This month has seen a flurry of activity in sublinear algorithms and a diverse collection of papers have come up, with topics ranging from differentially private sublinear algorithms to local testers for multiplicity codes. Apologies to the readers for the delay in putting this post together!

**Almost-Optimal Sublinear-Time Edit Distance in the Low Distance Regime** by Karl Bringmann, Alejandro Cassis, Nick Fischer and Vasileios Nakos (arXiv)

This paper considers the problem of gap edit distance, i.e., of determining if the edit distance between two strings \(x\) and \(y\) is at most \(k\) or at least \(K\). Their main result is an algorithm that runs in time \(O(n/k + \text{poly}(k))\) and solves the problem for \(K = k \cdot 2^{\tilde{O}(\sqrt{\log k})}\). The paper improves upon earlier results of Goldenberg, Krauthgamer and Saha (2019) and Kociumaka and Saha (2020) who solved the problem for \(K = k^2\) with the same asymptotic guarantee on the query complexity.

One of the interesting takeaways from the paper is that the complexity of solving the gap Hamming distance and gap edit distance are similar in the low distance regime. For both, the complexity of solving the \((k, k^{1+o(1)})\)-gap problem is \(n/k^{1 \pm o(1)}\). This needs to be contrasted with the fact that solving \((k, \Omega(n))\)-gap edit distance requires \(\Omega(\sqrt{k})\) queries as shown by Batu, Ergun, Kilian, Magen and Raskhodnikova (2003), whereas \((k, \Omega(n))\)-gap Hamming distance can be solved in \(O(1)\) time.

These results are incomparable to those obtained by Goldenberg, Kociumaka, Krauthgamer and Saha (which was discussed in our November 2021 post), where they give a nonadaptive algorithm with complexity \(O(n/k^{1.5})\) for \((k, k^2)\)-gap edit distance problem. The algorithm in the present paper is adaptive and works faster for smaller values of \(k\).

**Privately Estimating Graph Parameters in Sublinear time** by Jeremiah Blocki, Elena Grigorescu, Tamalika Mukherjee (arXiv)

Differentially private approximation algorithms for optimization problems on graphs is a well-studied topic. This paper opens up an exciting research direction by initiating a systematic study of the design of differentially private sublinear-time algorithms. The setting is that graphs are viewed as databases and two graphs are neighboring if they differ in an edge (or a node). An algorithm \(A\) is \(\epsilon\)-differentially private if for every pair of edge-neighboring(or node-neighboring) graphs \(G, G’\) and for every subset \(S\) of outputs, \(\Pr[A(G) \in S] \leq \exp(\epsilon) \cdot \Pr[A(G’) \in S]\).

The paper presents \(\epsilon\)-differentially private sublinear-time algorithms for well-studied problems such as estimating the average degree, the size of a min vertex cover and the size of a maximum matching. These algorithms access the input graphs via *neighbor queries* and *degree queries*.

In addition to providing a strong privacy guarantee, their algorithms nearly match the approximation and complexity guarantees of their non-differentially private counterparts. The main idea seems to be the formalization of a sensitivity notion, which they refer to as Global Coupled Sensitivity, and bounding it for the known sublinear-time algorithms for the aforementioned problems. Finally, they add Laplace noise calibrated with this sensitivity value to the output of the algorithms to make them differentially private.

**Testability and Local Certification of Monotone Properties in Minor-closed Classes** by Louis Esperet And Sergey Norin (arXiv)

One of the major interests in graph property testing is to characterize which properties are testable, i.e, can be \(\epsilon\)-tested with query complexity that depends only on the parameter \(\epsilon\). The question of testability is well-understood in the dense graph model as well as the bounded degree model. This paper concerns itself with testability questions in the general model or the sparse model of graph property testing, where graphs are represented as adjacency lists with no bound on the maximum degree.

The authors prove that every monotone property of** **minor-closed graph classes is testable with one-sided error, where a property is monotone if it is closed under taking subgraphs and a graph class is minor-closed if it is closed under taking minors. A crucial fact to be noted here is that a tester is allowed to make only uniformly random nerighbor queries.

This result is a significant generalization of a 2019 result by Czumaj and Sohler, who proved that for every finite set of graphs \(\mathcal{H}\), every \(\mathcal{H}\)-free property of minor-closed graph classes is testable with one-sided error, where a graph satisfies \(\mathcal{H}\)-freeness if none of its subgraphs belong to \(\mathcal{H}\).

They show an interesting consequence of their results to designing a short local certification scheme for monotone properties of minor-closed graph classes. Roughly speaking, they show the existence of a prover-verifier system for the aforementioned testing problem where proofs of length \(O(\log n)\) are assigned to each vertex and that verifier needs to observe only the proofs assigned to a vertex and its neighbors.

**The plane test is a local tester for Multiplicity Codes** by Dan Karliner, Roie Salama, and Amnon Ta-Shma (ECCC)

Multiplicity codes are a generalization of Reed-Muller codes and was first studied by Guruswami and Wang (2013) and Kopparty, Saraf and Yekhanin (2014). The messages here are polynomials of degree \(d\) over \(m\) variables and the codeword corresponding to a polynomial \(p\) is the evaluation of \(p\) and of all of its directional derivatives of order upto \(s\) over all the points in \(\mathbb{F}_q^m\), where \(q\) is a prime power.

Even though multiplicity codes are known to be locally decodable, it was open whether they are locally testable. Local testers for Reed Muller codes work by restricting the evaluations to a uniformly random line in \(\mathbb{F}_q^m\) and checking whether it corresponds to the evaluations of a degree \(d\) univariate polynomial. The authors first show that such a tester does not work for the case of multiplicity codes when \(d\) is large. They then show that a *plane test* is a good local tester for multiplicity codes even for larger values of \(d\). Specifically, a plane test checks whether the restriction of a given word, which is purportedly the evaluation of a polynomial of degree \(d\) and of its derivatives, to a uniformly random plane in \(\mathbb{F}_q^m\) is a bivariate multiplicity code of degree \(d\).

We conclude the post with a short note from Nader Bshouty and Oded Goldreich on a fundamental characterization result in property testing.

**On properties that are non-trivial to test** by Nader H. Bshouty and Oded Goldreich (ECCC)

A property on binary strings is nontrivial if for infinitely many \(n\), the property contains at least one string of length \(n\) and at most \(2^{n – \Omega(n)}\) strings of length \(n\). The note shows that every nontrivial property requires \(\Omega(1/\epsilon)\) queries to \(\epsilon\)-test.