Predecessor on the Ultra-Wide Word RAM

We consider the predecessor problem on the ultra-wide word RAM model of computation, which extends the word RAM model with 'ultrawords' consisting of $w^2$ bits [TAMC, 2015]. The model supports arithmetic and boolean operations on ultrawords, in addition to 'scattered' memory operations that access or modify $w$ (potentially non-contiguous) memory addresses simultaneously. The ultra-wide word RAM model captures (and idealizes) modern vector processor architectures. Our main result is a simple, linear space data structure that supports predecessor in constant time and updates in amortized, expected constant time. This improves the space of the previous constant time solution that uses space in the order of the size of the universe. Our result holds even in a weaker model where ultrawords consist of $w^{1+ε}$ bits for any $ε> 0 $. It is based on a new implementation of the classic $x$-fast trie data structure of Willard [Inform. Process. Lett. 17(2), 1983] combined with a new dictionary data structure that supports fast parallel lookups.