[论文解读] A deamortization approach for dynamic spanner and dynamic maximal matching

Aaron Bernstein, Sebastian Forster+1 more|arXiv (Cornell University)|Jan 6, 2019

Complexity and Algorithms in Graphs被引用 32

一句话总结

本文首次为动态聚补图（dynamic spanner）和动态最大匹配（dynamic maximal matching）提供了对数时间复杂度的高概率最坏情况更新时间界，对于 (2k−1)-聚补图实现 O(1)k log³(n) 的最坏情况时间，对于最大匹配实现 O(log⁵(n))。本文提出一种从最坏情况期望时间到高概率最坏情况时间的黑箱归约方法，从而首次在动态图设置中实现了具有强操作保证的拉斯维加斯算法。

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 O(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 + 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 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.

研究动机与目标

• 为解决动态图算法中缺乏最坏情况更新时间保证的问题，特别是针对实时系统。
• 弥合随机化动态图算法中摊还期望性能与强最坏情况性能之间的差距。
• 开发一种通用方法，将摊还或最坏情况期望的算法转换为具有高概率最坏情况更新时间的算法。
• 首次实现动态聚补图和动态最大匹配的对数时间复杂度高概率最坏情况更新时间。
• 为基本的动态图问题提供具有强操作保证的拉斯维加斯算法。

提出的方法

• 引入一种新的中间保证：最坏情况期望更新时间，其强度介于摊还与高概率最坏情况之间。
• 设计一种黑箱归约方法，将任何具有最坏情况期望更新时间的数据结构转换为具有高概率最坏情况更新时间的版本。
• 应用该归约方法，更新时间产生 O(log²(n)) 的乘法开销，同时保持查询时间不变。
• 首先将现有的摊还期望对数时间复杂度的动态聚补图和最大匹配算法转换为最坏情况期望版本。
• 利用黑箱归约将这些算法升级为高概率最坏情况更新时间，从而实现对数时间复杂度的界。
• 通过保持正确性概率为 1，确保所有算法均为拉斯维加斯算法。

实验结果

研究问题

• RQ1 我们能否为具有 2k−1 背离度的动态聚补图（边数为 O(n¹⁺¹ᐟᵏ)）实现高概率最坏情况更新时间？
• RQ2 是否可能在动态图中维护最大匹配，使其具有对数时间复杂度的高概率最坏情况更新时间？
• RQ3 我们能否系统性地将摊还或最坏情况期望的动态图算法转换为具有高概率最坏情况更新时间的算法？
• RQ4 此类转换在更新时间上的开销是多少，同时保持查询时间不变？
• RQ5 该方法能否为基本动态图问题首次提供具有强操作保证的拉斯维加斯算法？

主要发现

• 本文实现了 O(1)k log³(n) 的高概率最坏情况更新时间，用于维护具有最优 O(n¹⁺¹ᐟᵏ) 条边的 (2k−1)-聚补图。
• 对于动态最大匹配，本文提出了 O(log⁵(n)) 的高概率最坏情况更新时间，这是最大匹配首次获得此类界。
• 本文首次为所有常数 k 的动态聚补图实现了对数时间复杂度的高概率最坏情况更新时间，解决了长期悬而未决的开放问题。
• 黑箱归约使更新时间增加 O(log²(n)) 倍，同时保持查询时间不变，从而实现了从最坏情况期望算法到高概率最坏情况算法的转换。
• 所有算法均为拉斯维加斯算法，保证正确性概率为 1，并具备强操作性能界。
• 该方法可推广至其他动态图问题，为去摊销化随机化数据结构提供了一套新框架。