A Deamortization Approach for Dynamic Spanner and Dynamic Maximal Matching

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 article, 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(n(3/4)) high-probability worstcase update time for maintaining a 3-spanner and O(n(5/9)) for maintaining a 5-spanner. We give a O(1)(k) log(3) (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 (O) over tilde (n(1+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 withO(log(5) (n)) high-probability worst-case time; similar worst-case bounds existed only for maintaining a matching that was (2 + epsilon)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 a if for every update s, the expected time to process s is at most a. 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 T(f (n)) time to process, for arbitrarily high f (n). In this article, 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(log(2) (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.