elkin: This research was supported by ISF grant No. (2344/19). Yuval Gitlitz and Ofer Neiman: This research was supported by ISF grant No. (970/2).
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Elkin, M., Gitlitz, Y. & Neiman, O. Improved weighted additive spanners. Distrib. Comput. 36, 385–394 (2023). https://doi.org/10.1007/s00446-022-00433-x
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DOI: https://doi.org/10.1007/s00446-022-00433-x