The online bisection problem requires maintaining a dynamic partition of $n$ nodes into two equal-sized clusters. Requests arrive sequentially as node pairs. If the nodes lie in different clusters, the algorithm pays unit cost. After each request, the algorithm may migrate nodes between clusters at unit cost per node. This problem models datacenter resource allocation where virtual machines must be assigned to servers, balancing communication costs against migration overhead. We study the variant where requests are restricted to edges of a ring network, an abstraction of ring-allreduce patterns in distributed machine learning. Despite this restriction, the problem remains challenging with an $Ω(n)$ deterministic lower bound. We present a randomized algorithm achieving $O(\varepsilon^{-3} \cdot \log^2 n)$ competitive ratio using resource augmentation that allows clusters of size at most $(3/4 + \varepsilon) \cdot n$. Our approach formulates the problem as a metrical task system with a restricted state space. By limiting the number of cut-edges (i.e., ring edges between clusters) to at most $2k$, where $k = Θ(1/\varepsilon)$, we reduce the state space from exponential to polynomial (i.e., $n^{O(k)}$). The key technical contribution is proving that this restriction increases cost by only a factor of $O(k)$. Our algorithm follows by applying the randomized MTS solution of Bubeck et al. [SODA 2019]. The best result to date for bisection with ring demands is the $O(n \cdot \log n)$-competitive deterministic online algorithm of Rajaraman and Wasim [ESA 2024] for the general setting. While prior work for ring-demands by Räcke et al. [SPAA 2023] achieved $O(\log^3 n)$ for multiple clusters, their approach employs a resource augmentation factor of $2+\varepsilon$, making it inapplicable to bisection.

翻译：在线二分问题要求动态地将 $n$ 个节点划分为两个规模相等的簇。请求以节点对的形式顺序到达。如果节点位于不同的簇中，算法需支付单位成本。每次请求后，算法可以以每节点单位成本在簇间迁移节点。该问题模拟了数据中心资源分配场景，其中虚拟机必须分配到服务器，以平衡通信成本与迁移开销。我们研究请求被限制在环网络边上的变体，这是对分布式机器学习中环全归约模式的一种抽象。尽管有此限制，该问题仍具有挑战性，存在 $Ω(n)$ 的确定性下界。我们提出一种随机化算法，利用资源增强（允许簇的规模至多为 $(3/4 + \varepsilon) \cdot n$）实现了 $O(\varepsilon^{-3} \cdot \log^2 n)$ 的竞争比。我们的方法将该问题表述为具有受限状态空间的度量任务系统。通过将割边（即簇间的环边）数量限制在最多 $2k$（其中 $k = Θ(1/\varepsilon)$），我们将状态空间从指数级减少到多项式级（即 $n^{O(k)}$）。关键的技术贡献在于证明此限制仅使成本增加 $O(k)$ 倍。我们的算法随后通过应用 Bubeck 等人 [SODA 2019] 的随机化 MTS 解决方案得到。迄今为止，针对环需求的二分问题的最佳结果是 Rajaraman 和 Wasim [ESA 2024] 在一般设置下提出的 $O(n \cdot \log n)$ 竞争确定性在线算法。尽管 Räcke 等人 [SPAA 2023] 先前针对环需求的工作在多个簇的情况下实现了 $O(\log^3 n)$，但他们的方法采用了 $2+\varepsilon$ 的资源增强因子，使其不适用于二分问题。