Representing Trees of Higher Degree
This paper focuses on space efficient representations of trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (rooted trees where each node has k positions each of which may have a reference to a child) and ordinal trees (the children of each node are simply ordered). Our representations use a number of bits within a lower order term of the information theoretic lower bound. For cardinal trees the structure supports finding the parent, child i or subtree size of a given node. For ordinal trees we support the operations of finding the degree, parent, ith child and subtree size. These operations provide a mapping from the n nodes of the tree onto the integers [1, n] and all are performed in constant time, except finding child i in cardinal trees. For k-ary cardinal trees, this operation takes O(lg lg k) time for the worst relationship between k and n, and constant time if k is much less than n.
KeywordsBinary Tree Suffix Tree Prefix Code Auxiliary Structure Balance String
Unable to display preview. Download preview PDF.
- 1.David Benoit. Compact Tree Representations. MMath thesis, U. Waterloo, 1998.Google Scholar
- 2.Andrej Brodnik and J. Ian Munro. Membership in constant time and almost minimum space. SIAM J. Computing, to appear.Google Scholar
- 3.David Clark. Compact Pat Trees. PhD thesis, U. Waterloo, 1996.Google Scholar
- 4.David R. Clark and J. Ian Munro. Efficient suffix trees on secondary storage. In Proc. ACM-SIAM Symposium on Discrete Algorithms, 383–391, 1996.Google Scholar
- 10.G. H. Gonnet, R.A. Baeza-Yates, and T. Snider. New indicies for text: PAT trees and PAT arrays. In Information Retrieval: Data Structures & Algorithms, 66–82, Prentice Hall, 1992.Google Scholar
- 11.Guy Jacobson. Space-efficient static trees and graphs. In Proc. 30th Annual Symposium on Foundations of Computer Science, 549–554, 1989.Google Scholar
- 12.Guy Jacobson. Succinct Static Data Structures. PhD thesis, CMU, 1989.Google Scholar
- 15.J. Ian Munro and Venkatesh Raman. Succinct representation of balanced parentheses, static trees and planar graphs. In Proc. 38th Annual Symposium on Foundations of Computer Science, 118–126, 1997.Google Scholar