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.
A local algorithm is a distributed algorithm that runs in constant time, independently of the size of the network. Being highly scalable and fault tolerant, such algorithms are ideal in the operation of large-scale distributed systems. Furthermore, even though the model of local algorithms is very limited, in recent years we have seen many positive results for nontrivial problems. This work surveys the state-of-the-art in the field, covering impossibility results, deterministic local algorithms, randomized local algorithms, and local algorithms for geometric graphs.