LIPIcs.AofA.2024.2.pdf
- Filesize: 0.68 MB
- 12 pages
Document
Enumeration and Succinct Encoding of AVL Trees
Authors
Jeremy Chizewer 
,
Stephen Melczer ,
J. Ian Munro ,
Ava Pun
-
Part of:
Volume:
35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-07-18
File
Document Identifiers
Author Details
Acknowledgements
The authors thank Andrew Odlyzko for discussions on the asymptotic behaviour of AVL trees and the growth constant α, and thank Sebastian Wild for alerting us to relevant references.
Cite AsGet BibTex
Jeremy Chizewer, Stephen Melczer, J. Ian Munro, and Ava Pun. Enumeration and Succinct Encoding of AVL Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.2
https://doi.org/10.4230/LIPIcs.AofA.2024.2
Abstract
We use a novel decomposition to create succinct data structures - supporting a wide range of operations on static trees in constant time - for a variety of tree classes, extending results of Munro, Nicholson, Benkner, and Wild. Motivated by the class of AVL trees, we further derive asymptotics for the information-theoretic lower bound on the number of bits needed to store tree classes whose generating functions satisfy certain functional equations. In particular, we prove that AVL trees require approximately 0.938 bits per node to encode.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Enumeration
Keywords
- AVL trees
- analytic combinatorics
- succinct data structures
- enumeration
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
-
G. Adelson-Velsky and E. Landis. An algorithm for the organization of information. In Proceedings of the USSR Academy of Sciences (in Russian), 1962.
- Clark, David. Compact PAT trees. PhD thesis, University of Waterloo, 1997. URL: http://hdl.handle.net/10012/64.
- Arash Farzan and J. Ian Munro. A uniform paradigm to succinctly encode various families of trees. Algorithmica, 68(1):16-40, January 2014. URL: https://doi.org/10.1007/s00453-012-9664-0.
-
Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge, 2009.
-
Meng He, J. Ian Munro, and S. Srinivasa Rao. Succinct ordinal trees based on tree covering. In Automata, Languages and Programming, pages 509-520. 2007.
-
G. Jacobson. Space-efficient static trees and graphs. In 30th Annual Symposium on Foundations of Computer Science, pages 549-554, 1989.
-
Stephen Melczer. An Invitation to Analytic Combinatorics: From One to Several Variables. Texts and Monographs in Symbolic Computation. Springer International Publishing, 2021.
- J. Ian Munro. Tables. In Vijay Chandru and V. Vinay, editors, Foundations of Software Technology and Theoretical Computer Science, 16th Conference, Hyderabad, India, December 18-20, 1996, Proceedings, volume 1180 of Lecture Notes in Computer Science, pages 37-42. Springer, 1996. URL: https://doi.org/10.1007/3-540-62034-6_35.
-
J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild. Hypersuccinct trees – new universal tree source codes for optimal compressed tree data structures and range minima. In 29th Annual European Symposium on Algorithm, pages 70:1-70:18, 2021.
- J. Ian Munro, Venkatesh Raman, and Adam J. Storm. Representing dynamic binary trees succinctly. In S. Rao Kosaraju, editor, Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA, pages 529-536. ACM/SIAM, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365526.
- J. Ian Munro and S. Srinivasa Rao. Succinct representation of data structures. In Dinesh P. Mehta and Sartaj Sahni, editors, Handbook of Data Structures and Applications. Chapman and Hall/CRC, 2004. URL: https://doi.org/10.1201/9781420035179.ch37.
-
Gonzalo Navarro. Compact Data Structures: A Practical Approach. Cambridge University Press, 2016.
-
A. M. Odlyzko. Some new methods and results in tree enumeration. In Proceedings of the thirteenth Manitoba conference on numerical mathematics and computing (Winnipeg, Man., 1983), volume 42, pages 27-52, 1984.