Red–black trees with constant update time

Abstract

We show how a few modifications to the red-black trees allow for O(1) worst-case update time (once the position of the inserted or deleted element is known). The resulting structure is based on relaxing some of the properties of the red-black trees while guaranteeing that the height remains logarithmic with respect to the number of nodes. Compared to the other search trees with constant update time, our tree is the first to provide a tailored deletion procedure without using the global rebuilding technique. In addition, our structure is very simple to implement and allows for a simpler proof of correctness than those alternative trees.

Regular numeral systems for data structures

“…To illustrate the connection between counting and tree balancing, we consider a recently introduced version [26] of red-black trees [33] that can support position-based Insert and Extract in constant worst-case time. A position-based operation takes as a parameter a pointer to the position where the operation is to executed.…”

“…In addition to the element and normal list pointers, each bucket node stores a pointer to its junction point so the nodes inside a bucket know their owner. For the technical details, how buckets can be split and fused in constant worst-case time, we refer to [26].…”

“…A black node whose black height is one less than the black height of its sibling is considered doubly black. Using a constant number of rotations and colour flips per operation, it was shown in [26] that all the invariants can be satisfied simultaneously.…”

“…To be able to execute the fix-up procedures incrementally, the junction point of each bucket should also store a pointer to the internal node up to which the current fix-up process has proceeded. To show that the objectives are achievable, the incremental fix-up procedures suggested in [26] guarantee the following invariants on any root-to-leaf path:…”

B-Treaps Revised: Write Efficient Randomized Block Search Trees with High Load

“…BB[a] trees), some with O(1) writes per update (e.g. [9]). Block Search Trees are generalizations of binary search trees that store in every tree node up to α keys and α + 1 child pointer, where the array size α of the blocks is a (typically) fixed parameter.…”