Sepideh Mahabadi

Diverse near neighbor problem

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

On Streaming and Communication Complexity of the Set Cover Problem

S

Towards Tight Bounds for the Streaming Set Cover Problem

(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

Fractional Set Cover in the Streaming Model

q

Nonlinear dimension reduction via outer Bi-Lipschitz extensions

is linear in

Composable core-sets for diversity and coverage maximization

n

Real-time recommendation of diverse related articles

. To the best of our knowledge, these are the first known algorithms of this type that offer provable guarantees.

Approximate nearest line search in high dimensions

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 log2 n logm) using O(41/δ ) passes/rounds while achieving an approximation factor of O(41/δ logn) in polynomial time (for δ = Ω(1/logn)). If we allow the algorithm/protocol to spend exponential time per pass/round, we achieve an approximation factor of O(41/δ ). Our approach uses randomization, which we show is necessary: no deterministic constant approximation is possible (even given exponential time) using o(m n) 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.

Proximity in the age of distraction: robust approximate nearest neighbor search

We consider the classic Set Cover problem in the data stream model. For n elements and m sets (m ≥ n) we give a O(1/δ)-pass algorithm with a strongly sub-linear ~O(mnδ) space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [10] 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 2δ}-1) passes must use ~Ω(mnδ) space in the regime of m=O(n). Furthermore, we consider the problem in the geometric setting where the elements are points in R2 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 ~O(n) space to find a logarithmic approximation in O(1/δ) 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., Ω(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.

Set cover in sub-linear time

We study the Fractional Set Cover problem in the streaming model. That is, we consider the relaxation of the set cover problem over a universe of n elements and a collection of m sets, where each set can be picked fractionally, with a value in [0,1]. We present a randomized (1+a)-approximation algorithm that makes p passes over the data, and uses O(polylog(m,n,1/a) (mn^(O(1/(pa)))+n)) memory space. The algorithm works in both the set arrival and the edge arrival models. To the best of our knowledge, this is the first streaming result for the fractional set cover problem. We obtain our results by employing the multiplicative weights update framework in the streaming settings.