Abstract
Various methods have been used to represent a tree of n nodes in essentially the information-theoretic minimum space while supporting various navigational operations in constant time, but different representations usually support different operations. Our main contribution is a succinct representation of ordinal trees, based on that of Geary et al. (7), that supports all the navigational operations supported by various succinct tree representations while requiring only 2n + o(n) bits. It also supports efficient level-order traversal, a useful ordering previously supported only with a very limited set of operations (8).
Our second contribution expands on the notion of a single succinct representation supporting more than one traversal ordering, by showing that our method supports two other encoding schemes as abstract data types. In particular, it supports extracting a word (\(O(\lg n)\) bits) of the balanced parenthesis sequence (11) or depth first unary degree sequence (3;4) in O(f(n)) time, using at most n/f(n) + o(n) additional bits, for any f(n) in \(O(\lg n)\) and Ω(1).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barbay, J., Rao, S.: Succinct encoding for XPath location steps. Technical Report CS-2006-10, University of Waterloo, Ontario, Canada (2006)
Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Proc. 7th Latin American Theoretical Informatics Symp., pp. 88–94 (2000)
Benoit, D., Demaine, E.D., Munro, J.I., Raman, R., Raman, V., Rao, S.S.: Representing trees of higher degree. Algorithmica 43(4), 275–292 (2005)
Benoit, D., Demaine, E.D., Munro, J.I., Raman, V.: Representing trees of higher degree. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 169–180. Springer, Heidelberg (1999)
Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications to graph encoding and graph drawing. In: Proc. 12th ACM-SIAM Symp. Discrete Algorithms, pp. 506–515 (2001)
Clark, D.R., Munro, J.I.: Efficient suffix trees on secondary storage. In: Proc. 7th ACM-SIAM Symp. Discrete Algorithms, pp. 383–391 (1996)
Geary, R.F., Raman, R., Raman, V.: Succinct ordinal trees with level-ancestor queries. ACM Trans. Algorithms 2(4), 510–534 (2006)
Jacobson, G.: Space-efficient static trees and graphs. In: Proc. 30th IEEE Symp. Found. Comput. Sci., pp. 549–554 (1989)
Jansson, J., Sadakane, K., Sung, W.-K.: Ultra-succinct representation of ordered trees. In: Proc. 18th ACM-SIAM Symp. Discrete Algorithms, pp. 575–584 (2007)
Lu, H.-I., Yeh, C.-C.: Balanced parentheses strike back. Accepted to ACM Trans. Algorithms (2007)
Munro, J.I., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM J. Comput. 31(3), 762–776 (2001)
Munro, J.I., Raman, V., Rao, S.S.: Space efficient suffix trees. J. Algorithms 39(2), 205–222 (2001)
Munro, J.I., Rao, S.S.: Succinct representations of functions. In: Proc. 31st Int. Colloquium Automata, Languages and Programming, pp. 1006–1015 (2004)
Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding k-ary trees and multisets. In: Proc. 13th ACM-SIAM Symp. Discrete Algorithms, pp. 233–242 (2002)
Sadakane, K.: Succinct representations of lcp information and improvements in the compressed suffix arrays. In: Proc. 13th ACM-SIAM Symp. Discrete Algorithms, pp. 225–232 (2002)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
He, M., Munro, J.I., Rao, S.S. (2007). Succinct Ordinal Trees Based on Tree Covering. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_45
Download citation
DOI: https://doi.org/10.1007/978-3-540-73420-8_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73419-2
Online ISBN: 978-3-540-73420-8
eBook Packages: Computer ScienceComputer Science (R0)