Correlation clustering is perhaps the most natural formulation of clustering. Given @math objects and a pairwise similarity measure, the goal is to cluster the objects so that, to the best possible extent, similar objects are put in the same cluster and dissimilar objects are put in different clusters. Despite its theoretical appeal, the practical relevance of correlation clustering still remains largely unexplored, mainly due to the fact that correlation clustering requires the @math pairwise similarities as input. In this paper we initiate the investigation into algorithms for correlation clustering. In we are given the identifier of a single object and we want to return the cluster to which it belongs in some globally consistent near-optimal clustering, using a small number of similarity queries. Local algorithms for correlation clustering open the door to algorithms, which are particularly useful when the similarity between items is costly to compute, as it is often the case in many practical application domains. They also imply @math distributed and streaming clustering algorithms, @math constant-time estimators and testers for cluster edit distance, and @math property-preserving parallel reconstruction algorithms for clusterability. Specifically, we devise a local clustering algorithm attaining a @math -approximation in time @math independently of the dataset size. An explicit approximate clustering for all objects can be produced in time @math (which is provably optimal). We also provide a fully additive @math -approximation with local query complexity @math and time complexity @math . The latter yields the fastest polynomial-time approximation scheme for correlation clustering known to date.
For input @math , let @math denote the set of outputs that are the "legal" answers for a computational problem @math . Suppose @math and members of @math are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input @math , support queries by a user to values of specified locations @math in a legal output @math . When more than one legal output @math exists for a given @math , the local computation algorithm should output in a way that is consistent with at least one such @math . Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of @math -wise independent random variables and Beck's analysis in his algorithmic approach to the Lov ' a sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in polylogarithmic time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying @math -SAT formulas.