Towards Tight Bounds for the Streaming Set Cover Problem
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(mn[superscript δ]) space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demai...
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| Format: | Article |
| Language: | en_US |
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Association for Computing Machinery
2018
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| Online Access: | http://hdl.handle.net/1721.1/113829 https://orcid.org/0000-0002-7983-9524 https://orcid.org/0000-0001-5049-7594 https://orcid.org/0000-0001-5004-8991 |
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| author | Har-Peled, Sariel Indyk, Piotr Mahabadi, Sepideh Vakilian, Ali |
| author2 | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory |
| author_facet | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Har-Peled, Sariel Indyk, Piotr Mahabadi, Sepideh Vakilian, Ali |
| author_sort | Har-Peled, Sariel |
| collection | MIT |
| description | 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(mn[superscript δ]) 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(mn[superscript δ]) space in the regime of m=O(n). Furthermore, we consider the problem in the geometric setting where the elements are points in R[superscript 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 ~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. |
| first_indexed | 2024-09-23T17:12:45Z |
| format | Article |
| id | mit-1721.1/113829 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:12:45Z |
| publishDate | 2018 |
| publisher | Association for Computing Machinery |
| record_format | dspace |
| spelling | mit-1721.1/1138292022-10-03T11:09:55Z Towards Tight Bounds for the Streaming Set Cover Problem Har-Peled, Sariel Indyk, Piotr Mahabadi, Sepideh Vakilian, Ali Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Indyk, Piotr Vakilian, Ali Mahabadi, Sepideh 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(mn[superscript δ]) 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(mn[superscript δ]) space in the regime of m=O(n). Furthermore, we consider the problem in the geometric setting where the elements are points in R[superscript 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 ~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. 2018-02-20T15:27:33Z 2018-02-20T15:27:33Z 2016-07 Article http://purl.org/eprint/type/ConferencePaper 978-1-4503-4191-2 http://hdl.handle.net/1721.1/113829 Har-Peled, et al. “Towards Tight Bounds for the Streaming Set Cover Problem.” Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems - PODS ’16 (2016), 26 June - 1 July, 2016, San Francisco, California, Association of Computing Machinery, 2016, pp. 371-383. https://orcid.org/0000-0002-7983-9524 https://orcid.org/0000-0001-5049-7594 https://orcid.org/0000-0001-5004-8991 en_US http://dx.doi.org/10.1145/2902251.2902287 Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems - PODS '16 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Association for Computing Machinery MIT Web Domain |
| spellingShingle | Har-Peled, Sariel Indyk, Piotr Mahabadi, Sepideh Vakilian, Ali Towards Tight Bounds for the Streaming Set Cover Problem |
| title | Towards Tight Bounds for the Streaming Set Cover Problem |
| title_full | Towards Tight Bounds for the Streaming Set Cover Problem |
| title_fullStr | Towards Tight Bounds for the Streaming Set Cover Problem |
| title_full_unstemmed | Towards Tight Bounds for the Streaming Set Cover Problem |
| title_short | Towards Tight Bounds for the Streaming Set Cover Problem |
| title_sort | towards tight bounds for the streaming set cover problem |
| url | http://hdl.handle.net/1721.1/113829 https://orcid.org/0000-0002-7983-9524 https://orcid.org/0000-0001-5049-7594 https://orcid.org/0000-0001-5004-8991 |
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