# News for February 2023

Despite being a short month, February 2023 has witnessed a significant amount of activity under the sublinear “regime”. Let us know if we have missed anything!

Dynamic $$(1 + \epsilon)$$-Approximate Matching Size in Truly Sublinear Update Time by Sayan Bhattacharya, Peter Kiss, and Thatchaphol Saranurak (arXiv). This work throws light on connections between the dynamic and query models of computation and uses them for making advances on approximating the size of a maximum cardinality matching (MCM) in a general graph. In particular, as the main technical ingredient in obtaining an improved dynamic algorithm for maintaining an approximation to the size of MCM, the authors provide a $$\pm \epsilon n$$ approximation algorithm for estimating the size of MCM in a general $$n$$-vertex graph by making $$n^{2 – \Omega_{\epsilon}(1)}$$ adjacency queries. Prior to this result, the state of the art (Behnezhad, Roghani & Rubinstein; STOC’23) was a $$n^{2 – \Omega(1)}$$-query algorithm for the same problem with a multiplicative approximation guarantee of $$1.5$$ and an additive guarantee of $$o(n)$$.

Uniformity Testing over Hypergrids with Subcube Conditioning by Xi Chen and Cassandra Marcussen (arXiv). As the name indicates, the paper studies the fundamental problem of testing uniformity of distributions supported over hypergrids $$[m]^n$$. The tester that they present make $$O(\text{poly}(m)\sqrt{n}/\epsilon^2)$$ queries to a conditional subcube sampling oracle, which, when given a subcube of $$[m]^n$$, returns a point sampled from the distribution conditioned on the point belonging to the subcube. The result is a generalization of the uniformity tester for distributions supported over the $$n$$-dimensional hypercube (Canonne, Chen, Kamath, Levi and Waingarten; SODA ’21).

Easy Testability for Posets by Panna Timea Fekete and Gabor Kun (arXiv). This paper deals with testing properties of directed graphs in the adjacency matrix model. The main characters of the story are posets, or directed acyclic graphs (DAGs) that are transitively closed. Given a family $$\mathcal{F}$$ of finite posets, let $$\mathcal{P}_\mathcal{F}$$ denote the set of all finite posets that do not contain any element of $$\mathcal{F}$$ as a subposet. The main result of the paper is an $$\epsilon$$-tester with query complexity $$\text{poly}(1/\epsilon)$$ for $$\mathcal{P}_\mathcal{F}$$. The authors obtain this result by proving a removal lemma for posets. The result is placed in the larger context of understanding what properties of graphs can be tested with query complexity that has a polynomial dependence on $$1/\epsilon$$ in the adjacency matrix model.

Compressibility-Aware Quantum Algorithms on Strings by Daniel Gibney and Sharma V. Thankachan (arXiv). Lastly, we have a paper on quantum string algorithms that run in sublinear time. In short, the authors present quantum algorithms with optimal running times for computing the Lempel-Ziv (LZ) encoding and Burrows Wheeler Transform (BWT) of highly compressible strings. A main consequence of these results is a faster quantum algorithm for computing the longest common subsequence (LCS) of two strings when the concatenation of the strings is highly compressible. It is to be noted that sublinear-time algorithms do not exist for these problems in the classical model of computation. More details follow.

Factoring a string into disjoint substrings (factors) in an specific manner is the main step in the LZ compression algorithm. The smaller the number of factors, the more compressible the string is. This paper gives a quantum algorithm for the problem of computing the LZ factorization of a string in time $$\tilde{O}(\sqrt{nz})$$, where $$z$$ is the number of factors in the string. They also show that their algorithm is optimal. Using this algorithm, they obtain a fast algorithm for computing the BWT of an input string, as well as an algorithm running in time $$\tilde{O}(\sqrt{nz})$$ to compute the LCS of two strings, where $$n$$ is the length and $$z$$ is the number of factors in the concatenation of the two strings. When $$z$$ is $$o(n^{1/3})$$, this algorithm gives an improvement over the previous best quantum algorithm running in time $$\tilde{O}(n^{2/3})$$.