Document Open Access Logo

Implicit Representations via the Polynomial Method

Authors Jean Cardinal , Micha Sharir

PDF

File

Thumbnail PDF
PDF
LIPIcs.WG.2026.10.pdf
  • Filesize: 1.05 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Semialgebraic graphs
  • Geometric intersection graphs
  • Visibility graphs
  • Adjacency labelings
  • Polynomial partitioning

Metrics

Abstract

Semialgebraic graphs are graphs whose vertices are points in {ℝ}^d, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs.
That is, we show that for any family of semialgebraic graphs, given a graph on n vertices in this family, we can assign a label consisting of O(n^{1-2/(d+1) + {ε}}) bits to each vertex (where {ε} > 0 can be made arbitrarily small and the constant of proportionality depends on {ε} and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of O(n^{1/3 + {ε}}) bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of O(n^{1-1/d}log n) for d-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-d polynomial. Our labeling scheme is efficient in the sense that not only adjacency between two vertices can be decided in time linear in the size of their labels, but the labels can be computed in subquadratic time on a real RAM from the input points and the semialgebraic adjacency predicate, using recent polynomial partitioning algorithms.
We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size O(log n). We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size O(log³ n).

Cite As Get BibTex

Jean Cardinal and Micha Sharir. Implicit Representations via the Polynomial Method. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.10

Author Details

Jean Cardinal
  • Université libre de Bruxelles (ULB), Belgium
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Cardinal, Jean: Work by Jean Cardinal was supported by the Fonds de la Recherche Scientifique-FNRS under Grant n° T003325F.
  • Sharir, Micha: Work by Micha Sharir was partially supported by ISF grant 495/23.

Acknowledgements

The first author would like to thank Noga Alon for a discussion on his results in [Alon, 2024], and Lena Yuditsky for sharing observations on semilinear families. The authors are also grateful to Pankaj Agarwal, Sariel Har-Peled, and the WG reviewers for insightful comments and references.

References

  1. Mikkel Abrahamsen, Stephen Alstrup, Jacob Holm, Mathias Bæk Tejs Knudsen, and Morten Stöckel. Near-optimal induced universal graphs for bounded degree graphs. In 44th International Colloquium on Automata, Languages, and Programming, volume 80, pages Art. No. 128, 14, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.128.
  2. Mikkel Abrahamsen, Stephen Alstrup, Jacob Holm, Mathias Bæk Tejs Knudsen, and Morten Stöckel. Near-optimal induced universal graphs for cycles and paths. Discrete Appl. Math., 282:1-13, 2020. URL: https://doi.org/10.1016/j.dam.2019.10.030.
  3. David Adjiashvili and Noy Rotbart. Labeling schemes for bounded degree graphs. In Automata, languages, and programming. Part II, volume 8573 of Lecture Notes in Comput. Sci., pages 375-386. Springer, Heidelberg, 2014. URL: https://doi.org/10.1007/978-3-662-43951-7_32.
  4. Pankaj K. Agarwal. Simplex range searching and its variants: a review. In A Journey Through Discrete Mathematics. A tribute to Jirǐ Matoušek, pages 1-30. Springer, Cham, 2017. URL: https://doi.org/10.1007/978-3-319-44479-6_1.
  5. Pankaj K. Agarwal, Noga Alon, Boris Aronov, and Subhash Suri. Can visibility graphs be represented compactly? Discrete Comput. Geom., 12(3):347-365, 1994. URL: https://doi.org/10.1007/BF02574385.
  6. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection queries for flat semi-algebraic objects in three dimensions and related problems. ACM Transactions on Algorithms, 21(3):25:1-25:59, 2025. URL: https://doi.org/10.1145/3721290.
  7. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. Efficient algorithm for generalized polynomial partitioning and its applications. SIAM J. Comput., 50(2):760-787, 2021. URL: https://doi.org/10.1137/19M1268550.
  8. Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Semi-algebraic off-line range searching and biclique partitions in the plane. In 40th International Symposium on Computational Geometry, pages 4:1-4:15, 2024. URL: https://doi.org/10.4230/LIPIcs.SOCG.2024.4.
  9. Pankaj K. Agarwal, Jiří Matoušek, and Micha Sharir. On range searching with semialgebraic sets. II. SIAM J. Comput., 42(6):2039-2062, 2013. URL: https://doi.org/10.1137/120890855.
  10. Pankaj K. Agarwal and Kasturi R. Varadarajan. Efficient algorithms for approximating polygonal chains. Discrete Comput. Geom., 23(2):273-291, 2000. URL: https://doi.org/10.1007/PL00009500.
  11. Noga Alon. Asymptotically optimal induced universal graphs. Geom. Funct. Anal., 27(1):1-32, 2017. URL: https://doi.org/10.1007/s00039-017-0396-9.
  12. Noga Alon. Implicit representation of sparse hereditary families. Discrete Comput. Geom., 72(2):476-482, 2024. URL: https://doi.org/10.1007/s00454-023-00521-0.
  13. Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank and the Vapnik-Chervonenkis dimension. Mat. Sb., 208(12):4-41, 2017. URL: https://doi.org/10.4213/sm8780.
  14. Noga Alon, János Pach, Rom Pinchasi, Radoš Radoičić, and Micha Sharir. Crossing patterns of semi-algebraic sets. J. Combin. Theory Ser. A, 111(2):310-326, 2005. URL: https://doi.org/10.1016/j.jcta.2004.12.008.
  15. Stephen Alstrup, Søren Dahlgaard, and Mathias Bæk Tejs Knudsen. Optimal induced universal graphs and adjacency labeling for trees. J. ACM, 64(4):Art. 27, 22, 2017. URL: https://doi.org/10.1145/3088513.
  16. Stephen Alstrup, Haim Kaplan, Mikkel Thorup, and Uri Zwick. Adjacency labeling schemes and induced-universal graphs. SIAM J. Discrete Math., 33(1):116-137, 2019. URL: https://doi.org/10.1137/16M1105967.
  17. Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir. Subquadratic algorithms for some 3SUM-hard geometric problems in the algebraic decision-tree model. Comput. Geom., 109:Paper No. 101945, 21, 2023. URL: https://doi.org/10.1016/j.comgeo.2022.101945.
  18. Aistis Atminas, Andrew Collins, Vadim Lozin, and Victor Zamaraev. Implicit representations and factorial properties of graphs. Discrete Math., 338(2):164-179, 2015. URL: https://doi.org/10.1016/j.disc.2014.09.008.
  19. Avah Banerjee. An adjacency labeling scheme based on a decomposition of trees into caterpillars. In Combinatorial algorithms, volume 13270 of Lecture Notes in Comput. Sci., pages 114-127. Springer, Cham, 2022. URL: https://doi.org/10.1007/978-3-031-06678-8_9.
  20. Abdul Basit, Artem Chernikov, Sergei Starchenko, Terence Tao, and Chieu-Minh Tran. Zarankiewicz’s problem for semilinear hypergraphs. Forum Math. Sigma, 9:Paper No. e59, 23, 2021. URL: https://doi.org/10.1017/fms.2021.52.
  21. Saugata Basu, Richard Pollack, and Marie-Françoise Roy. On the number of cells defined by a family of polynomials on a variety. Mathematika, 43(1):120-126, 1996. URL: https://doi.org/10.1112/S0025579300011621.
  22. Peter Braß and Christian Knauer. On counting point-hyperplane incidences. Computational Geometry, 25(1-2):13-20, 2003. URL: https://doi.org/10.1016/S0925-7721(02)00127-X.
  23. Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, and Christian Knauer. Geometric matching and bottleneck problems. In 40th International Symposium on Computational Geometry, volume 293, pages 31:1-31:15, 2024. URL: https://doi.org/10.4230/LIPIcs.SOCG.2024.31.
  24. Jean Cardinal and Micha Sharir. Improved algebraic degeneracy testing. Discrete Comput. Geom., 74(1):177-195, 2025. URL: https://doi.org/10.1007/s00454-024-00673-7.
  25. Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025), volume 351 of Leibniz International Proceedings in Informatics (LIPIcs), pages 67:1-67:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.ESA.2025.67.
  26. Timothy M. Chan. Dynamic subgraph connectivity with geometric applications. SIAM Journal on Computing, 36(3):681-694, 2006. URL: https://doi.org/10.1137/S009753970343912X.
  27. Timothy M. Chan. Optimal partition trees. Discrete Comput. Geom., 47(4):661-690, 2012. URL: https://doi.org/10.1007/s00454-012-9410-z.
  28. Maurice Chandoo. Logical labeling schemes. Discrete Math., 346(10):Paper No. 113565, 21, 2023. URL: https://doi.org/10.1016/j.disc.2023.113565.
  29. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom., 9(2):145-158, 1993. URL: https://doi.org/10.1007/BF02189314.
  30. Bernard Chazelle and Emo Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete Comput. Geom., 4(5):467-489, 1989. URL: https://doi.org/10.1007/BF02187743.
  31. Fan R. K. Chung, Paul Erdős, and Joel Spencer. On the decomposition of graphs into complete bipartite subgraphs. In Studies in pure mathematics: To the Memory of Paul Turán, pages 95-101. Birkhäuser, Basel, 1983. URL: https://doi.org/10.1007/978-3-0348-5438-2_10.
  32. David Conlon, Jacob Fox, János Pach, Benny Sudakov, and Andrew Suk. Ramsey-type results for semi-algebraic relations. Trans. Amer. Math. Soc., 366(9):5043-5065, 2014. URL: https://doi.org/10.1090/S0002-9947-2014-06179-5.
  33. B. Courcelle and R. Vanicat. Query efficient implementation of graphs of bounded clique-width. Discrete Appl. Math., 131(1):129-150, 2003. The Second International Colloquium "Journées de l'Informatique Messine" (Metz, 2000). URL: https://doi.org/10.1016/S0166-218X(02)00421-3.
  34. James Davies, Tomasz Krawczyk, Rose McCarty, and Bartosz Walczak. Coloring polygon visibility graphs and their generalizations. J. Combin. Theory Ser. B, 161:268-300, 2023. URL: https://doi.org/10.1016/j.jctb.2023.02.010.
  35. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Visibility graphs. In Computational Geometry. Algorithms and Applications, Third Edition, chapter 15. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  36. Thao Do. Representation complexities of semialgebraic graphs. SIAM J. Discrete Math., 33(4):1864-1877, 2019. URL: https://doi.org/10.1137/18M1221606.
  37. Vida Dujmović, Louis Esperet, Cyril Gavoille, Gwenaël Joret, Piotr Micek, and Pat Morin. Adjacency labelling for planar graphs (and beyond). J. ACM, 68(6):Art. 42, 33, 2021. URL: https://doi.org/10.1145/3477542.
  38. Paul Erdős and László Pyber. Covering a graph by complete bipartite graphs. Discrete Math., 170(1-3):249-251, 1997. URL: https://doi.org/10.1016/S0012-365X(96)00124-0.
  39. Jeff Erickson. New lower bounds for Hopcroft’s problem. Discrete Comput. Geom., 16(4):389-418, 1996. URL: https://doi.org/10.1007/BF02712875.
  40. Tomás Feder and Rajeev Motwani. Clique partitions, graph compression and speeding-up algorithms. J. Comput. System Sci., 51(2):261-272, 1995. URL: https://doi.org/10.1006/jcss.1995.1065.
  41. Matthew Fitch. Implicit representation conjecture for semi-algebraic graphs. Discrete Appl. Math., 259:53-62, 2019. URL: https://doi.org/10.1016/j.dam.2018.11.030.
  42. Jacob Fox, Mikhail Gromov, Vincent Lafforgue, Assaf Naor, and János Pach. Overlap properties of geometric expanders. J. Reine Angew. Math., 671:49-83, 2012. URL: https://doi.org/10.1515/crelle.2011.157.
  43. Jacob Fox, János Pach, Adam Sheffer, Andrew Suk, and Joshua Zahl. A semi-algebraic version of Zarankiewicz’s problem. J. Eur. Math. Soc. (JEMS), 19(6):1785-1810, 2017. URL: https://doi.org/10.4171/JEMS/705.
  44. Jacob Fox, János Pach, and Andrew Suk. A polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing. SIAM J. Comput., 45(6):2199-2223, 2016. URL: https://doi.org/10.1137/15M1007355.
  45. Jacob Fox, János Pach, and Andrew Suk. Ramsey-Turán numbers for semi-algebraic graphs. Electron. J. Combin., 25(4):Paper No. 4.61, 5, 2018. URL: https://doi.org/10.37236/7988.
  46. Subir K. Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511543340.
  47. Jacob E. Goodman, Richard Pollack, and Bernd Sturmfels. Coordinate representation of order types requires exponential storage. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 405-410, 1989. URL: https://doi.org/10.1145/73007.73046.
  48. Larry Guth. Polynomial partitioning for a set of varieties. Math. Proc. Cambridge Philos. Soc., 159(3):459-469, 2015. URL: https://doi.org/10.1017/S0305004115000468.
  49. Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Ann. of Math. (2), 181(1):155-190, 2015. URL: https://doi.org/10.4007/annals.2015.181.1.2.
  50. Hamed Hatami and Pooya Hatami. The implicit graph conjecture is false. In 63rd Annual IEEE Symposium on Foundations of Computer Science, pages 1134-1137. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00109.
  51. Saarik Kalia, Micha Sharir, Noam Solomon, and Ben Yang. Generalizations of the Szemerédi-Trotter theorem. Discrete Comput. Geom., 55(3):571-593, 2016. URL: https://doi.org/10.1007/s00454-016-9759-5.
  52. Sampath Kannan, Moni Naor, and Steven Rudich. Implicit representation of graphs. SIAM J. Discrete Math., 5(4):596-603, 1992. URL: https://doi.org/10.1137/0405049.
  53. Haim Kaplan, Jiří Matoušek, Zuzana Safernová, and Micha Sharir. Unit distances in three dimensions. Combin. Probab. Comput., 21(4):597-610, 2012. URL: https://doi.org/10.1017/S0963548312000144.
  54. Haim Kaplan, Jiří Matoušek, and Micha Sharir. Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique. Discrete Comput. Geom., 48(3):499-517, 2012. URL: https://doi.org/10.1007/s00454-012-9443-3.
  55. Matthew J. Katz and Micha Sharir. An expander-based approach to geometric optimization. SIAM J. Comput., 26(5):1384-1408, 1997. URL: https://doi.org/10.1137/S0097539794268649.
  56. Jan Kratochvíl and Jiří Matoušek. Intersection graphs of segments. J. Combin. Theory Ser. B, 62(2):289-315, 1994. URL: https://doi.org/10.1006/jctb.1994.1071.
  57. Jiří Matoušek. Efficient partition trees. Discrete Comput. Geom., 8(3):315-334, 1992. URL: https://doi.org/10.1007/BF02293051.
  58. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete Comput. Geom., 10(2):157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
  59. Jiří Matoušek and Zuzana Patáková. Multilevel polynomial partitions and simplified range searching. Discrete Comput. Geom., 54(1):22-41, 2015. URL: https://doi.org/10.1007/s00454-015-9701-2.
  60. Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. J. Combin. Theory Ser. B, 103(1):114-143, 2013. URL: https://doi.org/10.1016/j.jctb.2012.09.004.
  61. John Milnor. On the Betti numbers of real varieties. Proc. Amer. Math. Soc., 15:275-280, 1964. URL: https://doi.org/10.2307/2034050.
  62. Nicolai E. Mnëv. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and Geometry - Rohlin Seminar, volume 1346 of Lecture Notes in Math., pages 527-543. Springer, Berlin, 1988. URL: https://doi.org/10.1007/BFb0082792.
  63. John H. Muller. Local structure in graph classes. PhD thesis, Georgia Institute of Technology, USA, 1988. Google Scholar
  64. Joseph O'Rourke. Visibility. In Handbook of Discrete and Computational Geometry, 3rd Edition, chapter 33. Chapman and Hall/CRC, 2017. URL: https://doi.org/10.1201/9781315119601.
  65. Natan Rubin. An efficient regularity lemma for semi-algebraic hypergraphs. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3611-3636. SIAM, Philadelphia, PA, 2025. URL: https://doi.org/10.1137/1.9781611978322.120.
  66. Lisa Sauermann. On the speed of algebraically defined graph classes. Adv. Math., 380:Paper No. 107593, 55, 2021. URL: https://doi.org/10.1016/j.aim.2021.107593.
  67. Micha Sharir and Noam Solomon. Incidences between points and lines in ℝ⁴. Discrete Comput. Geom., 57(3):702-756, 2017. URL: https://doi.org/10.1007/s00454-016-9822-2.
  68. Peter W. Shor. Stretchability of pseudolines is NP-hard. In Applied Geometry and Discrete Mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 531-554. Amer. Math. Soc., Providence, RI, 1991. URL: https://doi.org/10.1090/dimacs/004/41.
  69. József Solymosi and Terence Tao. An incidence theorem in higher dimensions. Discrete Comput. Geom., 48(2):255-280, 2012. URL: https://doi.org/10.1007/s00454-012-9420-x.
  70. Jeremy P. Spinrad. Efficient graph representations, volume 19 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 2003. URL: https://doi.org/10.1090/fim/019.
  71. René Thom. Sur l'homologie des variétés algébriques réelles. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 255-265. Princeton Univ. Press, Princeton, NJ, 1965. URL: https://doi.org/10.1515/9781400874842-016.
  72. István Tomon. Ramsey properties of semilinear graphs. Israel J. Math., 254(1):113-139, 2023. URL: https://doi.org/10.1007/s11856-022-2390-7.
  73. Hugh E. Warren. Lower bounds for approximation by nonlinear manifolds. Trans. Amer. Math. Soc., 133:167-178, 1968. URL: https://doi.org/10.2307/1994937.
  74. Emo Welzl. Partition trees for triangle counting and other range searching problems. In Proceedings of the 4th Annual Symposium on Computational Geometry, pages 23-33, 1988. URL: https://doi.org/10.1145/73393.73397.
  75. Joshua Zahl. An improved bound on the number of point-surface incidences in three dimensions. Contrib. Discrete Math., 8(1):100-121, 2013. URL: https://doi.org/10.11575/CDM.V8I1.62120.
  76. Günter M. Ziegler. Nonrational configurations, polytopes, and surfaces. Math. Intelligencer, 30(3):36-42, 2008. URL: https://doi.org/10.1007/BF02985377.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact