Dear readers, please welcome the new additions to our moderator group, Clément Canonne and Gautam Kamath! It’s great that more property testing researchers are helping take PTReview further. PTReview is becoming really popular these days! (Or so we hope…)
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.
(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