@article{11663,
  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 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(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(log 5 (n)) high-probability worst-case time; similar worst-case bounds existed only for maintaining a matching that was (2+ϵ)-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 ɑ if for every update σ, the expected time to process σ is at most ɑ. 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 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.},
  author       = {Bernstein, Aaron and Forster, Sebastian and Henzinger, Monika H},
  issn         = {1549-6333},
  journal      = {ACM Transactions on Algorithms},
  number       = {4},
  publisher    = {Association for Computing Machinery},
  title        = {{A deamortization approach for dynamic spanner and dynamic maximal matching}},
  doi          = {10.1145/3469833},
  volume       = {17},
  year         = {2021},
}