# Call for feedback: upcoming “Introduction to Property Testing”

Forwarding an announcement by Oded Goldreich:

I would like to seek the help of this community regarding my forthcoming book Introduction to Property Testing (see http://www.wisdom.weizmann.ac.il/~oded/pt-intro.html). Specifically, I would be grateful for feedback regarding any aspect and part of the current text, although detailed comments are likely to be more helpful at this time than suggestions for drastic and/or structural changes. Please do not shy from indicating relevant works that may be mentioned (rather than described in detail), since it is quite possible that I was not aware of them or forgot to mention them (rather than chose not to include a detailed description due to various considerations).

Since I am likely to continue revising the text and posting the revisions on the website, please indicate the text-locations to which you refer in a manner that is most robust to revision (e.g., indicate relative location with respect to sectioning and/or theorem-environment numbers rather than with respect to page numbers).

# 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 $$\tilde O(h \log (d)/\varepsilon)$$.