Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2102.08243

Online matching in lossless expanders

Marius Zimand
View original source ↗Read source document ↗Export metadata ↓

Abstract

Bauwens and Zimand [BZ 2019] have shown that lossless expanders have an interesting online matching property. The result appears in an implicit form in [BZ 2019]. We present an explicit version of this property which is directly amenable to typical applications, prove it in a self-contained manner that clarifies the role of some parameters, and give two applications. A $(K, \epsilon)$ lossless expander is a bipartite graph such that any subset $S$ of size at most $K$ of nodes on the left side of the bipartition has at least $(1-\epsilon) D |S|$ neighbors, where $D$ is the left degree.The main result is that any such graph, after a slight modification, admits $(1-O(\epsilon)D, 1)$ online matching up to size $K$. This means that for any sequence $S=(x_1, \ldots, x_K)$ of nodes on the left side of the bipartition, one can assign in an online manner to each node $x_i$ in $S$ a set $A_i$ consisting of $(1-O(\epsilon))$ fraction of its neighbors so that the sets $A_1, \ldots, A_K$ are pairwise disjoint. "Online manner" refers to the fact that, for every $i$, the set of nodes assigned to $x_i$ only depends on the nodes assigned to $x_1, \ldots, x_{i-1}$. The first application concerns storage schemes for representing a set $S$, so that a membership query "Is $x \in S$?" can be answered probabilistically by reading a single bit. All the previous one-probe storage schemes were for a static set $S$. We show that a lossless expander can be used to construct a one-probe storage scheme for dynamic sets, i.e., sets in which elements can be inserted and deleted without affecting the representation of other elements. The second application is about non-blocking networks.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Marius Zimand. 2021-02-16. Online matching in lossless expanders. https://arxiv.org/abs/2102.08243

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗