Zip-zip Trees: Making Zip Trees More Balanced, Biased, Compact, or Persistent

TL;DR

Zip-zip trees are a new variant of zip trees that are more balanced, space-efficient, and can be biased or persistent, with theoretical and empirical analysis showing competitive depth bounds and low metadata overhead.

Contribution

We introduce zip-zip trees, a simple variant of zip trees that improves balance, reduces metadata, and supports biasing and persistence, with rigorous analysis and practical variants.

Findings

Expected node depth is at most 1.3863 log n - 1

Uses only O(log log n) bits of metadata per node

Supports biasing and partial persistence efficiently

Abstract

We define simple variants of zip trees, called zip-zip trees, which provide several advantages over zip trees, including overcoming a bias that favors smaller keys over larger ones. We analyze zip-zip trees theoretically and empirically, showing, e.g., that the expected depth of a node in an $n$-node zip-zip tree is at most $1.3863\log n-1+o(1)$, which matches the expected depth of treaps and binary search trees built by uniformly random insertions. Unlike these other data structures, however, zip-zip trees achieve their bounds using only $O(\log\log n)$ bits of metadata per node, w.h.p., as compared to the $\Theta(\log n)$ bits per node required by treaps. In fact, we even describe a ``just-in-time'' zip-zip tree variant, which needs just an expected $O(1)$ number of bits of metadata per node. Moreover, we can define zip-zip trees to be strongly history independent, whereas treaps are generally only weakly history independent. We also introduce \emph{biased zip-zip trees}, which have an explicit bias based on key weights, so the expected depth of a key, $k$, with weight, $w_k$, is $O(\log (W/w_k))$, where $W$ is the weight of all keys in the weighted zip-zip tree. Finally, we show that one can easily make zip-zip trees partially persistent with only $O(n)$ space overhead w.h.p.