Succinct Representations of Separable Graphs

  • Guy E. Blelloch
  • Arash Farzan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6129)


We consider the problem of highly space-efficient representation of separable graphs while supporting queries in constant time in the RAM with logarithmic word size. In particular, we show constant-time support for adjacency, degree and neighborhood queries. For any monotone class of separable graphs, the storage requirement of the representation is optimal to within lower order terms.

Separable graphs are those that admit a O(n c )-separator theorem where c < 1. Many graphs that arise in practice are indeed separable. For instance, graphs with a bounded genus are separable. In particular, planar graphs (genus 0) are separable and our scheme gives the first succinct representation of planar graphs with a storage requirement that matches the information-theory minimum to within lower order terms with constant time support for the queries.

We, furthers, show that we can also modify the scheme to succinctly represent the combinatorial planar embedding of planar graphs (and hence encode planar maps).


Planar Graph Storage Requirement Lower Order Term Graph Label Neighborhood Query 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Gulli, A., Signorini, A.: The indexable web is more than 11.5 billion pages. In: WWW 2005: Special interest tracks and posters of the 14th international conference on World Wide Web, pp. 902–903. ACM, New York (2005)CrossRefGoogle Scholar
  2. 2.
    Claude, F., Navarro, G.: A fast and compact web graph representation. In: Ziviani, N., Baeza-Yates, R. (eds.) SPIRE 2007. LNCS, vol. 4726, pp. 118–129. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web. Comput. Netw. 33(1-6), 309–320 (2000)CrossRefGoogle Scholar
  4. 4.
    Adler, M., Mitzenmacher, M.: Towards compressing web graphs. In: DCC 2001: Proceedings of the Data Compression Conference, Washington, DC, USA, p. 203. IEEE Computer Society, Los Alamitos (2001)CrossRefGoogle Scholar
  5. 5.
    Suel, T., Yuan, J.: Compressing the graph structure of the web. In: DCC 2001: Data Compression Conference, p. 213. IEEE, Los Alamitos (2001)CrossRefGoogle Scholar
  6. 6.
    Munro, J.I.: Succinct data structures. Electronic Notes in Theoretical Computer Science 91, 3 (2004)CrossRefGoogle Scholar
  7. 7.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36(2), 177–189 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Miller, G.L., Teng, S.H., Thurston, W., Vavasis, S.A.: Separators for sphere-packings and nearest neighbor graphs. J. ACM 44(1), 1–29 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Leighton, T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: FOCS 1988: Foundations of Computer Science, pp. 422–431. IEEE, Los Alamitos (1988)Google Scholar
  11. 11.
    Blandford, D.K., Blelloch, G.E., Kash, I.A.: Compact representations of separable graphs. In: SODA: ACM-SIAM Symposium on Discrete Algorithms (2003)Google Scholar
  12. 12.
    Raman, R., Raman, V., Satti, S.R.: Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans. Algorithms 3(4), 43 (2007)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Farzan, A., Munro, J.I.: Succinct representations of arbitrary graphs. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 393–404. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Lu, H.I.: Linear-time compression of bounded-genus graphs into information-theoretically optimal number of bits. In: SODA 2002: Proceedings of ACM-SIAM symposium on Discrete algorithms, pp. 223–224 (2002)Google Scholar
  15. 15.
    Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM J. Discrete Math. 5(4), 596–603 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Turán, G.: On the succinct representation of graphs. Discrete Applied Mathematics 8, 289–294 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Keeler, W.: Short encodings of planar graphs and maps. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 58 (1995)Google Scholar
  18. 18.
    He, X., Kao, M.Y., Lu, H.I.: A fast general methodology for information-theoretically optimal encodings of graphs. SIAM Journal on Computing 30(3), 838–846 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jacobson, G.: Space-efficient static trees and graphs. In: 30th Annual Symposium on Foundations of Computer Science, 1989, October 30 – November 1, pp. 549–554 (1989)Google Scholar
  20. 20.
    Munro, J.I., Raman, V.: Succinct representation of balanced parentheses, static trees and planar graphs. In: IEEE Symposium on Foundations of Computer Science, pp. 118–126 (1997)Google Scholar
  21. 21.
    Chuang, R.C.N., Garg, A., He, X., Kao, M.Y., Lu, H.I.: Compact encodings of planar graphs via canonical orderings and multiple parentheses. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 118–129. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  22. 22.
    Chiang, Y.T., Lin, C.C., Lu, H.I.: Orderly spanning trees with applications to graph encoding and graph drawing. In: SODA 2001: ACM-SIAM symposium on Discrete algorithms, pp. 506–515 (2001)Google Scholar
  23. 23.
    Devillers, L.C.A.O., Schaeffer, G.: Succinct representations of planar maps. Theor. Comput. Sci. 408(2-3), 174–187 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM Journal on Numerical Analysis 16, 346–358 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Liskovets, V.A., Walsh, T.R.: Ten steps to counting planar graphs. Congressus Numerantium 60, 269–277 (1987)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Guy E. Blelloch
    • 1
  • Arash Farzan
    • 2
  1. 1.Computer Science DepartmentCarnegie Mellon University 
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

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