Online Discrepancy Minimization via Persistent Self-Balancing Walks
David Arbour, Drew Dimmery, Tung Mai, Anup Rao
备注：
The proof of Lemma 7 is incorrect. There is a serious issue that we don't know how to fix at the moment. We thank Yang, Nikhil and collaborators for bringing it to our attention
We study the online discrepancy minimization problem for vectors in $\mathbb{R}^d$ in the oblivious setting where an adversary is allowed fix the vectors $x_1, x_2, \ldots, x_n$ in arbitrary order ahead of time. We give an algorithm that maintains $O(\sqrt{\log(nd/δ)})$ discrepancy with probability $1-δ$, matching the lower bound given in [Bansal et al. 2020] up to an $O(\sqrt{\log \log n})$ factor in the high-probability regime. We also provide results for the weighted and multi-color versions of the problem.
We introduce a novel relaxation of combinatorial discrepancy called Gaussian discrepancy, whereby binary signings are replaced with correlated standard Gaussian random variables. This relaxation effectively reformulates an optimization problem over the Boolean hypercube into one over the space of correlation matrices. We show that Gaussian discrepancy is a tighter relaxation than the previously studied vector and spherical discrepancy problems, and we construct a fast online algorithm that achieves a version of the Banaszczyk bound for Gaussian discrepancy. This work also raises new questions …
We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Komlós conjecture.
A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of $T$ unit vectors in $\mathbb{R}^d$, find $\pm$ signs for each of them such that the signed sum vector along any prefix has a small $\ell_\infty$-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Komlós problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an $O(\sqrt{\log d+ \log T})$ bound for the prefix discrepancy pr…
We consider an online vector balancing question where $T$ vectors, chosen from an arbitrary distribution over $[-1,1]^n$, arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the discrepancy small as possible. A concrete example is the online interval discrepancy problem where T points are sampled uniformly in [0,1], and the goal is to immediately color them $\pm$ such that every sub-interval remains nearly balanced. As random coloring incurs $Ω(T^{1/2})$ discrepancy, while the offline bounds are $Θ(\sqrt{n \log (T/n)})$ for vector balancing and $1$ for interval ba…
We consider the problem of deterministic load balancing of tokens in the discrete model. A set of $n$ processors is connected into a $d$-regular undirected network. In every time step, each processor exchanges some of its tokens with each of its neighbors in the network. The goal is to minimize the discrepancy between the number of tokens on the most-loaded and the least-loaded processor as quickly as possible. Rabani et al. (1998) present a general technique for the analysis of a wide class of discrete load balancing algorithms. Their approach is to characterize the deviation between the actu…
The vector-balancing problem is a fundamental problem in discrepancy theory: given T vectors in $[-1,1]^n$, find a signing $σ(a) \in \{\pm 1\}$ of each vector $a$ to minimize the discrepancy $\| \sum_{a} σ(a) \cdot a \|_{\infty}$. This problem has been extensively studied in the static/offline setting. In this paper we initiate its study in the fully-dynamic setting with recourse: the algorithm sees a stream of T insertions and deletions of vectors, and at each time must maintain a low-discrepancy signing, while also minimizing the amortized recourse (the number of times any vector changes its…
In the stochastic online vector balancing problem, vectors $v_1,v_2,\ldots,v_T$ chosen independently from an arbitrary distribution in $\mathbb{R}^n$ arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to $\mathsf{polylog}(nT)$ factors. This substantially generalizes and improves upon the…
Iterative load balancing algorithms for indivisible tokens have been studied intensively in the past. Complementing previous worst-case analyses, we study an average-case scenario where the load inputs are drawn from a fixed probability distribution. For cycles, tori, hypercubes and expanders, we obtain almost matching upper and lower bounds on the discrepancy, the difference between the maximum and the minimum load. Our bounds hold for a variety of probability distributions including the uniform and binomial distribution but also distributions with unbounded range such as the Poisson and geom…
In a balancing network each processor has an initial collection of unit-size jobs (tokens) and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially affects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly-random assignment rule, and the arbitrary one (in the spirit of smoothed-analysis). We start with an arbitrary assignment of balancer directions and then flip e…
We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time $O(n \log n + kn \log k)$ for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$ tasks with given start- and endtimes hav…
