  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 $w^{1+\epsilon}$ bits for any $\epsilon > 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.
