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Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

Authors Ainesh Bakshi, Nadiia Chepurko, David P. Woodruff

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Author Details

Ainesh Bakshi
  • Carnegie Mellon University, Pittsburgh, PA, USA
Nadiia Chepurko
  • MIT, Cambridge, MA, USA
David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

Ainesh Bakshi and David Woodruff acknowledge support in part from NSF No. CCF-1815840.

Acknowledgements

Part of this work was done while Ainesh Bakshi and David Woodruff were visiting the Simons Institute for the Theorem of Computing.

Cite AsGet BibTex

Ainesh Bakshi, Nadiia Chepurko, and David P. Woodruff. Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 64:1-64:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.64

Abstract

We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let α be the cardinality of the largest independent set. Our goal is to estimate α in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating β, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating α and β. For unit-length intervals, we obtain a (2+ε)-approximation to α and β in poly(log(n)/ε) space. We also show a matching lower bound. Combined with the 3/2-approximation for insertion-only streams by Cabello and Perez-Lanterno [Cabello and Pérez-Lantero, 2017], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a (8√3/π)-approximation to α and β in poly(log(n)/ε) space, which is closely related to the hexagonal circle packing constant. Finally, we provide algorithms for estimating α for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating α and β, where the parameter is the ratio of maximum to minimum interval length.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Streaming models
Keywords
  • Weighted Maximum Independent Set
  • Geometric Graphs
  • Turnstile Streams

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References

  1. Pankaj K Agarwal, Marc Van Kreveld, and Subhash Suri. Label placement by maximum independent set in rectangles. Computational Geometry, 11(3-4):209-218, 1998. Google Scholar
  2. Yuqing Ai, Wei Hu, Yi Li, and David P. Woodruff. New characterizations in turnstile streams with applications. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 20:1-20:22, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.20.
  3. Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 20-29. ACM, 1996. Google Scholar
  4. Baruch Awerbuch, Yair Bartal, Amos Fiat, and Adi Rosén. Competitive non-preemptive call control. In SODA, volume 94, pages 312-320, 1994. Google Scholar
  5. Yossi Azar and Oren Gilon. Buffer management for packets with processing times. In Algorithms-ESA 2015, pages 47-58. Springer, 2015. Google Scholar
  6. Unnar Th Bachmann, Magnús M Halldórsson, and Hadas Shachnai. Online scheduling intervals and t-intervals. Full version, 2010. Google Scholar
  7. Ziv Bar-Yossef, TS Jayram, Ravi Kumar, D Sivakumar, and Luca Trevisan. Counting distinct elements in a data stream. In International Workshop on Randomization and Approximation Techniques in Computer Science, pages 1-10. Springer, 2002. Google Scholar
  8. Radu Berinde, Graham Cormode, Piotr Indyk, and Martin J. Strauss. Space-optimal heavy hitters with strong error bounds. In Proceedings of the Twenty-Eigth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2009, June 19 - July 1, 2009, Providence, Rhode Island, USA, pages 157-166, 2009. URL: https://doi.org/10.1145/1559795.1559819.
  9. Jarosław Błasiok. Optimal streaming and tracking distinct elements with high probability. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2432-2448. SIAM, 2018. Google Scholar
  10. Sergio Cabello and Pablo Pérez-Lantero. Interval selection in the streaming model. In Algorithms and Data Structures - 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings, pages 127-139, 2015. URL: https://doi.org/10.1007/978-3-319-21840-3_11.
  11. Sergio Cabello and Pablo Pérez-Lantero. Interval selection in the streaming model. Theoretical Computer Science, 702:77-96, 2017. Google Scholar
  12. Ran Canetti and Sandy Irani. Bounding the power of preemption in randomized scheduling. SIAM Journal on Computing, 27(4):993-1015, 1998. Google Scholar
  13. Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms, pages 892-901. Society for Industrial and Applied Mathematics, 2009. Google Scholar
  14. Timothy M Chan. Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms, 46(2):178-189, 2003. Google Scholar
  15. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: https://doi.org/10.1007/s00454-012-9417-5.
  16. Hai-Chau Chang and Lih-Chung Wang. A simple proof of thue’s theorem on circle packing. arXiv preprint, 2010. URL: http://arxiv.org/abs/1009.4322.
  17. Moses Charikar, Kevin C. Chen, and Martin Farach-Colton. Finding frequent items in data streams. In Automata, Languages and Programming, 29th International Colloquium, ICALP 2002, Malaga, Spain, July 8-13, 2002, Proceedings, pages 693-703, 2002. URL: https://doi.org/10.1007/3-540-45465-9_59.
  18. Julia Chuzhoy and Alina Ene. On approximating maximum independent set of rectangles. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 820-829. IEEE, 2016. Google Scholar
  19. Brent N Clark, Charles J Colbourn, and David S Johnson. Unit disk graphs. Discrete mathematics, 86(1-3):165-177, 1990. Google Scholar
  20. Peter Clifford and Ioana Cosma. A simple sketching algorithm for entropy estimation over streaming data. In Artificial Intelligence and Statistics, pages 196-206, 2013. Google Scholar
  21. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2009. Google Scholar
  22. Graham Cormode, Jacques Dark, and Christian Konrad. Independent set size approximation in graph streams. CoRR, abs/1702.08299, 2017. URL: http://arxiv.org/abs/1702.08299.
  23. Graham Cormode, Jacques Dark, and Christian Konrad. Independent sets in vertex-arrival streams. arXiv preprint, 2018. URL: http://arxiv.org/abs/1807.08331.
  24. Yuval Emek, Magnús M. Halldórsson, and Adi Rosén. Space-constrained interval selection. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, pages 302-313, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_26.
  25. Thomas Erlebach and Jirı Fiala. The maximum independent set problem in unit disk graphs, 2003. Google Scholar
  26. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal on Computing, 34(6):1302-1323, 2005. Google Scholar
  27. Cristian Estan, George Varghese, and Mike Fisk. Bitmap algorithms for counting active flows on high speed links. In Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement, pages 153-166. ACM, 2003. Google Scholar
  28. Philippe Flajolet, Éric Fusy, Olivier Gandouet, and Frédéric Meunier. Hyperloglog: the analysis of a near-optimal cardinality estimation algorithm. In Discrete Mathematics and Theoretical Computer Science, pages 137-156. Discrete Mathematics and Theoretical Computer Science, 2007. Google Scholar
  29. Philippe Flajolet and G Nigel Martin. Probabilistic counting algorithms for data base applications. Journal of computer and system sciences, 31(2):182-209, 1985. Google Scholar
  30. Robert J Fowler, Michael S Paterson, and Steven L Tanimoto. Optimal packing and covering in the plane are np-complete. Information processing letters, 12(3):133-137, 1981. Google Scholar
  31. Fănică Gavril. Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM Journal on Computing, 1(2):180-187, 1972. Google Scholar
  32. Phillip B Gibbons and Srikanta Tirthapura. Estimating simple functions on the union of data streams. In Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures, pages 281-291. ACM, 2001. Google Scholar
  33. William K Hale. Frequency assignment: Theory and applications. Proceedings of the IEEE, 68(12):1497-1514, 1980. Google Scholar
  34. Johan Håstad. Clique is hard to approximate within n^1-epsilon. In 37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996, pages 627-636, 1996. URL: https://doi.org/10.1109/SFCS.1996.548522.
  35. Petr Hliněnỳ. Contact graphs of line segments are np-complete. Discrete Mathematics, 235(1-3):95-106, 2001. Google Scholar
  36. Dorit S Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and vlsi. Journal of the ACM (JACM), 32(1):130-136, 1985. Google Scholar
  37. Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. Journal of algorithms, 4(4):310-323, 1983. Google Scholar
  38. Hossein Jowhari, Mert Sağlam, and Gábor Tardos. Tight bounds for lp samplers, finding duplicates in streams, and related problems. In Proceedings of the thirtieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pages 49-58. ACM, 2011. Google Scholar
  39. Daniel M Kane, Jelani Nelson, and David P Woodruff. On the exact space complexity of sketching and streaming small norms. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 1161-1178. Society for Industrial and Applied Mathematics, 2010. Google Scholar
  40. Daniel M. Kane, Jelani Nelson, and David P. Woodruff. An optimal algorithm for the distinct elements problem. In Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2010, June 6-11, 2010, Indianapolis, Indiana, USA, pages 41-52, 2010. URL: https://doi.org/10.1145/1807085.1807094.
  41. Jon Kleinberg and Eva Tardos. Algorithm design. Pearson Education India, 2006. Google Scholar
  42. Antoon WJ Kolen, Jan Karel Lenstra, Christos H Papadimitriou, and Frits CR Spieksma. Interval scheduling: A survey. Naval Research Logistics (NRL), 54(5):530-543, 2007. Google Scholar
  43. Christian Konrad. Maximum matching in turnstile streams. In Algorithms-ESA 2015, pages 840-852. Springer, 2015. Google Scholar
  44. Ping Li and Cun-Hui Zhang. A new algorithm for compressed counting with applications in shannon entropy estimation in dynamic data. In Proceedings of the 24th Annual Conference on Learning Theory, pages 477-496, 2011. Google Scholar
  45. Ewa Malesinska. Graph theoretical models for frequency assignment problems. Shaker, 1997. Google Scholar
  46. Andrew McGregor. Graph stream algorithms: a survey. ACM SIGMOD Record, 43(1):9-20, 2014. Google Scholar
  47. Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. J. Comput. Syst. Sci., 57(1):37-49, 1998. URL: https://doi.org/10.1006/jcss.1998.1577.
  48. Kevin Mote. Fast point-feature label placement for dynamic visualizations. Information Visualization, 6(4):249-260, 2007. Google Scholar
  49. Xiaoming Sun, David P Woodruff, Guang Yang, and Jialin Zhang. Querying a matrix through matrix-vector products. arXiv preprint, 2019. URL: http://arxiv.org/abs/1906.05736.
  50. Gerhard J Woeginger. On-line scheduling of jobs with fixed start and end times. Theoretical Computer Science, 130(1):5-16, 1994. Google Scholar
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