Document

# Distance Queries over Dynamic Interval Graphs

## File

LIPIcs.ISAAC.2023.18.pdf
• Filesize: 0.82 MB
• 19 pages

## Cite As

Jingbang Chen, Meng He, J. Ian Munro, Richard Peng, Kaiyu Wu, and Daniel J. Zhang. Distance Queries over Dynamic Interval Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.18

## Abstract

We design the first dynamic distance oracles for interval graphs, which are intersection graphs of a set of intervals on the real line, and for proper interval graphs, which are intersection graphs of a set of intervals in which no interval is properly contained in another. For proper interval graphs, we design a linear space data structure which supports distance queries (computing the distance between two query vertices) and vertex insertion or deletion in O(lg n) worst-case time, where n is the number of vertices currently in G. Under incremental (insertion only) or decremental (deletion only) settings, we design linear space data structures that support distance queries in O(lg n) worst-case time and vertex insertion or deletion in O(lg n) amortized time, where n is the maximum number of vertices in the graph. Under fully dynamic settings, we design a data structure that represents an interval graph G in O(n) words of space to support distance queries in O(n lg n/S(n)) worst-case time and vertex insertion or deletion in O(S(n)+lg n) worst-case time, where n is the number of vertices currently in G and S(n) is an arbitrary function that satisfies S(n) = Ω(1) and S(n) = O(n). This implies an O(n)-word solution with O(√{nlg n})-time support for both distance queries and updates. All four data structures can answer shortest path queries by reporting the vertices in the shortest path between two query vertices in O(lg n) worst-case time per vertex. We also study the hardness of supporting distance queries under updates over an intersection graph of 3D axis-aligned line segments, which generalizes our problem to 3D. Finally, we solve the problem of computing the diameter of a dynamic connected interval graph.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Data structures design and analysis
• Information systems → Data structures
##### Keywords
• interval graph
• proper interval graph
• intersection graph
• geometric intersection graph
• distance oracle
• distance query
• shortest path query
• dynamic graph

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Ittai Abraham and Cyril Gavoille. On approximate distance labels and routing schemes with affine stretch. In David Peleg, editor, Distributed Computing - 25th International Symposium, DISC 2011, Rome, Italy, September 20-22, 2011. Proceedings, volume 6950 of Lecture Notes in Computer Science, pages 404-415. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-24100-0_39.
2. Hüseyin Acan, Sankardeep Chakraborty, Seungbum Jo, and Srinivasa Rao Satti. Succinct encodings for families of interval graphs. Algorithmica, 83(3):776-794, 2021. URL: https://doi.org/10.1007/s00453-020-00710-w.
3. Josh Alman, Timothy Chu, Aaron Schild, and Zhao Song. Algorithms and hardness for linear algebra on geometric graphs. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 541-552. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00057.
4. Stephen Alstrup, Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Maintaining information in fully-dynamic trees with top trees. ACM Transactions on Algorithms, 1, December 2003. URL: https://doi.org/10.1145/1103963.1103966.
5. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. J. ACM, 48(5):1069-1090, 2001. URL: https://doi.org/10.1145/502102.502107.
6. Michael A. Bender and Martin Farach-Colton. The level ancestor problem simplified. Theor. Comput. Sci., 321(1):5-12, 2004. URL: https://doi.org/10.1016/j.tcs.2003.05.002.
7. Karl Bringmann, Sándor Kisfaludi-Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian. Towards sub-quadratic diameter computation in geometric intersection graphs. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.21.
8. Timothy M. Chan. Finding triangles and other small subgraphs in geometric intersection graphs. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 1777-1805. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch68.
9. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in geometric intersection graphs. J. Comput. Geom., 10(1):27-41, 2019. URL: https://doi.org/10.20382/jocg.v10i1a2.
10. Timothy M. Chan and Dimitrios Skrepetos. Approximate shortest paths and distance oracles in weighted unit-disk graphs. J. Comput. Geom., 10(2):3-20, 2019. URL: https://doi.org/10.20382/jocg.v10i2a2.
11. Danny Z. Chen, D. T. Lee, R. Sridhar, and Chandra N. Sekharan. Solving the all-pair shortest path query problem on interval and circular-arc graphs. Networks, 31(4):249-258, 1998. URL: https://doi.org/10.1002/(SICI)1097-0037(199807)31:4<249::AID-NET5>3.0.CO;2-D.
12. Jonathan B. Conroy and Csaba D. Tóth. Hop-spanners for geometric intersection graphs. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 30:1-30:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.30.
13. Christophe Crespelle. Fully dynamic representations of interval graphs. Theoretical Computer Science, 759:14-49, 2019. URL: https://doi.org/10.1016/j.tcs.2019.01.007.
14. Camil Demetrescu and Giuseppe F. Italiano. A new approach to dynamic all pairs shortest paths. J. ACM, 51(6):968-992, 2004. URL: https://doi.org/10.1145/1039488.1039492.
15. Hicham El-Zein, Moshe Lewenstein, J Ian Munro, Venkatesh Raman, and Timothy M Chan. On the succinct representation of equivalence classes. Algorithmica, 78:1020-1040, 2017.
16. Jie Gao and Li Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM J. Comput., 35(1):151-169, 2005. URL: https://doi.org/10.1137/S0097539703436357.
17. Cyril Gavoille and Christophe Paul. Optimal distance labeling for interval graphs and related graph families. SIAM J. Discret. Math., 22(3):1239-1258, 2008. URL: https://doi.org/10.1137/050635006.
18. Meng He, J. Ian Munro, Yakov Nekrich, Sebastian Wild, and Kaiyu Wu. Distance oracles for interval graphs via breadth-first rank/select in succinct trees. In Yixin Cao, Siu-Wing Cheng, and Minming Li, editors, 31st International Symposium on Algorithms and Computation, ISAAC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 181 of LIPIcs, pages 25:1-25:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2020.25.
19. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, STOC '15, pages 21-30, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746609.
20. Hung Le and Christian Wulff-Nilsen. Optimal approximate distance oracle for planar graphs. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 363-374. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00044.
21. Yaowei Long and Seth Pettie. Planar distance oracles with better time-space tradeoffs. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10-13, 2021, pages 2517-2537. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.149.
22. J. Ian Munro and Corwin Sinnamon. Time and space efficient representations of distributive lattices. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 550-567. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.36.
23. J. Ian Munro and Kaiyu Wu. Succinct data structures for chordal graphs. In Wen-Lian Hsu, Der-Tsai Lee, and Chung-Shou Liao, editors, 29th International Symposium on Algorithms and Computation, ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan, volume 123 of LIPIcs, pages 67:1-67:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
24. Mihai Pǎtraşcu and Liam Roditty. Distance oracles beyond the thorup-zwick bound. SIAM J. Comput., 43(1):300-311, 2014. URL: https://doi.org/10.1137/11084128X.
25. Gaurav Singh, N. S. Narayanaswamy, and G. Ramakrishna. Approximate distance oracle in o(n²) time and o(n) space for chordal graphs. In M. Sohel Rahman and Etsuji Tomita, editors, WALCOM: Algorithms and Computation - 9th International Workshop, WALCOM 2015, Dhaka, Bangladesh, February 26-28, 2015. Proceedings, volume 8973 of Lecture Notes in Computer Science, pages 89-100. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-15612-5_9.
26. Mikkel Thorup. Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In Torben Hagerup and Jyrki Katajainen, editors, Algorithm Theory - SWAT 2004, 9th Scandinavian Workshop on Algorithm Theory, Humlebaek, Denmark, July 8-10, 2004, Proceedings, volume 3111 of Lecture Notes in Computer Science, pages 384-396. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27810-8_33.
27. Antti Ukkonen, Carlos Castillo, Debora Donato, and Aristides Gionis. Searching the wikipedia with contextual information. In James G. Shanahan, Sihem Amer-Yahia, Ioana Manolescu, Yi Zhang, David A. Evans, Aleksander Kolcz, Key-Sun Choi, and Abdur Chowdhury, editors, Proceedings of the 17th ACM Conference on Information and Knowledge Management, CIKM 2008, Napa Valley, California, USA, October 26-30, 2008, pages 1351-1352. ACM, 2008. URL: https://doi.org/10.1145/1458082.1458274.
28. Monique V. Vieira, Bruno M. Fonseca, Rodrigo Damazio, Paulo Braz Golgher, Davi de Castro Reis, and Berthier A. Ribeiro-Neto. Efficient search ranking in social networks. In Mário J. Silva, Alberto H. F. Laender, Ricardo A. Baeza-Yates, Deborah L. McGuinness, Bjørn Olstad, Øystein Haug Olsen, and André O. Falcão, editors, Proceedings of the Sixteenth ACM Conference on Information and Knowledge Management, CIKM 2007, Lisbon, Portugal, November 6-10, 2007, pages 563-572. ACM, 2007. URL: https://doi.org/10.1145/1321440.1321520.
29. Harry Wiener. Structural determination of paraffin boiling points. J. Am. Chem. Soc., 69(1):17-20, 1947.
30. Peisen Zhang, Eric A. Schon, Stuart G. Fischer, Eftihia Cayanis, Janie Weiss, Susan Kistler, and Philip E. Bourne. An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. Comput. Appl. Biosci., 10(3):309-317, 1994. URL: https://doi.org/10.1093/bioinformatics/10.3.309.