Reut Levi

Approximating and testing k-histogram distributions in sub-linear time

A discrete distribution <i>p</i>, over <i>[n]</i>, is a <i>k</i> histogram if its probability distribution function can be represented as a piece-wise constant function with <i>k</i> pieces. Such a function is represented by a list of <i>k</i> intervals and <i>k</i> corresponding values. We consider the following problem: given a collection of samples from a distribution <i>p</i>, find a <i>k</i>-histogram that (approximately) minimizes the l ₂ distance to the distribution <i>p</i>. We give time and sample efficient algorithms for this problem. We further provide algorithms that distinguish distributions that have the property of being a <i>k</i>-histogram from distributions that are ε-far from any <i>k</i>-histogram in the l ₁ distance and l ₂ distance respectively.

Testing Properties of Collections of Distributions

We propose a framework for studying property testing of collections of distributions, where the number of distributions in the collection is a parameter of the problem. Previous work on property testing of distributions considered single distributions or pairs of distributions. We suggest two models that differ in the way the algorithm is given access to samples from the distributions. In one model the algorithm may ask for a sample from any distribution of its choice, and in the other the choice of the distribution is random. Our main focus is on the basic problem of distinguishing between the case that all the distributions in the collection are the same (or very similar), and the case that it is necessary to modify the distributions in the collection in a non-negligible manner so as to obtain this property. We give almost tight upper and lower bounds for this testing problem, as well as study an extension to a clusterability property. One of our lower bounds directly implies a lower bound on testing independence of a joint distribution, a result which was left open by previous work.

A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient partition oracles. A partition oracle is a procedure that, given access to the incidence lists representation of a bounded-degree graph G= (V,E) and a parameter ε, when queried on a vertex v ∈ V, returns the part (subset of vertices) that v belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most ε |V|. In this work, we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/ε, improving on the result of Hassidim et al. (Proceedings of FOCS 2009), who gave a partition oracle with query complexity exponential in 1/ε. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.

Non-local Probes Do Not Help with Many Graph Problems

This work bridges the gap between distributed and centralised models of computing in the context of sublinear-time graph algorithms. A priori, typical centralised models of computing (e.g., parallel decision trees or centralised local algorithms) seem to be much more powerful than distributed message-passing algorithms: centralised algorithms can directly probe any part of the input, while in distributed algorithms nodes can only communicate with their immediate neighbours. We show that for a large class of graph problems, this extra freedom does not help centralised algorithms at all: efficient stateless deterministic centralised local algorithms can be simulated with efficient distributed message-passing algorithms. In particular, this enables us to transfer existing lower bound results from distributed algorithms to centralised local algorithms.

A quasi-polynomial time partition oracle for graphs with an excluded minor

Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient partition oracles. A partition oracle is a procedure that, given access to the incidence lists representation of a bounded-degree graph G=(V,E) and a parameter e, when queried on a vertex v∈V, returns the part (subset of vertices) which v belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most e|V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/e, thus improving on the result of Hassidim et al. (Proceedings of FOCS 2009) who gave a partition oracle with query complexity exponential in 1/e. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.

Testing Similar Means

We consider the problem of testing a basic property of collections of distributions: having similar means. Namely, the algorithm should accept collections of distributions in which all distributions have means that do not differ by more than some given parameter and should reject collections that are relatively far from having this property. By “far” we mean that it is necessary to modify the distributions in a relatively significant manner (according to some predetermined distance measure) so as to obtain the property. We study this problem in two models. In the first model (the query model) the algorithm may ask for samples from any distribution of its choice, and in the second model (the sampling model) the distributions from which it gets samples are selected randomly. We provide upper and lower bounds in both models. In particular, in the query model, the complexity of the problem is polynomial in

A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error.

1/\epsilon

Three Notes on Distributed Property Testing

(where

Local Algorithms for Sparse Spanning Graphs

\epsilon

Testing Properties of Collections of Distributions.

is the given distance parameter), while in the sampling model, the comple...