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An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling

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Sujoy Bhore, Guangping Li, and Martin Nöllenburg. An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.19

Abstract

Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set (MIS) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types - (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time. This breaks the natural barrier of Ω(Δ) update time (where Δ is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes. For uniform rectangles, we first give an algorithm that maintains a 4-approximate Max-IS with O(1) update time. In a subsequent algorithm, we establish the trade-off between approximation quality 2(1+1/k) and update time O(k²log n), for k ∈ ℕ. We conclude with an algorithm that maintains a 2-approximate Max-IS for dynamic sets of unit-height and arbitrary-width rectangles with O(ω log n) update time, where ω is the maximum size of an independent set of rectangles stabbed by any horizontal line. We have implemented our algorithms and report the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances.

Subject Classification

ACM Subject Classification
• Theory of computation → Computational geometry
• Theory of computation → Dynamic graph algorithms
Keywords
• Independent Sets
• Dynamic Algorithms
• Rectangle Intersection Graphs
• Approximation Algorithms
• Experimental Evaluation

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References

1. Amir Abboud, Raghavendra Addanki, Fabrizio Grandoni, Debmalya Panigrahi, and Barna Saha. Dynamic set cover: improved algorithms and lower bounds. In Symposium on Theory of Computing (STOC'19), pages 114-125. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316376.
2. Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In Foundations of Computer Science (FOCS'13), pages 400-409. IEEE, 2013. URL: https://doi.org/10.1109/FOCS.2013.50.
3. 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. URL: https://doi.org/10.1016/S0925-7721(98)00028-5.
4. Sepehr Assadi, Krzysztof Onak, Baruch Schieber, and Shay Solomon. Fully dynamic maximal independent set with sublinear update time. In Symposium on Theory of Computing (STOC'18), pages 815-826, 2018. URL: https://doi.org/10.1145/3188745.3188922.
5. Sepehr Assadi, Krzysztof Onak, Baruch Schieber, and Shay Solomon. Fully dynamic maximal independent set with sublinear in n update time. In Symposium on Discrete Algorithms (SODA'19), pages 1919-1936. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.116.
6. Egon Balas and Chang Sung Yu. Finding a maximum clique in an arbitrary graph. SIAM Journal on Computing, 15(4):1054-1068, 1986. URL: https://doi.org/10.1137/0215075.
7. Ken Been, Martin Nöllenburg, Sheung-Hung Poon, and Alexander Wolff. Optimizing active ranges for consistent dynamic map labeling. Comput. Geom. Theory Appl., 43(3):312-328, 2010. URL: https://doi.org/10.1016/j.comgeo.2009.03.006.
8. Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Cliff Stein, and Madhu Sudan. Fully dynamic maximal independent set with polylogarithmic update time. In Foundations of Computer Science (FOCS'19), pages 382-405. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00032.
9. Aaron Bernstein, Sebastian Forster, and Monika Henzinger. A deamortization approach for dynamic spanner and dynamic maximal matching. In Symposium on Discrete Algorithms (SODA'19), pages 1899-1918. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.115.
10. Sayan Bhattacharya, Deeparnab Chakrabarty, and Monika Henzinger. Deterministic fully dynamic approximate vertex cover and fractional matching in o(1) amortized update time. In Integer Programming and Combinatorial Optimization (IPCO'17), volume 10328 of LNCS, pages 86-98. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59250-3_8.
11. Sayan Bhattacharya, Deeparnab Chakrabarty, Monika Henzinger, and Danupon Nanongkai. Dynamic algorithms for graph coloring. In Symposium on Discrete Algorithms (SODA'18), pages 1-20. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.1.
12. Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Symposium on Discrete Algorithms (SODA'09), pages 892-901. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.97.
13. 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.
14. Timothy M. Chan and Konstantinos Tsakalidis. Dynamic orthogonal range searching on the RAM, revisited. In Boris Aronov and Matthew J. Katz, editors, Computational Geometry (SoCG'17), volume 77 of LIPIcs, pages 28:1-28:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.28.
15. Shiri Chechik and Tianyi Zhang. Fully dynamic maximal independent set in expected poly-log update time. In Foundations of Computer Science (FOCS'19), pages 370-381. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00031.
16. Julia Chuzhoy and Alina Ene. On approximating maximum independent set of rectangles. In Foundations of Computer Science (FOCS'16), pages 820-829. IEEE, 2016. URL: https://doi.org/10.1109/FOCS.2016.92.
17. Graham Cormode, Jacques Dark, and Christian Konrad. Independent sets in vertex-arrival streams. In International Colloquium on Automata, Languages, and Programming (ICALP'19), volume 132 of LIPIcs, pages 45:1-45:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.45.
18. David Eppstein, Zvi Galil, and Giuseppe F. Italiano. Dynamic graph algorithms. In Mikhail J. Atallah, editor, Algorithms and Theory of Computation Handbook, chapter 8. CRC Press, 1999. URL: https://doi.org/10.1.1.43.8372.
19. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal on Computing, 34(6):1302-1323, 2005. URL: https://doi.org/10.1137/s0097539702402676.
20. Michael Formann and Frank Wagner. A packing problem with applications to lettering of maps. In Symposium on Computational Geometry (SoCG'91), pages 281-288. ACM, 1991. URL: https://doi.org/10.1145/109648.109680.
21. Robert J. Fowler, Mike Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett., 12(3):133-137, 1981. URL: https://doi.org/10.1016/0020-0190(81)90111-3.
22. Edith Gabriel. Spatio-temporal point pattern analysis and modeling. In Shashi Shekhar, Hui Xiong, and Xun Zhou, editors, Encyclopedia of GIS, pages 1-8. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-23519-6_1646-1.
23. Buddhima Gamlath, Michael Kapralov, Andreas Maggiori, Ola Svensson, and David Wajc. Online matching with general arrivals. In Foundations of Computer Science (FOCS'19), pages 26-37, 2019. URL: https://doi.org/10.1109/FOCS.2019.00011.
24. Alexander Gavruskin, Bakhadyr Khoussainov, Mikhail Kokho, and Jiamou Liu. Dynamic algorithms for monotonic interval scheduling problem. Theoretical Computer Science, 562:227-242, 2015. URL: https://doi.org/10.1016/j.tcs.2014.09.046.
25. Andreas Gemsa, Martin Nöllenburg, and Ignaz Rutter. Consistent labeling of rotating maps. J. Computational Geometry, 7(1):308-331, 2016. URL: https://doi.org/10.20382/jocg.v7i1a15.
26. U. I. Gupta, D. T. Lee, and Joseph Y.-T. Leung. Efficient algorithms for interval graphs and circular-arc graphs. Networks, 12(4):459-467, 1982. URL: https://doi.org/10.1002/net.3230120410.
27. Johan Håstad. Clique is hard to approximate within n^1-ε. Acta Mathematica, 182(1):105-142, 1999. URL: https://doi.org/10.1007/BF02392825.
28. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. In Sergio Cabello and Danny Z. Chen, editors, Symposium on Computational Geometry (SoCG 2020), volume 164 of LIPIcs, pages 51:1-51:14. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.51.
29. Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and vlsi. J. ACM, 32(1):130-136, 1985. URL: https://doi.org/10.1145/2455.214106.
30. John E Hopcroft and Richard M Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225-231, 1973. URL: https://doi.org/10.1137/0202019.
31. Richard M. Karp. Reducibility among combinatorial problems. In R. E. Miller, J. W. Thatcher, and J. D. Bohlinger, editors, Complexity of Computer Computations, pages 85-103, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
32. Fabian Klute, Guangping Li, Raphael Löffler, Martin Nöllenburg, and Manuela Schmidt. Exploring semi-automatic map labeling. In Advances in Geographic Information Systems (SIGSPATIAL'19), pages 13-22. ACM, 2019. URL: https://doi.org/10.1145/3347146.3359359.
33. Nathan Linial. Distributive graph algorithms global solutions from local data. In Foundations of Computer Science (SFCS'87), pages 331-335. IEEE, 1987. URL: https://doi.org/10.1109/SFCS.1987.20.
34. Kurt Mehlhorn and Stefan Näher. Dynamic fractional cascading. Algorithmica, 5(1-4):215-241, 1990. URL: https://doi.org/10.1007/BF01840386.
35. Kurt Mehlhorn and Stefan Näher. The LEDA Platform of Combinatorial and Geometric Computing. Cambridge University Press, 1999. URL: https://doi.org/10.1145/204865.204889.
36. Huy N Nguyen and Krzysztof Onak. Constant-time approximation algorithms via local improvements. In Foundations of Computer Science (FOCS'08), pages 327-336. IEEE, 2008. URL: https://doi.org/10.1109/FOCS.2008.81.
37. Panos M Pardalos and Jue Xue. The maximum clique problem. Journal of Global Optimization, 4(3):301-328, 1994. URL: https://doi.org/10.1007/BF01098364.
38. René van Bevern, Matthias Mnich, Rolf Niedermeier, and Mathias Weller. Interval scheduling and colorful independent sets. Journal of Scheduling, 18(5):449-469, 2015. URL: https://doi.org/10.1007/s10951-014-0398-5.
39. Marc J. van Kreveld, Tycho Strijk, and Alexander Wolff. Point set labeling with sliding labels. In Ravi Janardan, editor, Proceedings of the Fourteenth Annual Symposium on Computational Geometry, Minneapolis, Minnesota, USA, June 7-10, 1998, pages 337-346. ACM, 1998. URL: https://doi.org/10.1145/276884.276922.
40. Frank Wagner and Alexander Wolff. A practical map labeling algorithm. Comput. Geom. Theory Appl., 7:387-404, 1997. URL: https://doi.org/10.1016/S0925-7721(96)00007-7.