So what did we see in February 2014?
Local Algorithms for Sparse Spanning Graphs by Reut Levi, Dana Ron, and Ronitt Rubinfeld (arXiv). Given a graph \(G = (V,E)\), the aim is the find a small sparse spanning subgraph \(G’\) in the local algorithm setting. So, given any edge \((u,v) \in E\), we must determine in sublinear time whether this edge exists in \(G’\). The main result is an \(\Omega(|V|)\) lower bound, and a matching algorithm when \(G\) is either an expander or hyperfinite.
A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems by Yuichi Yoshida (arXiv). This result gives a complete characterization of (two-sided) locally testable affine-invariant properties. These are based on existing decomposition theorems for functions into structured and pseudorandom parts. This result is analogous to that of Alon, Fischer, Newman, and Shapira (link) on characterizations for testability over dense graphs.
Testing probability distributions underlying aggregated data by Clément Canonne and Ronitt Rubinfeld (arXiv). There has been much recent work on distribution testing, and this work discusses stronger models where better algorithms can be obtained. Given a distribution \(D\) over \([n]\), we can get samples from this distribution, and also query the probability \(D(i)\) of any \(i \in [n]\). With these “dual” queries, one gets distribution testers for a variety of problems surpassing previous lower bounds. Interestingly, this is stronger than the conditional model introduced recently (here and here).
Strong Locally Testable Codes with Relaxed Local Decoders by Oded Goldreich, Tom Gur, and Ilan Komargodski (ECCC). For a locally testable code (LTC), the property of being a codeword is efficiently (constant time) testable. A locally decodable code (LDC) is a code from which individual bits of the original message can be recovered in constant time. A relaxed local decoder is allowed to declare failure for recovered a small fraction of message bits. This allows for bypassing the difficult (and prominent) open problem of constructing small LDCs. This result gives a construction of near-linear LTCs with a relaxed local decoder. This result leads to a property that is hard to test but easy to verify (in the MA proofs of proximity sense, refer to last month’s open problem).