Nearest neighbor searching systems are an integral part of many online applications, including but not limited to pattern recognition, plagiarism detection, and recommender systems. With increasingly larger data sets, scalability has become an important issue. Many of the most space and running time efficient algorithms are based on locality-sensitive hashing. Here, we view the data set as an n by lUl matrix where each row corresponds to one of n users and the columns correspond to items drawn from a universe U. The de-facto standard approach to quickly answer nearest neighbor queries on such a data set is usually a form of min-hashing. Not only is min-hashing very fast, but it is also space efficient and can be implemented in many computational models aimed at dealing with large data sets such as MapReduce and streaming. However, a significant drawback is that minhashing and related methods are only able to handle insertions to user profiles and tend to perform poorly when items may be removed. We initiate the study of scalable locality-sensitive hashing (LSH) for fully dynamic data-streams. Specifically, using the Jaccard index as similarity measure, we design (1) a collaborative filtering mechanism maintainable in dynamic data streams and (2) a sketching algorithm for similarity estimation. Our algorithms have little overhead in terms of running time compared to previous LSH approaches for the insertion only case, and drastically outperform previous algorithms in case of deletions.

We show that for n points in d-dimensional Euclidean space, a data oblivious random projection of the columns onto m∈ O((logk+loglogn)ε−6log1/ε) dimensions is sufficient to approximate the cost of all k-means clusterings up to a multiplicative (1±ε) factor. The previous-best upper bounds on m are O(logn· ε−2) given by a direct application of the Johnson-Lindenstrauss Lemma, and O(kε−2) given by [Cohen et al.-STOC’15].

Recommender systems are an integral part of many web applications. With increasingly larger user bases, scalability has become an important issue. Many of the most scalable algorithms with respect to both space and running times are based on locality sensitive hashing. However, a significant drawback is that these methods are only able to handle insertions to user profiles and tend to perform poorly when items may be removed. We initiate the study of scalable locality sensitive hashing (LSH) for dynamic input. Specifically, using the Jaccard index as similarity measure, we design (1) a sketching algorithm for similarity estimation via a black box reduction to $\ell_0$ norm estimation and (2) a locality sensitive hashing scheme maintainable in fully dynamic data streams that quickly filters out low-similarity pairs. Our algorithms have little to no overhead in terms of running time compared to previous LSH approaches for the insertion only case, and drastically outperform previous algorithms in case of deletions.

We initiate the study of finding the Jaccard center of a given collection N of sets. For two sets X,Y, the Jaccard index is defined as |X\cap Y|/|X\cup Y| and the corresponding distance is 1-|X\cap Y|/|X\cup Y|. The Jaccard center is a set C minimizing the maximum distance to any set of N. We show that the problem is NP-hard to solve exactly, and that it admits a PTAS while no FPTAS can exist unless P = NP. Furthermore, we show that the problem is fixed parameter tractable in the maximum Hamming norm between Jaccard center and any input set. Our algorithms are based on a compression technique similar in spirit to coresets for the Euclidean 1-center problem. In addition, we also show that, contrary to the previously studied median problem by Chierichetti et al. (SODA 2010), the continuous version of the Jaccard center problem admits a simple polynomial time algorithm.