The construction of optimum multiway search trees for n keys, n key weights and n+1 gap weights, is investigated. A new general optimality principle is presented, which can be "tuned" for specific applications. Moreover we consider the affects of three additional constraints, namely height, structural and node search restrictions, which lead to a number of new construction algorithms. In particular we concentrate on the construction of optimum t-ary search trees with linear and binary search within their nodes for which we obtain O(n 3 t) and O(n 3log2 t) time algorithms, respectively. Whether these algorithms are or are not optimal remains an important open problem, even in the binary case.

Optimum multiway search trees

Optimal binary split trees

The definition of binary split trees is generalized by removing the condition of decreasing frequency. It is shown that the access time of generalized split trees is less than that of split trees in general. The optimal representation of generalized split trees is studied. A polynomial time algorithm to construct such optimal tree structures is given. The relationship among several classes of binary trees are also discussed.

Generalized binary split trees

Operations research applications of dichotomous search

This paper considers the construction of optimal search trees for a sequence of n keys of varying sizes, under various cost measures. Constructing optimal search cost multiway trees is NP-hard, although it can be done in pseudo-polynomial time O3 and space O2, where L is the page size limit. An optimal space multiway search tree is obtained in O3 time and O2 space, while an optimal height tree in O(n2 log2n) time and O(n) space both having additionally minimal root sizes. The monotonicity principle does not hold for the above cases. Finding optimal search cost weak B-trees is NP-hard, but a weak B-tree of height 2 and minimal root size can be constructed in O(n log n) time. In addition, if its root is restricted to contain M keys then a different algorithm is applied, having time complexity O(nM log n). The latter solves a problem posed by McCreight.

Optimal multiway search trees for variable size keys

Given Nweighted keysN+1 missing-key weights and a branching factor t the application of dynamic programming yields algorithms for constructing optimal binary search trees (t = 2), optimal multi-way search trees (t>2), and optimal leaf search trees (or alphabetic code trees) with leaf weights only. The basic running time in all cases is O(N 3)(in terms of the number of keys), but it can be reduced to O(N 2) for binary search trees by a “monotonicity” principle which restricts the number of candidates for the root at each step in the construction. This principle can also be applied for multiway search trees when the missing-key weights are zero. However it does not extend to optimal multiway search trees in general, as we demonstrate; in particular, there is no monotonicity principle for alphabetic code trees, contrary to what is claimed in [5].

D. Wood

Papers

The construction of optimal multiway search trees and the monotonicity principle