Online Bisection with Ring Demands

TL;DR

This paper analyzes online bisection with ring demands, where requests are edges of a cycle and the algorithm must keep two $n/2$-balanced clusters while paying for cross-cluster requests and migrations. It recasts the problem as a metrical task system with a sparsified state space by restricting to partitions induced by at most $2k$ cut-edges, where $k=\Theta(1/\varepsilon)$, and shows this sparsification inflates cost by only a factor $O(k)$. Leveraging the randomized MTS solution of Bubeck et al., the authors obtain an online $O(k^2\log^2 n)$-competitive algorithm against offline solutions in the restricted class, which translates to an $O(\varepsilon^{-3}\log^2 n)$-competitive, $(3/2+\varepsilon)$-augmented algorithm for ring demands. This advances the understanding of online bisection under ring constraints and provides a practical, near-optimal approach for ring-allreduce-like communication patterns, by marrying state-space sparsification with modern MTS techniques.

Abstract

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.

Figures (2)

• Figure 1: An example coloring of a cycle with the set of cut-edges $\mathcal{C} = {c_1, c_2, \dots, c_{10}}$. The cycle contains 22 blue and 14 red nodes; hence the less frequent color is red, and $\textsc{less}(\mathcal{C}) = 14$.
• Figure 2: An example configuration of Opt and Off with $\textsc{cut}(\textsc{Opt}\xspace) = {c_1, c_2, \dots, c_{12}}$ and $\textsc{cut}(\textsc{Off}\xspace)={c_1, c_3, c_7, c_{12}}$. Cut-edges of Off are shown with thicker lines. The coloring outside the circle corresponds to Opt, while the coloring inside corresponds to Off. The set $\Phi$ consists of nodes indicated by green sectors. The endpoints of these sectors coincide with $\textsc{cut}(\textsc{Opt}\xspace) \setminus \textsc{cut}(\textsc{Off}\xspace)$.

Theorems & Definitions (7)

• Lemma 0: Global rebalancing procedure
• Theorem 1
• Theorem 2
• Corollary 3
• Definition 4
• Definition 5
• Lemma 6