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 [72]. The original van Emde Boas structure had size , making it impractical for large integers.

The and data structures were discovered by Willard [75]. The structure is closely related to van Emde Boas trees; for instance, the hash tables in an 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 [32]. This structure can store -bit integers in space so that the operation runs in time. By using a fusion tree when and a when , one obtains an space data structure that can implement the operation in time. Recent lower-bound results of Ptracu and Thorup [57] show that these results are more or less optimal, at least for structures that use only space.

still runs in time.

Hint: Each node in your data structure should store a hash table that is indexed by character values.

- Design and implement a modified version of the operation in an that runs in expected time. Hint: The hash table contains all the values, , such that , so that would be a good place to start.
- Design and implement a modified version of the operation in an that runs in expected time.

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