A Lock-free Binary Trie

A binary trie is a sequential data structure for a dynamic set on the universe $\{0,\dots,u-1\}$ supporting Search with $O(1)$ worst-case step complexity, and Insert, Delete, and Predecessor operations with $O(\log u)$ worst-case step complexity. We give a wait-free implementation of a relaxed binary trie, using read, write, CAS, and ($\log u$)-bit AND operations. It supports all operations with the same worst-case step complexity as the sequential binary trie. However, Predecessor operations may not return a key when there are concurrent update operations. We use this as a component of a lock-free, linearizable implementation of a binary trie. It supports Search with $O(1)$ worst-case step complexity and Insert, Delete and Predecessor with $O(c^2 + \log u)$ amortized step complexity, where $c$ is a measure of the contention. A lock-free binary trie is challenging to implement as compared to many other lock-free data structures because Insert and Delete operations perform a non-constant number of modifications to the binary trie in the worst-case to ensure the correctness of Predecessor operations.