Comparison-efficient and write-optimal searching and sorting

Abstract

We consider the problem of updating a binary search tree in O(log n) amortized time while using as few comparisons as possible. We show that a tree of height [log(n + 1) + 1/√log(n + 1)] can be maintained in O(log n) amortized time such that the difference between the longest and shortest paths from the root to an external node is at most 2.

We also study the problem of sorting and searching in the slow write model of computation, where we have a constant size cache of fast memory and a large amount of memory with a much slower writing time than reading time. In such a model, it is important to sort using only θ(n) writes into the slower memory. We say that such algorithms are write optimal, and we introduce a O(n log n) time, write-optimal sorting algorithm that requires only n log n + O(n) comparisons in the worst case. No previous sorting algorithm that performs n log n + o(n log n) comparisons in the worst case had previously been shown to be write optimal.

The above results are based on a class of trees called k-stratum trees, which can be viewed as a generalization of stratified search trees.