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Implicit Representation of Relations

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Abstract

We consider implicit representation of an arbitrary family of relations on finite sets. We derive upper and lower bounds for the general cases and for a number of restricted subfamilies, in particular for sparse and symmetric relations, and for relations first-order definable from families for which labeling schemes are already known. Our work extends existing work on implicit representation of graphs in two ways: (i) the known upper and lower bounds for many standard families of graphs are special cases of the results we derive; (ii) we allow families of relations to relate elements on both distinct sets and on multiple copies of the same set, and for different relations in the same family to have different arities, and to be defined on distinct or overlapping sets. The present paper is the first to study bounds on the size of labeling schemes for relations (including graphs) defined from existing relations using basic operations such as first-order logic. The techniques used to prove new results in this setting may be of independent interest.

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Notes

  1. It is interesting to note that [33] did not consider a scheme worthy of being called a “labeling scheme” unless the decoder worked in polynomial time, but this demand seems to occur very rarely in the modern literature. For trade-offs between decoding time and label size for graph labeling schemes, see [12, 13].

  2. The literature on variations of these families and history of successive improvements of labeling schemes with smaller label sizes is so voluminous that we only cite a few examples.

  3. The quantity \( \left| {\bigcup _{i=1}^m V_i}\right| \) is sometimes called the base of a relation [29]; we refrain from this to avoid confusion with the notion of “base relation” in database theory. Also note that the sets \(V_1,\ldots ,V_m\) may have non-empty pairwise intersections and that some elements might thus occur in several distinct sets among \(V_1,\ldots ,V_m\)–but the definition \(\text {sz}({V_1 \times \cdots \times V_m})\) uses the union without repetitions.

  4. We whimsically call this the “sloppy comma padding” as the padded string avoids the string 11 at any even index, but can contain 11 as a substring at odd indices (unlike traditional comma codes).

  5. This bound corresponds to the original \(n/2 + O(\log n)\) bound for simple undirected graphs by Moon [40]; later refinements have improved the bound for undirected graphs to \(n/2 + O(1)\) [4].

  6. We may have \(\vert R \vert \le f(n) \ll n^k\) due to f-sparsity, but the particular elements of R cannot necessarily be enumerated fast without further assumptions.

  7. For other graph families, e.g., the set of subgraphs of hypercubes, the best-known arboricity-based labeling schemes are not quite optimal, but asymptotically optimal schemes can be found by other methods, see e.g. [26]

  8. Observe that the degree of the polynomial may depend on the size of the formula—see also the notion of height in Definition 23 below.

References

  1. Alstrup, S., Dahlgaard, S., Knudsen, M.B.T.: Optimal induced universal graphs and adjacency labeling for trees. J. ACM. 64(4), 27:1–27:22 (2017)

  2. Alstrup, S., Dahlgaard, S., Knudsen, M.B.T., Porat, E.: Sublinear distance labeling. In: 24th Annual european symposium on algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pp. 5:1–5:15 (2016)

  3. Alstrup, S., Gørtz, I.L., Halvorsen, E.B., Porat, E.: Distance labeling schemes for trees. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pp. 132:1–132:16 (2016)

  4. Alstrup, S., Kaplan, H., Thorup, M., Zwick, U.: Adjacency labeling schemes and induced-universal graphs. SIAM J. Discrete Math. 33(1), 116–137 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andrews, G.E.: The theory of partitions. Cambridge University Press, Encyclopedia of mathematics and its applications (1984)

    Book  MATH  Google Scholar 

  6. Beeri, C., Bernstein, P.A.: Computational problems related to the design of normal form relational schemas. ACM Transactions on Database Systems (TODS) 4(1), 30–59 (1979)

    Article  Google Scholar 

  7. Bækgaard, L., Mark, L.: Incremental computation of nested relational query expressions. ACM Trans. Database Syst. 20, 111–148 (1995)

    Article  Google Scholar 

  8. Breuer, M.: Coding the vertexes of a graph. IEEE Trans. Inf. Theor. 12, 148–153 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  9. Breuer, M.A., Folkman, J.: An unexpected result in coding the vertices of a graph. J. Math. Anal. Appl. 20(3), 583–600 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brodal, G.S., Fagerberg, R.: Dynamic representation of sparse graphs. In: Algorithms and data structures, 6th International Workshop, WADS ’99, Vancouver, British Columbia, Canada, August 11-14, 1999, Proceedings, pp. 342–351 (1999)

  11. Cao, B., Badia, A.: Sql query optimization through nested relational algebra. ACM Trans. Database Syst. 32, 18 (2007)

    Article  Google Scholar 

  12. Chandoo, M.: On the implicit graph conjecture. In: P. Faliszewski, A. Muscholl, R. Niedermeier (eds.) 41st International symposium on mathematical foundations of computer science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, LIPIcs, vol. 58, pp. 23:1–23:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

  13. Chandoo, M.: Computational complexity aspects of implicit graph representations. Ph.D. thesis, Leibniz Universität Hannover (2018)

  14. Chepoi, V., Dragan, F.F., Estellon, B., Habib, M., Vaxès, Y., Xiang, Y.: Additive spanners and distance and routing labeling schemes for hyperbolic graphs. Algorithmica 62(3–4), 713–732 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chepoi, V., Dragan, F.F., Vaxès, Y.: Distance and routing labeling schemes for non-positively curved plane graphs. J. Algorithms 61(2), 60–88 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chepoi, V., Labourel, A., Ratel, S.: On density of subgraphs of cartesian products. J. Graph Theor. 93(1), 64–87 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chin, L.H., Tarski, A.: Remarks on projective algebras. In: Bulletin of the american mathematical society, vol. 54, pp. 80–81 (1948)

  18. Chu, E., Beckmann, J.L., Naughton, J.F.: The case for a wide-table approach to manage sparse relational data sets. In: C.Y. Chan, B.C. Ooi, A. Zhou (eds.) Proceedings of the ACM SIGMOD international conference on management of data, Beijing, China, June 12-14, 2007, pp. 821–832. ACM (2007)

  19. Codd, E.F.: Relational completeness of data base sublanguages. In: Database systems, pp. 65–98. Prentice-Hall (1972)

  20. Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Labeling schemes for tree representation. Algorithmica 53(1), 1–15 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Courcelle, B.: The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

  22. Courcelle, B., Gavoille, C., Kanté, M.M.: Compact labelings for efficient first-order model-checking. J. Comb. Optim. 21(1), 19–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dechter, R.: Decomposing a relation into a tree of binary relations. J. Comput. Syst. Sci. 41(1), 2–24 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dey, D., Storey, V.C., Barron, T.M.: Improving database design through the analysis of relationships. ACM Trans. Database Syst. 24(4), 453–486 (1999)

    Article  Google Scholar 

  25. Enderton, H.B.: A mathematical introduction to logic, 2 edn. Academic Press (2001)

  26. Esperet, L., Harms, N., Zamaraev, V.: Optimal adjacency labels for subgraphs of cartesian products (2022)

  27. Fagin, R.: Generalized first-order spectra, and polynomial. time recognizable sets. SIAM-AMS Proc. 7, 43–73 (1974)

  28. Flajolet, P., Sedgewick, R.: Analytic combinatorics. Cambridge University Press (2009)

  29. Fraïssé, R.: Theory of relations, studies in logic and the foundations of mathematics, vol. 145. Elsevier (2000)

  30. Gavoille, C., Labourel, A.: Shorter implicit representation for planar graphs and bounded treewidth graphs. In: L. Arge, M. Hoffmann, E. Welzl (eds.) Algorithms - ESA 2007, 15th Annual european symposium, Eilat, Israel, October 8-10, 2007, proceedings, lecture notes in computer science, vol. 4698, pp. 582–593. Springer (2007)

  31. Gavoille, C., Peleg, D.: Compact and localized distributed data structures. Distrib. Comput. 16(2), 111–120 (2003)

    Article  MATH  Google Scholar 

  32. Gavoille, C., Peleg, D., Pérennes, S., Raz, R.: Distance labeling in graphs. J. Algorithms 53(1), 85–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM J. Discrete Math. 5(4), 596–603 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kim, W.: On optimizing an sql-like nested query. ACM Trans. Database Syst. 7(3), 443–469 (1982)

    Article  MATH  Google Scholar 

  35. Korman, A., Peleg, D.: Labeling schemes for weighted dynamic trees. Inf. Comput. 205(12), 1721–1740 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Korman, A., Peleg, D.: Compact separator decompositions in dynamic trees and applications to labeling schemes. Distrib Comput. 21(2), 141–161 (2008)

    Article  MATH  Google Scholar 

  37. Korman, A., Peleg, D., Rodeh, Y.: Labeling schemes for dynamic tree networks. Theor. Comput. Syst. 37(1), 49–75 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, M., Vitányi, P.: An introduction to kolmogorov complexity and its applications, 3 edn. Springer-Verlag (2008)

  39. van Lint, P.H.: Introduction to coding theory, 3rd edition edn. Springer-Verlag (1999)

  40. Moon, J.: On minimal \(n\)-universal graphs. Proc. Glasg. Math. Assoc. 7(1), 32–33 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  41. Müller, J.: Local structure in graph classes. Ph.D. thesis, Georgia Tech (1988)

  42. Öszu, M.T., Valduriez, P.: Principles of distributed database systems, 3rd edition edn. Springer-Verlag (2011)

  43. Petersen, C., Rotbart, N., Simonsen, J.G., Wulff-Nilsen, C.: Near optimal adjacency labeling schemes for power-law graphs. In: I. Chatzigiannakis, M. Mitzenmacher, Y. Rabani, D. Sangiorgi (eds.) 43rd International colloquium on automata, languages, and programming, ICALP 2016, July 11-15, 2016, Rome, Italy, LIPIcs, vol. 55, pp. 133:1–133:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

  44. Ross, K.A., Stoyanovich, J.: Symmetric relations and cardinality-bounded multisets in database systems. In: Proceedings of the 38th International conference on very large data bases - vol. 30, VLDB ’04, pp. 912–923. VLDB Endowment (2004)

  45. Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. J. ACM 51(6), 993–1024 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Thorup, M., Zwick, U.: Compact routing schemes. In: Proceedings of the 13th Annual ACM Symposium on parallel algorithms and architectures, SPAA 2001, Heraklion, Crete Island, Greece, July 4-6, 2001, pp. 1–10 (2001)

  47. Weimann, O., Peleg, D.: A note on exact distance labeling. Inf. Process. Lett. 111(14), 671–673 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Computer Science, University of Copenhagen (DIKU), Copenhagen, Denmark

    Vladan Glončák, Jarl Emil Erla Munkstrup & Jakob Grue Simonsen

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Glončák, V., Munkstrup, J.E.E. & Grue Simonsen, J. Implicit Representation of Relations. Theory Comput Syst 67, 1156–1196 (2023). https://doi.org/10.1007/s00224-023-10141-z

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