arXiv:2305.01420 · v6

An Improved Subquadratic Bound for Online Bisection

首次提交 2023-05-02 21:49

版本更新 2026-11-12 02:10

雷达观测 2026-11-13 14:16

摘要

The online bisection problem is a natural dynamic variant of the classic bisection problem, where one has to dynamically maintain a partition of $n$ elements into two clusters of cardinality $n/2$. During runtime, an online algorithm is given a sequence of requests, each being a pair of elements: an inter-cluster request costs one unit while an intra-cluster one is free. The algorithm may change the partition, paying a unit cost for each element that changes its cluster. This natural problem admits a simple deterministic $O(n^2)$-competitive algorithm [Avin et al., DISC 2016]. While several significant improvements over this result have been obtained since the original work, all of them either limit the generality of the input or assume some form of resource augmentation (e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the best known competitive ratio even if randomization is allowed. In this paper, we present the first randomized online algorithm that breaks this natural quadratic barrier and achieves a competitive ratio of $\tilde{O}(n^{7/4})$ without resource augmentation and for an arbitrary sequence of requests.

作者备注：Full version of the paper published in STACS 2024, with competitive ratio improved to $O(n^{7/4} \log^{1/4} n)$