Zip Trees

Abstract

We introduce the zip tree, (Zip: “To move very fast.”) a form of randomized binary search tree that integrates previous ideas into one practical, performant, and pleasant-to-implement package. A zip tree is a binary search tree in which each node has a numeric rank and the tree is (max)-heap-ordered with respect to ranks, with ties broken in favor of smaller keys. Zip trees are essentially treaps [8], except that ranks are drawn from a geometric distribution instead of a uniform distribution, and we allow rank ties. These changes enable us to use fewer random bits per node.

We perform insertions and deletions by unmerging and merging paths (unzipping and zipping) rather than by doing rotations, which avoids some pointer changes and improves efficiency. The methods of zipping and unzipping take inspiration from previous top-down approaches to insertion and deletion by Stephenson [10], Martínez and Roura [5], and Sprugnoli [9].

From a theoretical standpoint, this work provides two main results. First, zip trees require only $O(\log \log n)$ bits (with high probability) to represent the largest rank in an n-node binary search tree; previous data structures require $O(\log n)$ bits for the largest rank. Second, zip trees are naturally isomorphic to skip lists [7], and simplify Dean and Jones’ mapping between skip lists and binary search trees [2].

Research at Princeton University partially supported by an innovation research grant from Princeton and a gift from Microsoft.