Fusion tree

In computer science, a fusion tree is a type of tree data structure that implements an associative array on $w$ -bit integers. When operating on a collection of $n$ key–value pairs, it uses $O (n)$ space and performs searches in $O (log w n)$ time, which is asymptotically faster than a traditional self-balancing binary search tree, and also better than the van Emde Boas tree for large values of $w$ . It achieves this speed by exploiting certain constant-time operations that can be done on a machine word. Fusion trees were invented in 1990 by Michael Fredman and Dan Willard.

Several advances have been made since Fredman and Willard's original 1990 paper. In 1999 it was shown how to implement fusion trees under a model of computation in which all of the underlying operations of the algorithm belong to AC⁰, a model of circuit complexity that allows addition and bitwise Boolean operations but disallows the multiplication operations used in the original fusion tree algorithm. A dynamic version of fusion trees using hash tables was proposed in 1996 which matched the original structure's $O (log w n)$ runtime in expectation. Another dynamic version using exponential tree was proposed in 2007 which yields worst-case runtimes of $O (log w n + log log u)$ per operation, where $u$ is the size of the largest key. It remains open whether dynamic fusion trees can achieve $O (log w n)$ per operation with high probability.

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