In this paper we consider efficient construction of "composable core-sets" for basic diversity and coverage maximization problems. A core-set for a point-set in a metric space is a subset of the point-set with the property that an approximate solution to the whole point-set can be obtained given the core-set alone. A composable core-set has the property that for a collection of sets, the approximate solution to the union of the sets in the collection can be obtained given the union of the composable core-sets for the point sets in the collection. Using composable core-sets one can obtain efficient solutions to a wide variety of massive data processing applications, including nearest neighbor search, streaming algorithms and map-reduce computation.Our main results are algorithms for constructing composable core-sets for several notions of "diversity objective functions", a topic that attracted a significant amount of research over the last few years. The composable core-sets we construct are small and accurate: their approximation factor almost matches that of the best "off-line" algorithms for the relevant optimization problems (up to a constant factor). Moreover, we also show applications of our results to diverse nearest neighbor search, streaming algorithms and map-reduce computation. Finally, we show that for an alternative notion of diversity maximization based on the maximum coverage problem small composable core-sets do not exist.

Motivated by the recent research on diversity-aware search, we investigate the k-diverse near neighbor reporting problem. The problem is defined as follows: given a query point q, report the maximum diversity set S of k points in the ball of radius r around q. The diversity of a set S is measured by the minimum distance between any pair of points in S (the higher, the better).We present two approximation algorithms for the case where the points live in a d-dimensional Hamming space. Our algorithms guarantee query times that are sub-linear in n and only polynomial in the diversity parameter k, as well as the dimension d. For low values of k, our algorithms achieve sub-linear query times even if the number of points within distance r from a query q is linear in n. To the best of our knowledge, these are the first known algorithms of this type that offer provable guarantees.

News articles typically drive a lot of traffic in the form of comments posted by users on a news site. Such usergenerated content tends to carry additional information such as entities and sentiment. In general, when articles are recommended to users, only popularity (e.g., most shared and most commented), recency, and sometimes (manual) editors' picks (based on daily hot topics), are considered. We formalize a novel recommendation problem where the goal is to find the closest most diverse articles to the one the user is currently browsing. Our diversity measure incorporates entities and sentiment extracted from comments. Given the realtime nature of our recommendations, we explore the applicability of nearest neighbor algorithms to solve the problem. Our user study on real opinion articles from aljazeera.net and reuters.com validates the use of entities and sentiment extracted from articles and their comments to achieve news diversity when compared to content-based diversity. Finally, our performance experiments show the real-time feasibility of our solution.

We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to $1$. Specifically, we show that any algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes must use $\tilde{\Omega}(mn^{\delta})$ space in the regime of $m=O(n)$. Furthermore, we consider the problem in the geometric setting where the elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal $\tilde{O}(n)$ space to find a logarithmic approximation in $O(1/\delta)$ passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201

Abstract. We develop the first streaming algorithm and the first two-party communication protocol that uses a constant number of passes/rounds and sublinear space/communication for logarithmic approximation to the classic Set Cover problem. Specifically, for n elements and m sets, our algorithm/protocol achieves a space bound of O(m · n δ log 2 n log m) using O(4 1/δ ) passes/rounds while achieving an approximation factor of O(4 1/δ log n) in polynomial time (for δ = Ω(1/ log n)). If we allow the algorithm/protocol to spend exponential time per pass/round, we achieve an approximation factor of O(4 1/δ ). Our approach uses randomization, which we show is necessary: no deterministic constant approximation is possible (even given exponential time) using o(mn) space. These results are some of the first on streaming algorithms and efficient two-party communication protocols for approximation algorithms. Moreover, we show that our algorithm can be applied to multi-party communication model.