A Uniform Approach Towards Succinct Representation of Trees
We propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decomposition of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct data structures and has been shown fruitful in succinct representation of ordinal trees [7,10] and dynamic binary trees [16,21]. We take an approach that simplifies the existing representation of ordinal trees while allowing the full set of navigational operations. The approach applied to cardinal (i.e. k-ary) trees yields a space-optimal succinct representation allowing cardinal-type operations (e.g. determining child labeled i) as well as the full set of ordinal-type operations (e.g. reporting the number of siblings to the left of a node). Existing space-optimal succinct representations had not been able to support both types of operations [2,19].
We demonstrate how the approach can be applied to obtain a space-optimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropy-based succinct representations. We show that our approach matches the degree-distribution entropy suggested by Jansson et al. . We discuss that our approach can be made adaptive to various other entropy measures.
KeywordsLookup Table Free Tree Dummy Node Recursive Decomposition Succinct Representation
Unable to display preview. Download preview PDF.
- 1.Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multi-labeled trees. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 680–689. ACM-SIAM, New York (2007)Google Scholar
- 4.Clark, D.R.: Compact pat trees. PhD thesis, Waterloo, Ontario. Canada (1998)Google Scholar
- 5.Clark, D.R., Munro, J.I.: Efficient suffix trees on secondary storage (extended abstract). In: SODA, pp. 383–391 (1996)Google Scholar
- 9.Harary, F., Palmer, E.M.: Graphical Enuemration. Academic Press, New York (1973)Google Scholar
- 11.Jacobson, G.: Space-efficient static trees and graphs. In: Foundations of Computer Science. 30th Annual Symposium on (30 October-1 November 1989), pp. 549–554 (1989)Google Scholar
- 12.Jacobson, G.J. Succinct static data structures. PhD thesis, Pittsburgh, PA, USA (1988)Google Scholar
- 13.Jansson, J., Sadakane, K., Sung, W.-K.: Ultra-succinct representation of ordered trees. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 575–584. SIAM, Philadelphia (2007)Google Scholar
- 14.Knuth, D.E.: The Art of Computer Programming, 3rd edn. vol. 1. Addison-Wesley, Reading (1997)Google Scholar
- 15.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
- 16.Munro, J.I., Raman, V., Storm, A.J.: Representing dynamic binary trees succinctly. In: SODA, pp. 529–536 (2001)Google Scholar
- 17.Odlyzko, A.M.: Some new methods and results in tree enumeration, (May 04, 1984)Google Scholar
- 19.Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding k-ary trees and multisets. In: SODA, pp. 233–242 (2002)Google Scholar
- 20.Rote, G.: Binary trees having a given number of nodes with 0,1, and 2 children. Séminaire Lotharingien de Combinatoire 38 (1997)Google Scholar
- 21.Storm, A.J. Representing dynamic binary trees succinctly. Master’s thesis, School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada (2000)Google Scholar