News for April 2016

April has been kind to us, providing us with a trio of new papers in sublinear algorithms. Also, in case you missed them, be sure to check out Oded Goldreich’s guest post and the associated open problem.

Sparse Fourier Transform in Any Constant Dimension with Nearly-Optimal Sample Complexity in Sublinear Time, by Michael Kapralov (ArXiv). There is a rich line of work focused on going beyond the celebrated Fast Fourier Transform algorithm by exploiting sparsity in the signal’s Fourier representation. While the one-dimensional case is very well understood, results in higher dimensions have either been lacking with respect to time complexity, sample complexity, or the approximation guarantee. This work is the first to give an algorithm which runs in sublinear time, has near-optimal sample complexity, and provides an arbitrarily accurate approximation guarantee for the multivariate case.

Sublinear Time Estimation of Degree Distribution Moments: The Arboricity Connection, by Talya Eden, Dana Ron, C. Seshadhri (ArXiv). This work revisits the problem of approximating the moments of the degree distribution of a graph. Prior work has shown that the \(s\)-th moment of the degree distribution can be computed with \(\tilde O(n^{1 – 1/(s+1)})\) queries, which is optimal up to \(\mathrm{poly}(\log n)\) factors. This work provides a new algorithm which requires only \(\tilde O(n^{1 – 1/s})\) queries on graphs of bounded arboricity, while still matching previous near-optimal bounds in the worst case. Impressively, the algorithm is incredibly simple, and can be stated in just a few lines of pseudocode.

A Local Algorithm for Constructing Spanners in Minor-Free Graphs, by Reut Levi, Dana Ron, Ronitt Rubinfeld (ArXiv). This paper addresses the problem of locally constructing a sparse spanning graph. For an edge \(e\) in a graph \(G\), one must determine whether or not \(e\) is in a sparse spanning graph \(G’\) in a manner that is consistent with previous answers. In general graphs, the complexity of this problem is \(\Omega(\sqrt{n})\), leading to the study of restricted families of graphs. In minor-free graphs (which include, for example, planar graphs), existing algorithms require \((d/\varepsilon)^{\mathrm{poly}(h)\log(1/\varepsilon)}\) queries and induce a stretch of \(\mathrm{poly}(d, h, 1/\varepsilon)\), where \(d\) is the maximum degree, \(h\) is the size of the minor, and \(G’\) is of size at most \((1 + \varepsilon)n\). The algorithm in this paper shows the complexity of this problem to be polynomial, requiring only \(\mathrm{poly}(d, h, 1/\varepsilon)\) queries and inducing a stretch of \(\tilde O(h \log (d)/\varepsilon)\).

Open problem for April 2016

As open problems of the month, we state here the two questions discussed in our recent guest post by Oded Goldreich:

Open Problem 1 (obtaining a one-sided error reduction): The reduction from testing affinity of a subspace to that of testing affinity of a function has two-sided error probability. We wonder whether a one-sided error reduction of similar complexity can be found.

Note that this will yield a one-sided error tester for affinity, which we believe is not known. Actually, we would also welcome a two-sided error reduction that, when combined with the linearity tester, yields a tester of complexity \(O(1/\epsilon)\) rather than \(\tilde{O}(1/\epsilon)\).

Turning to the task of testing monomials, we recall that the solution of [PRS02] is based on employing self-correction to the following test that refers to the case that \(H=\{x:AX=b\}\) is an \((\ell-k)\)-dimensional affine space. Basically, the test selects a random \(x\in H\) and a random \(y\in\{0,1\}^\ell\), and checks whether it holds that \(y\in H\) if and only if \(x\wedge y \in H\), where \(\wedge\) denotes the bit-by-bit product of vectors. It is painfully shown in [5] that if \(H\) is not a translation by \(1^\ell\) of an axis-aligned linear space, then the check fails with probability \(\Omega(2^{-k})\). Furthermore, it is shown that for a constant fraction of the \(x\)’s (in \(H\)), the check fails on a \(\Omega(2^{-k})\) fraction of the \(y\)’s (in \(\{0,1\}^\ell\)). This strengthening is important, since selecting \(x\in H\) uniformly requires \(\Theta(2^k)\) trials. Recall that proving the foregoing assertion for \(k=1\) is quite easy (cf. [5]), which leads us to ask

Open Problem 2 (can a simpler proof be found for the case of \(k>1\)): Is there a relatively simple reduction of the foregoing claim for general \(k\) to the special case of \(k=1\)?

[PRS02] M. Parnas, D. Ron, and A. Samorodnitsky. Testing Basic Boolean Formulae. SIAM Journal on Disc. Math. and Alg., Vol. 16 (1), pages 20–46, 2002.

Guest Post: Oded Goldreich, “Reducing testing affine spaces to testing linearity”

[We are delighted to have this month a blog post by Oded Goldreich, on a reduction from testing affinity of subspaces to testing linearity of Boolean functions. Oded also gave us open problems directly related to this post.]

We consider the task of testing whether a Boolean function \(f\colon \{0,1\}^\ell\to\{0,1\}\) is the indicator function of an \((\ell-k)\)-dimensional space. An optimal tester for this property was presented by Parnas, Ron, and Samorodnitsky (SIDMA, 2002), by mimicking the celebrated linearity tester (of Blum, Luby and Rubinfeld, JCSS, 1993) and its analysis. We show that this task can be reduced to testing the linearity of a related function \(g\colon\{0,1\}^\ell\to\{0,1\}^k\), yielding an almost optimal tester.

Continue reading

News for March 2016

While March 2016 has been rather low in terms of property testing, we did see a new paper appear:

A Note on Tolerant Testing with One-Sided Error, by Roei Tell (ECCC). A natural generalization of property testing is that of tolerant testing, as introduced by Parnas, Ron, and Rubinfeld [PRR06]: where the tester still must reject all objects that are far from satisfying the property, but now also has to accept those that are sufficiently close (all that with constant probability). In this work is considered the question of one-sidedness of tolerant testers: namely, is it possible to only err on the farness side, but accept close output with probability one? As it turns out, it is not — the author shows that any such one-sided tolerant tester, for basically any property of interest, must essentially query the whole input…

Universal Locally Testable Codes, by Oded Goldreich and Tom Gur (ECCC). In this work, the authors introduce and initiate the study of an extension of locally testable codes they name universal locally testable codes (universal-LTC). At a high-level, a universal-LTC (with regard to a family of functions \(\cal F\)) is a locally testable code \(C\) “for which the restrictions (subcodes) of \(C\) by functions in \(\cal F\) are also locally testable.” In other terms, one is then able to test efficiently, given an encoded string \(w\), if (i) \(w=C(x)\) for some \(x\); but also, for any \(f\in \cal F\), if (ii) \(w=C(x)\) for some \(x\) that satisfies \(f(x)=1\).

Edit (04/06): added the work of Goldreich and Gur, which was overlooked in our first version of the article.

News For February 2016

News for February 2016! Another take on distribution testing, exciting stuff about generating random graphs, and distributed property testing.

Sublinear Random Access Generators for Preferential Attachment Graphs by Guy Even, Reut Levi, Moti Medina, and Adi Rosén (ArXiv). A major aspect of research in “network science” and applied large graph analysis is random graph models. A classic example is the Barabasi-Albert preferential attachment (PA) model, that attempts to model graphs that appear in the real-world. The model is inherently sequential, in that vertices “arrive” in order and connect to other vertices according to a specified distribution. It was not clear how to generate a PA graph in parallel, or how one can construct a local view of a PA graph without constructing all of it. Not clear, that is, until this paper. This really cool result shows the following. Suppose we want to construct an \(n\) vertex PA graph. One can basically query the adjacency list of any vertex in \(poly(\log n)\) time, without constructing the graph! The PA graph is implicitly constructed through the queries. This is really interesting because PA graphs (at least conceptually) are created sequentially, and the order of vertex arrival fundamentally affects the graph structure.

 

The Uniform Distribution is Complete with Respect to Testing Identity to a Fixed Distribution by Oded Goldreich (ECCC) . This is a follow up to a recent result by Diakonikolas and Kane (covered last month!). This paper reduces testing equality (of a distribution) to a fixed distribution, to the special case to when the fixed distribution is uniform. The notions of reduction in distribution testing were exploited by Diakonikolas and Kane to get optimal testers. While this new paper does not give new bounds, it gives a clean and simplified approach for testing distribution equality. This paper came out of Oded’s lecture notes, and has a nice expository feel to it.

 

Fast Distributed Algorithms for Testing Graph Properties by Keren Censor-Hillel, Eldar Fischer, Gregory Schwartzman, and Yadu Vasudev (ArXiv) . Connections between distributed algorithms and property testing have been unearthed in the past (most explicitly by Parnas and Ron). This paper directly solves graph property testing problems in the distributed setting. For dense graph testing, it is known that any (constant-query) testing algorithm algorithm can be made non-adaptive, and thus can be simulated in the distributed setting. The results for testing bipartiteness in the sparse graph model are quite interesting, because existing property testers use random walks. This paper gives a distributed implementation of multiple random walks from all vertices, and controls the total congestion of the implementation. This leads to a \(O(\log n)\)-round bipartiteness tester.

Lecture Notes on Property Testing

(Update: Krzysztof pointed out the resources at sublinear.info, which we forgot to mention. It makes for sense for all of this information to be there. We’ve added links there, and would urge our readers to post more information there. Beats having it in half-baked blogposts!)

Despite the relative maturity of property testing (more than a decade old!), we still lack a good textbook on the subject. At least, I definitely feel the need when I’m talking to interested students. I end up pointing to a bunch of papers, all with their own notation. We now have numerous courses being taught, so that certainly helps. And Oded Goldreich just pointed out his notes. So here’s a summary of stuff that I have found. Please let us know if you have other sources, including your own courses!

Oded Goldreich’s lecture notes
Ronitt Rubinfeld’s collection of surveys
Sofya Raskhodnikova’s course notes at Penn State
Eric Price’s course notes at UT Austin
Grigory Yaroslavtsev’s crash course at the University of Buenos Aires
Rocco Servedio’s course notes at Columbia

News for January 2016

In the first month of 2016 we had a couple of interesting papers in property testing.

Sublinear-Time Algorithms for Counting Star Subgraphs with Applications to Join Selectivity Estimation by Maryam Aliakbarpour, Amartya Shankha Biswas, Themistoklis Gouleakis, John Peebles, Ronitt Rubinfeld and Anak Yodpinyanee (ArXiv) . Estimating the number of subgraphs of a fixed form in a given graph has been a very well studied problem in the subject of property testing.  The classical example is counting the number of triangles in a graph. Usually we assume we can sample the vertices uniformly at random. In this paper a new model of query has been considered where one can sample the edges uniformly at random. Under this model it is shown that the query complexity can be reduced when the goal is to estimate the number of p-stars in a graph. It would be interesting to see if this model helps to get better query complexity for other problems also. Depending on the application different query models can be relevant and hence understanding these different query models are essential.

 

A New Approach for Testing Properties of Discrete Distributions by Ilias Diakonikolas and Daniel M. Kane (ArXiv) . Testing properties of distribution has been a central topic in our field. Not only are these problems interesting they also has been applied as subroutines to many other testing algorithms. In the literature a number of different properties like testing whether a distribution is uniform or testing whether two distributions are identical or testing if two distributions are independent and many more has been studied. In most of the properties the distance measure is the L_1 measure. In this paper a unifying technique to obtain testing algorithms for properties of distributions in the L_1 distance has been given by converting them to the problem of testing in the L_2 distance and applying standard testing algorithms in the L_2 distance. This approach would hopefully help us simplify the various algorithms in distribution testing.

News for December 2015

Greetings from the exciting Workshop on Sublinear Algorithms at John Hopkins University! As this workshop and the upcoming SODA and ITCS conferences get 2016 to a roaring start, let us take one last look back at property testing news from last year. In December, one work in particular caught my eye:

 Non-Local Probes Do Not Help with Graph Problems by Mika Göös, Juho Hirvonen, Reut Levi, Moti Medina, and Jukka Suomela (arXiv). A generalization of property testing that has recently seen some fascinating developments in the past few years is the local computation algorithms (LCA) model, in which the algorithm is asked to answer some local query (such as “what is the color of this vertex in some fixed, legal coloring of the graph?”) in sublinear-time. This paper relates the LCA model to message-passing models and in the process gives a powerful new tool for establishing lower bounds in LCAs.