A deamortization approach for dynamic spanner and dynamic maximal matching
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.
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https://arxiv.org/abs/1810.10932
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Author
Bernstein, Aaron;
Forster, Sebastian;
Henzinger, MonikaISTA
Abstract
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.
In this paper we present the first polylogarithmic high-probability worst-case time bounds for the dynamic spanner and the dynamic maximal matching problem.
1.
For 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.)
2.
For 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.
Our 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)).
Thus 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.
Publishing Year
Date Published
2019-01-01
Proceedings Title
30th Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Page
1899-1918
Conference
SODA: Symposium on Discrete Algorithms
Conference Location
San Diego, CA, United States
Conference Date
2019-01-06 – 2019-01-09
IST-REx-ID
Cite this
Bernstein A, Forster S, Henzinger MH. A deamortization approach for dynamic spanner and dynamic maximal matching. In: 30th Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics; 2019:1899-1918. doi:10.1137/1.9781611975482.115
Bernstein, A., Forster, S., & Henzinger, M. H. (2019). A deamortization approach for dynamic spanner and dynamic maximal matching. In 30th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 1899–1918). San Diego, CA, United States: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611975482.115
Bernstein, Aaron, Sebastian Forster, and Monika H Henzinger. “A Deamortization Approach for Dynamic Spanner and Dynamic Maximal Matching.” In 30th Annual ACM-SIAM Symposium on Discrete Algorithms, 1899–1918. Society for Industrial and Applied Mathematics, 2019. https://doi.org/10.1137/1.9781611975482.115.
A. Bernstein, S. Forster, and M. H. Henzinger, “A deamortization approach for dynamic spanner and dynamic maximal matching,” in 30th Annual ACM-SIAM Symposium on Discrete Algorithms, San Diego, CA, United States, 2019, pp. 1899–1918.
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.
Bernstein, Aaron, et al. “A Deamortization Approach for Dynamic Spanner and Dynamic Maximal Matching.” 30th Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2019, pp. 1899–918, doi:10.1137/1.9781611975482.115.
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