The threshold for online balancing of i.i.d. binary vectors

2026-09-13

AI Summary

研究解决了随机二进制向量在线平衡问题，通过高效在线算法确定了在不同稀疏度下最优前缀不一致性阈值。

Abstract

Consider the task of online vector balancing for stochastic arrivals $X_1,\ldots,{X_T}$, where the $X_i$ are independent uniformly random $d$--sparse binary vectors in $\{0,1\}^n$. This is a random analogue of the online Beck--Fiala problem. We show that uniformly for $2\le d\le n/2$ and $T = Θ(n)$, the optimal online prefix discrepancy $\max\limits_{t\leq T}\left\|\sum_{i=1}^tσ_i X_i\right\|_\infty$ is of order \[ Θ\big(\max\{\sqrt d,\log\log n\}\big). \] The upper bound is achieved by an efficient online algorithm. Thus, for $d\le(\log\log n)^2$, the optimal discrepancy is $Θ(\log\log n)$ and is independent of the sparsity up to constant factors, whereas above this scale it is $Θ(\sqrt d)$, matching the order of the offline discrepancy. This identifies the threshold at which sparsity begins to govern the online discrepancy of the random Beck--Fiala model.

Problem

• online balancing
• i.i.d. binary vectors
• sparsity
• discrepancy
• threshold

Innovation

• online vector balancing
• stochastic arrivals
• prefix discrepancy
• sparsity threshold