The first data structure to provide time , , and operations was proposed by van Emde Boas and has since become known as the van Emde Boas (or stratified) tree. The original van Emde Boas structure had size , so was impractical for large integers.
The XFastTrie and YFastTrie data structures were discovered by Willard [64]. The XFastTrie structure is very closely related to van Emde Boas trees. One view of this is that the hash tables in an XFastTrie replace arrays in a van Emde Boas tree. That is, instead of storing the hash table , a van Emde Boas tree stores an array of length .
Another structure for storing integers is Fredman and Willard's fusion trees [24]. This structure can store -bit integers in space so that the operation runs in time. By using a fusion tree when and a YFastTrie when one obtains an space data structure that can implement the operation in time. Recent lower-bound results of Ptracu and Thorup [50] show that these results are optimal, at least for structures that use only space.
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