{"intvolume":"        45","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/1308.0776","open_access":"1"}],"quality_controlled":"1","scopus_import":"1","title":"Dynamic approximate all-pairs shortest paths: Breaking the O(mn) barrier and derandomization","issue":"3","oa_version":"Preprint","publisher":"Society for Industrial & Applied Mathematics","article_type":"original","year":"2016","day":"01","date_published":"2016-05-01T00:00:00Z","date_updated":"2023-02-17T14:21:40Z","volume":45,"extern":"1","type":"journal_article","publication_identifier":{"issn":["0097-5397"],"eissn":["1095-7111"]},"_id":"11891","doi":"10.1137/140957299","author":[{"first_name":"Monika H","full_name":"Henzinger, Monika H","orcid":"0000-0002-5008-6530","last_name":"Henzinger","id":"540c9bbd-f2de-11ec-812d-d04a5be85630"},{"full_name":"Krinninger, Sebastian","first_name":"Sebastian","last_name":"Krinninger"},{"last_name":"Nanongkai","first_name":"Danupon","full_name":"Nanongkai, Danupon"}],"publication":"SIAM Journal on Computing","publication_status":"published","abstract":[{"text":"We study dynamic (1+𝜖)-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected 𝑛-node 𝑚-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of 𝑂̃ (𝑚𝑛/𝜖) and constant query time by Roditty and Zwick [SIAM J. Comput., 41 (2012), pp. 670--683]. The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [J. ACM, 28 (1981), pp. 1--4]; it has a total update time of 𝑂(𝑚𝑛2) and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of 𝑂̃ (𝑛5/2/𝜖) and constant query time that has an additive error of 2 in addition to the 1+𝜖 multiplicative error. This beats the previous 𝑂̃ (𝑚𝑛/𝜖) time when 𝑚=Ω(𝑛3/2). Note that the additive error is unavoidable since, even in the static case, an 𝑂(𝑛3−𝛿)-time (a so-called truly subcubic) combinatorial algorithm with 1+𝜖 multiplicative error cannot have an additive error less than 2−𝜖, unless we make a major breakthrough for Boolean matrix multiplication [D. Dor, S. Halrepin, and U. Zwick, SIAM J. Comput., 29 (2000), pp. 1740--1759] and many other long-standing problems [V. Vassilevska Williams and R. Williams, Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 645--654]. The algorithm can also be turned into a (2+𝜖)-approximation algorithm (without an additive error) with the same time guarantees, improving the recent (3+𝜖)-approximation algorithm with 𝑂̃ (𝑛5/2+𝑂(log(1/𝜖)/log𝑛√)) running time of Bernstein and Roditty [Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011, pp. 1355--1365] in terms of both approximation and time guarantees. (2) We present a deterministic algorithm with a total update time of 𝑂̃ (𝑚𝑛/𝜖) and a query time of 𝑂(loglog𝑛). The algorithm has a multiplicative error of 1+𝜖 and gives the first improved deterministic algorithm since 1981. It also answers an open question raised by Bernstein in [Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, 2013, pp. 725--734]. The deterministic algorithm can be turned into a deterministic fully dynamic (1+𝜖)-approximation with an amortized update time of 𝑂̃ (𝑚𝑛/(𝜖𝑡)) and a query time of 𝑂̃ (𝑡) for every 𝑡≤𝑛√. In order to achieve our results, we introduce two new techniques: (i) A monotone Even--Shiloach tree algorithm which maintains a bounded-distance shortest-paths tree on a certain type of emulator called a locally persevering emulator. (ii) A derandomization technique based on moving Even--Shiloach trees as a way to derandomize the standard random set argument. These techniques might be of independent interest.","lang":"eng"}],"article_processing_charge":"No","citation":{"short":"M.H. Henzinger, S. Krinninger, D. Nanongkai, SIAM Journal on Computing 45 (2016) 947–1006.","mla":"Henzinger, Monika H., et al. “Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(Mn) Barrier and Derandomization.” <i>SIAM Journal on Computing</i>, vol. 45, no. 3, Society for Industrial &#38; Applied Mathematics, 2016, pp. 947–1006, doi:<a href=\"https://doi.org/10.1137/140957299\">10.1137/140957299</a>.","ama":"Henzinger MH, Krinninger S, Nanongkai D. Dynamic approximate all-pairs shortest paths: Breaking the O(mn) barrier and derandomization. <i>SIAM Journal on Computing</i>. 2016;45(3):947-1006. doi:<a href=\"https://doi.org/10.1137/140957299\">10.1137/140957299</a>","ieee":"M. H. Henzinger, S. Krinninger, and D. Nanongkai, “Dynamic approximate all-pairs shortest paths: Breaking the O(mn) barrier and derandomization,” <i>SIAM Journal on Computing</i>, vol. 45, no. 3. Society for Industrial &#38; Applied Mathematics, pp. 947–1006, 2016.","chicago":"Henzinger, Monika H, Sebastian Krinninger, and Danupon Nanongkai. “Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(Mn) Barrier and Derandomization.” <i>SIAM Journal on Computing</i>. Society for Industrial &#38; Applied Mathematics, 2016. <a href=\"https://doi.org/10.1137/140957299\">https://doi.org/10.1137/140957299</a>.","ista":"Henzinger MH, Krinninger S, Nanongkai D. 2016. Dynamic approximate all-pairs shortest paths: Breaking the O(mn) barrier and derandomization. SIAM Journal on Computing. 45(3), 947–1006.","apa":"Henzinger, M. H., Krinninger, S., &#38; Nanongkai, D. (2016). Dynamic approximate all-pairs shortest paths: Breaking the O(mn) barrier and derandomization. <i>SIAM Journal on Computing</i>. Society for Industrial &#38; Applied Mathematics. <a href=\"https://doi.org/10.1137/140957299\">https://doi.org/10.1137/140957299</a>"},"month":"05","date_created":"2022-08-17T08:37:00Z","status":"public","page":"947-1006","external_id":{"arxiv":["1308.0776"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1}