TY - GEN

T1 - Towards tight bounds for the streaming set cover problem

AU - Har-Peled, Sariel

AU - Indyk, Piotr

AU - Mahabadi, Sepideh

AU - Vakilian, Ali

N1 - Publisher Copyright:
© 2016 ACM.

PY - 2016/6/15

Y1 - 2016/6/15

N2 - 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 Õ(mnδ) space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [11] 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 (1/2δ - 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 Õ(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.

AB - 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 Õ(mnδ) space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [11] 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 (1/2δ - 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 Õ(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.

UR - http://www.scopus.com/inward/record.url?scp=84978476749&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978476749&partnerID=8YFLogxK

U2 - 10.1145/2902251.2902287

DO - 10.1145/2902251.2902287

M3 - Conference contribution

AN - SCOPUS:84978476749

T3 - Proceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems

SP - 371

EP - 383

BT - PODS 2016 - Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

PB - Association for Computing Machinery

T2 - 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2016

Y2 - 26 June 2016 through 1 July 2016

ER -