Online Bisection with Ring Demands

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.

💡 Research Summary

The paper studies the online bisection problem under the restriction that all requests correspond to edges of a ring (cycle) network. In the classic formulation, n (even) nodes must be kept in two clusters of exactly n/2 nodes each. When a request (a pair of nodes) arrives, a cost of 1 is incurred if the two nodes belong to different clusters; otherwise the request is free. After serving a request the algorithm may migrate any number of nodes between clusters, paying one unit per migrated node. This models the trade‑off between communication cost and VM migration in data centers.

Previous work showed that even with ring‑restricted requests a deterministic Ω(n) lower bound holds, and the best known algorithm for the general (unrestricted) case is a deterministic O(n log n)‑competitive algorithm with (1+ε) resource augmentation. For multiple clusters, Räcke et al. achieved O(log³ n) competitiveness but required an augmentation factor of 2+ε, which is too large for the bisection setting.

The authors introduce a randomized online algorithm that works with a modest augmentation factor of (3/2 + ε) (i.e., each cluster may contain up to (3/4 + ε)·n nodes) and attains a competitive ratio of O(ε⁻³·log² n). The key ideas are:

1. Metrical Task System (MTS) formulation – each feasible partition is a state, and the distance between states equals the number of nodes that must be recolored (migrated) to move from one partition to another.

2. State‑space reduction via cut‑edge limitation – In a ring, a partition is uniquely determined (up to swapping colors) by the set of “cut‑edges” where the two colors meet. The authors restrict attention to partitions that use at most 2k cut‑edges, where k = Θ(1/ε). This reduces the number of states from exponential (≈ C(n, n/2) ≈ 2ⁿ) to polynomial n^{O(k)}.

3. Bounding the cost of restriction – The central technical contribution (Theorem 2) shows that limiting to at most 2k cut‑edges increases the optimal offline cost by at most a factor O(k). The proof constructs an offline algorithm Off that “chases” the true optimal Opt. Off repeatedly sparsifies Opt’s cut‑edge set to a subset of size ≤ 2k while maintaining an α‑balanced partition with α = 3/2 + 1/k. Whenever this balance is violated, a deterministic global rebalancing procedure (Lemma 1) selects a (1+1/k)‑balanced subset of cut‑edges, incurring only O(n) additional cost per phase. The amortized analysis shows that the extra cost per phase is proportional to Opt’s cost, yielding the O(k) factor.

4. Applying a randomized MTS algorithm – With the reduced state space, the authors invoke the randomized MTS algorithm of Bubeck et al. (SODA 2019), which achieves O(log |S|) competitiveness. Since |S| = n^{O(k)}, the factor becomes O(k·log n). Accounting for the O(k) loss from the restriction gives an overall O(k²·log² n) competitive ratio against any offline algorithm in the same restricted class (Theorem 3).

5. Choosing parameters – Setting k = ⌈1/ε⌉ gives α = 3/2 + ε, satisfying the desired augmentation. Substituting k yields the final competitive ratio O(ε⁻³·log² n).

The paper also discusses how its approach differs from prior multi‑cluster work: there the number of cut‑edges equals the number of clusters and their positions are fixed, whereas here cut‑edges are a tunable parameter and may appear anywhere on the ring, providing greater flexibility.

Overall, the work demonstrates that modest resource augmentation combined with a careful restriction of the state space enables a randomized online algorithm that dramatically improves over the best deterministic bounds for the ring‑demand bisection problem. The techniques may be applicable to other online graph‑partitioning settings where the underlying demand graph has a simple structure.