Dynamic Trees with Almost-Optimal Access Cost

TL;DR

This paper introduces a method for maintaining an almost optimal binary search tree with an additive approximation to the optimal search cost, and applies it to efficient adaptive alphabetic coding.

Contribution

It presents a new technique for dynamically maintaining an almost optimal weighted binary search tree with additive approximation, improving adaptive alphabetic coding efficiency.

Findings

Maintains a tree with leaf depth within a constant of the optimal logarithmic bound.

Provides an $O(m)$ time algorithm for adaptive alphabetic coding with near-entropy bounds.

First efficient constant-time per symbol adaptive alphabetic coding algorithm.

Abstract

An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost. In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element $e_i$ does not exceed $\min(\log(W/w_i),\log n)+O(1)$ where $w_i$ is the number of times $e_i$ was accessed and $W$ is the total length of the access sequence. Our techniques can also be used to encode a sequence of $m$ symbols with a dynamic alphabetic code in $O(m)$ time so that the encoding length is bounded by $m(H+O(1))$, where $H$ is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol.