Abstract
The Optimal Alphabetic Binary Tree (OABT) problem is equivalent to the Optimal Binary Search Tree problem with the restriction that all data are in the leaves. The problem can be solved in O(n log n) time, while the best known lower bound is Ω(n).
The main result of this paper is an O(n√log n)-time algorithm for the integer OABT problem. As a side effect we obtain an O(n log k)-time algorithm for the general OABT problem, where k is a number at most as large as the number of local minima. This algorithm gives rise to linear time algorithms for some special cases. As a corollary, we obtain an O(nL)-time algorithm for the integer case of the optimal height-limited alphabetic tree problem, where L is the height limitation.
Preview
Unable to display preview. Download preview PDF.
References
K. Abrahamson, N. Dadoun, D. G. Kirkpatrick, T. Przytycka. A simple parallel tree contraction algorithm, Journal of Algorithms 10, (1989) pp. 287–302.
H. N. Gabow, J. L. Bentley, R. E. Tarjan, Scaling and related techniques for geometry problems, Proceedings of the 16th ACM Symposium on Theory of Computing (1984), pp. 135–143.
H. N. Gabow and R. E. Tarjan, A linear time algorithm for a special case of disjoint set union, J. Comput. System Sci. 30 (1985), 209–221.
A. M. Garsia and M. L. Wachs, A New algorithm for minimal binary search trees, SIAM Journal of Computing 6 (1977), pp. 622–642.
T. C. Hu. A new proof of the T-C algorithm, SIAM Journal of Applied Mathematics 25 (1973), pp. 83–94.
T. C. Hu and A. C. Tucker, Optimal computer search trees and variable length alphabetic codes, SIAM Journal of Applied Mathematics 21 (1971), pp. 514–532.
M. M. Klawe and B. Mumey, Upper and Lower Bounds on Constructing Alphabetic Binary Trees, Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms (1993), pp. 185–193.
L. L. Larmore, and D. S. Hirschberg, Length-limited coding, Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms (1990), pp. 310–318.
L. L. Larmore, and T. M. Przytycka, Parallel construction of trees with optimal weighted path length, Proceedings of the 3rd ACM Symposium on Parallel Algorithms and Architectures (1991), pp. 71–80.
L. L. Larmore, and T. M. Przytycka, A Fast algorithm for optimum height limited alphabetic binary trees, SIAM Journal on Computing to appear.
L. L. Larmore, and T. M. Przytycka, A parallel algorithm for almost optimal alphabetic trees, manuscript, submitted for publication.
L. L. Larmore, T. M. Przytycka, and W. Rytter, Parallel construction of optimal alphabetic trees, Proceedings of the 5th ACM Symposium on Parallel Algorithms and Architectures (1993).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Przytycka, T.M., Larmore, L.L. (1994). The Optimal Alphabetic Tree problem revisited. In: Abiteboul, S., Shamir, E. (eds) Automata, Languages and Programming. ICALP 1994. Lecture Notes in Computer Science, vol 820. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58201-0_73
Download citation
DOI: https://doi.org/10.1007/3-540-58201-0_73
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58201-4
Online ISBN: 978-3-540-48566-7
eBook Packages: Springer Book Archive