Grandchildren-weight-balanced binary search trees

TL;DR

This paper introduces grand-children balanced trees, a new class of weight-balanced binary search trees with reduced height, and extends existing algorithms for maintaining these trees efficiently across a broader range of parameters.

Contribution

It strengthens the concept of weight-balanced trees by requiring larger grand-children weights, reducing tree height, and adapts maintenance algorithms to these new trees for improved efficiency.

Findings

Grand-children balanced trees have up to 6% smaller height.

All such trees with n nodes have height less than 2 log2(n).

Algorithms maintain trees with constant amortised rebalancing operations.

Abstract

We revisit weight-balanced trees, also known as trees of bounded balance. This class of binary search trees was invented by Nievergelt and Reingold in 1972. Such trees are obtained by assigning a weight to each node and requesting that the weight of each node should be quite larger than the weights of its children, the precise meaning of ``quite larger'' depending on a real-valued parameter~$\gamma$. Blum and Mehlhorn then showed how to maintain these trees in a recursive (bottom-up) fashion when~$2/11 \leqslant \gamma \leqslant 1-1/\sqrt{2}$, their algorithm requiring only an amortised constant number of tree rebalancing operations per update (insertion or deletion). Later, in 1993, Lai and Wood proposed a top-down procedure for updating these trees when~$2/11 \leqslant \gamma \leqslant 1/4$. Our contribution is two-fold. First, we strengthen the requirements of Nievergelt and Reingold, by also requesting that each node should have a substantially larger weight than its grand-children, thereby obtaining what we call grand-children balanced trees. Grand-children balanced trees are not harder to maintain than weight-balanced trees, but enjoy a smaller node depth, both in the worst case (with a 6~\% decrease) and on average (with a 1.6~\% decrease). In particular, unlike standard weight-balanced trees, all grand-children balanced trees with $n$ nodes are of height less than $2 \log_2(n)$. Second, we adapt the algorithm of Lai and Wood to all weight-balanced trees, i.e., to all parameter values~$\gamma$ such that~$2/11 \leqslant \gamma \leqslant 1-1/\sqrt{2}$. More precisely, we adapt it to all grand-children balanced trees for which~$1/4 < \gamma \leqslant 1 - 1/\sqrt{2}$. Finally, we show that, except in critical cases, all these algorithms result in making a constant amortised number of tree rebalancing operations per tree update.