Some new classes of balanced trees, defined by very simple balance criteria, are introduced. Those trees can be maintained by partial rebuilding at lower update cost than previously used weight-balanced trees. The used balance criteria also allow us to maintain a balanced tree without any balance information stored in the nodes.

Abstract. As a contribution to the recent debate on simple implementations of dictionaries, we present new maintenance algorithms for balanced trees. In terms of code simplicity, our algorithms compare favourably with those for deterministic and probabilistic skip lists.

We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the nodes. Thus, in the case when amortized bounds are sufficient, there is no need for sophisticated balance criteria. The maintenance algorithms use partial rebuilding. This is important for certain applications and has previously been used with weight-balanced trees. We show that the amortized cost incurred by general balanced trees is lower than what has been shown for weight-balanced trees. � 1999 Academic Press 1.

Trees of optimal and near-optimal height may be represented as a pointer-free structure in an array of size O(n). In this way we obtain an array implementation of a dictionary with O(log n) search cost and O(log2 n) update cost, allowing interpolation search to improve the expected search time. 1 Introduction The binary search tree is a fundamental and well studied data structure, commonly used in computer applications to implement the abstract data type dictionary. In a comparison-based model of computation, the lower bound on the three basic operations insert, delete and search is dlog(n + 1)e comparisons per operation. This bound may be achieved by storing the set in a binary search tree of optimal height. Definition 1 A binary tree has optimal height if and only if the height of the tree is dlog(n + 1)e. A special case of a tree of optimal height is an optimally balanced tree, as defined below. Definition 2 A binary tree is optimally balanced if and only if the difference in length between the longest and shortest paths is at most one.

First we present a generalization of symmetric binary B-trees, SBB(k)- trees. The obtained structure has a height of only \Sigma (1 + 1k) log(n + 1)\Upsilon, where k may be chosen to be any positive integer. The maintenance algorithms require only a constant number of rotations per updating operation in the worst case. These properties together with the fact that the structure is relatively simple to implement makes it a useful alternative to other search trees in practical applications. Then, by using an SBB(k)-tree with a varying k we achieve a structure with a logarithmic amortized cost per update and a height of log n + o(log n). This result is an improvement of the upper bound on the height of a dynamic binary search tree. By maintaining two trees simultaneously the amortized cost is transformed into a worstcase cost. Thus, we have improved the worst-case complexity of the dictionary problem.

A new data structure for the dictionary problem is presented. Updates are performed in \Theta(log n) time in the worst case and the number of comparisons per operation is dlog n + 1 + ffle, where ffl is an arbitrary positive constant. 1 Introduction One of the fundamental and most studied problems in computer science is the dictionary problem, that is the problem of how to maintain a set of data during the operations search, insert and delete. It is well known that in a comparison-based model the lower bound on these operations is dlog(n + 1)e comparisons both in the average and in the worst case. This bound can be achieved by storing the set in an array or in a perfectly balanced binary search tree. However, for both these data structures the overhead cost per update is high, \Theta(n) in the worst case. An efficient dynamic data structure for the dictionary problem should have a worst case cost of \Theta(log n) per operation. The first efficient solution was presented by Adelson-Vel...

The balance criterion defining the class of ff-balanced trees states that the ratio between the shortest and longest paths from a node to a leaf be at least ff. We show that a straight-forward use of partial rebuilding for maintenance of ff-balanced trees requires an amortized cost of \Omega\Gamma p n) per update. By slight modifications of the maintenance algorithms the cost can be reduced to O(log n) for any value of ff, 0 ! ff ! 1. KEY WORDS ff-balanced trees, partial rebuilding, search trees. CR CATEGORIES: E.1, F.2, I.1.2. 1 Introduction In his thesis Olivie [9] introduced a class of binary search trees, which he calls ff-balanced trees, or ffBB-trees. Let h(v) denote the length for the longest path from a node v to a leaf and let s(v) denote the length of the shortest path. We give a formal definition of ff-balanced trees below. Definition 1 A binary tree is ff-balanced if the following is true for each node v in the tree: s(v) h(v) ff; h(v) 1 1 \Gamma ff (1) h(v) \...

Two chapters of this thesis analyze expert consulting problems via game theoretic models; the first points out a close connection between the problem of consulting a set of experts and the problem of searching. The last chapter presents a solution to the dictionary problem of supporting and update (Insert and Delete) operations on a set of key values. The first chapter shows...