The Optimal Alphabetic Tree problem revisited
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.
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- 1.K. Abrahamson, N. Dadoun, D. G. Kirkpatrick, T. Przytycka. A simple parallel tree contraction algorithm, Journal of Algorithms 10, (1989) pp. 287–302.Google Scholar
- 2.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.Google Scholar
- 3.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.Google Scholar
- 4.A. M. Garsia and M. L. Wachs, A New algorithm for minimal binary search trees, SIAM Journal of Computing 6 (1977), pp. 622–642.Google Scholar
- 5.T. C. Hu. A new proof of the T-C algorithm, SIAM Journal of Applied Mathematics 25 (1973), pp. 83–94.Google Scholar
- 6.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.Google Scholar
- 7.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.Google Scholar
- 8.L. L. Larmore, and D. S. Hirschberg, Length-limited coding, Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms (1990), pp. 310–318.Google Scholar
- 9.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.Google Scholar
- 10.L. L. Larmore, and T. M. Przytycka, A Fast algorithm for optimum height limited alphabetic binary trees, SIAM Journal on Computing to appear.Google Scholar
- 11.L. L. Larmore, and T. M. Przytycka, A parallel algorithm for almost optimal alphabetic trees, manuscript, submitted for publication.Google Scholar
- 12.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).Google Scholar