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Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algorithmic research is to design efficient algorithms, where efficiency is typicallymeasured as a function of the length of the input. For instance, the elementary school algorithm for multiplying two n digit integers takes roughly n2 steps, while more sophisticated algorithmshave been devised which run in less than n log2 n steps. It is still not known whether a linear time algorithm is achievable for integer multiplication. Obviously any algorithm for this task, as for anyother nontrivial task, would need to take at least linear time in n, since this is what it would take to read the entire input and write the output. Thus, showing the existence of a linear time algorithmfor a problem was traditionally considered to be the gold standard of achievement. Nevertheless, due to the recent tremendous increase in computational power that is inundatingus with a multitude of data, we are now encountering a paradigm shift from traditional computational models. The scale of these data sets, coupled with the typical situation in which there is verylittle time to perform our computations, raises the issue of whether there is time to consider any more than a miniscule fraction of the data in our computations? Analogous to the reasoning thatwe used for multiplication, for most natural problems, an algorithm which runs in sublinear time must necessarily use randomization and must give an answer which is in some sense imprecise.Nevertheless, there are many situations in which a fast approximate solution is more useful than a slower exact solution.

Property testing algorithms are “ultra”-efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this article we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: • The self-correcting approach, which was mainly applied in the study of property testing of algebraic properties; • The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the dense-graphs model), as well as in other contexts;

Abstract—We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of bounded-degree graphs with an excluded minor, and in general, for any hyperfinite class of bounded-degree graphs. These oracles utilize only local computation to consistently answer queries about a global partition that breaks the graph into small connected components by removing only a small fraction of the edges. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance: • We give constant-time approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor. • We show a simple proof that any minor-closed graph property is testable in constant time in the bounded degree model. • We prove that it is possible to approximate the distance to almost any hereditary property in any bounded degree hereditary families of graphs. Hereditary properties of interest include bipartiteness, k-colorability, and perfectness. 1.

We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameter, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximity-oblivious testers. While proximity-oblivious testers were studied before – most notably in the algebraic setting – the current study seems to be the first one to focus on graph properties. We provide a mix of positive and negative results, and in particular characterizations of the graph properties that have constant-query proximity-oblivious testers in the two standard models (i.e., the adjacency matrix and the bounded-degree models). Furthermore, we show that constant-query proximity-oblivious testers do not exist for many easily testable properties, and that even when proximity-oblivious testers exist, repeating them does not necessarily yield the best standard testers for the corresponding property.

We investigate the problem of monotonicity reconstruction, as defined by Ailon, Chazelle, Comandur and Liu (2004) in a localized setting. We have oracle access to a nonnegative realvalued function f defined on the domain [n] d = {1,..., n} d (where d is viewed as a constant). We would like to closely approximate f by a monotone function g. This should be done by a procedure (a filter) that given as input a point x ∈ [n] d outputs the value of g(x), and runs in time that is polylogarithmic in n. The procedure can (indeed must) be randomized, but we require that all of the randomness be specified in advance by a single short random seed. We construct such an implementation where the time and space per query is (log n) O(1) and the size of the seed is polynomial in log n and d. Furthermore, with high probability, the ratio of the (Hamming) distance between g and f to the minimum possible Hamming distance between a monotone function and f is bounded above by a function of d (independent of n). This allows for a local implementation: one can initialize many copies of the filter with the same short random seed, and they can autonomously handle queries, while producing outputs that are consistent with the same approximating function g.

We consider the problem of online sublinear expander reconstruction and its relation to random walks in "noisy" expanders. Given access to an adjacency list representation of a bounded-degree graph G, we want to convert this graphinto a bounded-degree expander G0 changing G as little as possible. The graph G0 will be output by a distributed filter: this is a sublinear time procedure that given a query vertex, outputs all its neighbors in G0, and can do so even in a distributed manner, ensuring consistency in all the answers. One of the main tools in our analysis is a result on the be-havior of random walks in graph that are almost expanders: graphs that are formed by arbitrarily connecting a small un-known graph (the noise) to a large expander. We show that a random walk from almost any vertex in the expander part will have fast mixing properties, in the general setting of irreducible finite Markov chains. We also design sublinear time procedures to distinguish vertices of the expander part from those in the noise part, and use this procedure in the reconstruction algorithm.

Referring to the query complexity of property testing, we prove the existence of a rich hierarchy of corresponding complexity classes. That is, for any relevant function q, we prove the existence of properties that have testing complexity Θ(q). Such results are proven in three standard domains often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. While in two cases the proofs are quite straightforward, the techniques employed in the case of the dense graph model seem significantly more involved. Specifically, problems that arise and are treated in the latter case include (1) the preservation of distances between graph under a blow-up operation, and (2) the

Gábor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every ǫ> 0 there is some finite k such that each graph G in the collection can be broken into connected components of size at most k by removing a set of edges of size at most ǫ |V (G)|. We presently extend this notion to a certain compactification of finite bounded-degree graphs, and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite. 1

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