{"abstract":[{"lang":"eng","text":"Many dynamic graph algorithms have an amortized update time, rather than a stronger worst-case guarantee. But amortized data structures are not suitable for real-time systems, where each individual operation has to be executed quickly. For this reason, there exist many recent randomized results that aim to provide a guarantee stronger than amortized expected. The strongest possible guarantee for a randomized algorithm is that it is always correct (Las Vegas), and has high-probability worst-case update time, which gives a bound on the time for each individual operation that holds with high probability.\r\n\r\nIn this paper we present the first polylogarithmic high-probability worst-case time bounds for the dynamic spanner and the dynamic maximal matching problem.\r\n\r\n1.\t\r\nFor dynamic spanner, the only known o(n) worst-case bounds were O(n3/4) high-probability worst-case update time for maintaining a 3-spanner, and O(n5/9) for maintaining a 5-spanner. We give a O(1)k log3(n) high-probability worst-case time bound for maintaining a (2k – 1)-spanner, which yields the first worst-case polylog update time for all constant k. (All the results above maintain the optimal tradeoff of stretch 2k – 1 and Õ(n1+1/k) edges.)\r\n\r\n2.\t\r\nFor dynamic maximal matching, or dynamic 2-approximate maximum matching, no algorithm with o(n) worst-case time bound was known and we present an algorithm with O(log5 (n)) high-probability worst-case time; similar worst-case bounds existed only for maintaining a matching that was (2 + ∊)-approximate, and hence not maximal.\r\n\r\nOur results are achieved using a new approach for converting amortized guarantees to worst-case ones for randomized data structures by going through a third type of guarantee, which is a middle ground between the two above: an algorithm is said to have worst-case expected update time α if for every update σ, the expected time to process σ is at most α. Although stronger than amortized expected, the worst-case expected guarantee does not resolve the fundamental problem of amortization: a worst-case expected update time of O(1) still allows for the possibility that every 1/f(n) updates requires Θ(f(n)) time to process, for arbitrarily high f(n). In this paper we present a black-box reduction that converts any data structure with worst-case expected update time into one with a high-probability worst-case update time: the query time remains the same, while the update time increases by a factor of O(log2(n)).\r\n\r\nThus we achieve our results in two steps: (1) First we show how to convert existing dynamic graph algorithms with amortized expected polylogarithmic running times into algorithms with worst-case expected polylogarithmic running times. (2) Then we use our black-box reduction to achieve the polylogarithmic high-probability worst-case time bound. All our algorithms are Las-Vegas-type algorithms."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1810.10932"}],"day":"01","publisher":"Society for Industrial and Applied Mathematics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"conference","related_material":{"record":[{"relation":"earlier_version","id":"11871","status":"public"}]},"title":"A deamortization approach for dynamic spanner and dynamic maximal matching","oa":1,"quality_controlled":"1","date_published":"2019-01-01T00:00:00Z","external_id":{"arxiv":["1810.10932"]},"citation":{"apa":"Bernstein, A., Forster, S., &#38; Henzinger, M. H. (2019). A deamortization approach for dynamic spanner and dynamic maximal matching. In <i>30th Annual ACM-SIAM Symposium on Discrete Algorithms</i> (pp. 1899–1918). San Diego, CA, United States: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611975482.115\">https://doi.org/10.1137/1.9781611975482.115</a>","ista":"Bernstein A, Forster S, Henzinger MH. 2019. A deamortization approach for dynamic spanner and dynamic maximal matching. 30th Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 1899–1918.","mla":"Bernstein, Aaron, et al. “A Deamortization Approach for Dynamic Spanner and Dynamic Maximal Matching.” <i>30th Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Society for Industrial and Applied Mathematics, 2019, pp. 1899–918, doi:<a href=\"https://doi.org/10.1137/1.9781611975482.115\">10.1137/1.9781611975482.115</a>.","short":"A. Bernstein, S. Forster, M.H. Henzinger, in:, 30th Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2019, pp. 1899–1918.","ama":"Bernstein A, Forster S, Henzinger MH. A deamortization approach for dynamic spanner and dynamic maximal matching. In: <i>30th Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Society for Industrial and Applied Mathematics; 2019:1899-1918. doi:<a href=\"https://doi.org/10.1137/1.9781611975482.115\">10.1137/1.9781611975482.115</a>","ieee":"A. Bernstein, S. Forster, and M. H. Henzinger, “A deamortization approach for dynamic spanner and dynamic maximal matching,” in <i>30th Annual ACM-SIAM Symposium on Discrete Algorithms</i>, San Diego, CA, United States, 2019, pp. 1899–1918.","chicago":"Bernstein, Aaron, Sebastian Forster, and Monika H Henzinger. “A Deamortization Approach for Dynamic Spanner and Dynamic Maximal Matching.” In <i>30th Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 1899–1918. Society for Industrial and Applied Mathematics, 2019. <a href=\"https://doi.org/10.1137/1.9781611975482.115\">https://doi.org/10.1137/1.9781611975482.115</a>."},"article_processing_charge":"No","doi":"10.1137/1.9781611975482.115","status":"public","_id":"11871","page":"1899-1918","publication":"30th Annual ACM-SIAM Symposium on Discrete Algorithms","author":[{"first_name":"Aaron","full_name":"Bernstein, Aaron","last_name":"Bernstein"},{"first_name":"Sebastian","last_name":"Forster","full_name":"Forster, Sebastian"},{"id":"540c9bbd-f2de-11ec-812d-d04a5be85630","orcid":"0000-0002-5008-6530","first_name":"Monika H","last_name":"Henzinger","full_name":"Henzinger, Monika H"}],"extern":"1","date_created":"2022-08-16T09:50:33Z","date_updated":"2023-02-21T16:31:21Z","conference":{"name":"SODA: Symposium on Discrete Algorithms","start_date":"2019-01-06","location":"San Diego, CA, United States","end_date":"2019-01-09"},"year":"2019","publication_identifier":{"eisbn":["978-1-61197-548-2"]},"language":[{"iso":"eng"}],"scopus_import":"1","oa_version":"Preprint","publication_status":"published","month":"01"}