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 introduce data-structural bootstrapping, a technique to design data structures recursively, and use it to design confluently persistent deques. Our data structure requires O(log 3 k) worstcase time and space per deletion, where k is the total number of deque operations, and constant worst-case time and space for other operations. Further, the data structure allows a purely functional implementation, with no side effects. This improves a previous result of Driscoll, Sleator, and Tarjan. 1 An extended abstract of this paper was presented at the 4th ACM-SIAM Symposium on Discrete Algorithms, 1993. 2 Supported by a Fannie and John Hertz Foundation fellowship, National Science Foundation Grant No. CCR-8920505, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) under NSF-STC88-09648. 3 Also affiliated with NEC Research Institute, 4 Independence Way, Princeton, NJ 08540. Research at Princeton University partially supported by the National Science Foundatio...

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.

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...

We prove that no algorithm for balanced binary search trees performing insertions and deletions in amortized time O(f(n)) can guarantee a height smaller than dlog(n + 1) + 1=f(n)e for all n. We improve the existing upper bound to dlog(n + 1) + log 2 (f(n))=f(n)e, thus almost matching our lower bound. We also improve the existing upper bound for worst case algorithms, and give a lower bound for the semi-dynamic case.

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) \...