13.4: Discussion and Exercises

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The first data structure to provide $O(\log\mathtt{w})$ time $\mathtt{add(x)}$, $\mathtt{remove(x)}$, and $\mathtt{find(x)}$ operations was proposed by van Emde Boas and has since become known as the van Emde Boas (or stratified) tree [74]. The original van Emde Boas structure had size $2^{\mathtt{w}}$, making it impractical for large integers.

The XFastTrie and YFastTrie data structures were discovered by Willard [77]. The XFastTrie structure is closely related to van Emde Boas trees; for instance, the hash tables in an XFastTrie replace arrays in a van Emde Boas tree. That is, instead of storing the hash table $\mathtt{t[i]}$, a van Emde Boas tree stores an array of length $2^{\mathtt{i}}$.

Another structure for storing integers is Fredman and Willard's fusion trees [32]. This structure can store $\mathtt{n}$ $\mathtt{w}$-bit integers in $O(\mathtt{n})$ space so that the $\mathtt{find(x)}$ operation runs in $O((\log \mathtt{n})/(\log\mathtt{w}))$ time. By using a fusion tree when $\log \mathtt{w} > \sqrt{\log \mathtt{n}}$ and a YFastTrie when $\log \mathtt{w} \le \sqrt{\log \mathtt{n}}$, one obtains an $O(\mathtt{n})$ space data structure that can implement the $\mathtt{find(x)}$ operation in $O(\sqrt{\log \mathtt{n}})$ time. Recent lower-bound results of P ${\v{a\/}}\kern.05em$ tra ${\c{s\/}}$ cu and Thorup [59] show that these results are more or less optimal, at least for structures that use only $O(\mathtt{n})$ space.

Exercise $\PageIndex{1}$

Design and implement a simplified version of a BinaryTrie that does not have a linked list or jump pointers, but for which $\mathtt{find(x)}$ still runs in $O(\mathtt{w})$ time.

Exercise $\PageIndex{2}$

Design and implement a simplified implementation of an XFastTrie that doesn't use a binary trie at all. Instead, your implementation should store everything in a doubly-linked list and $\mathtt{w}+1$ hash tables.

Exercise $\PageIndex{3}$

We can think of a BinaryTrie as a structure that stores bit strings of length $\mathtt{w}$ in such a way that each bitstring is represented as a root to leaf path. Extend this idea into an SSet implementation that stores variable-length strings and implements $\mathtt{add(s)}$, $\mathtt{remove(s)}$, and $\mathtt{find(s)}$ in time proporitional to the length of $\mathtt{s}$.

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

Exercise $\PageIndex{4}$

For an integer $\mathtt{x}\in\{0,\ldots2^{\mathtt{w}}-1\}$, let $d(\mathtt{x})$ denote the difference between $\mathtt{x}$ and the value returned by $\mathtt{find(x)}$ [if $\mathtt{find(x)}$ returns $\mathtt{null}$, then define $d(\mathtt{x})$ as $2^\mathtt{w}$]. For example, if $\mathtt{find(23)}$ returns 43, then $d(23)=20$.

Design and implement a modified version of the $\mathtt{find(x)}$ operation in an XFastTrie that runs in $O(1+\log d(\mathtt{x}))$ expected time. Hint: The hash table $t[\mathtt{w}]$ contains all the values, $\mathtt{x}$, such that $d(\mathtt{x})=0$, so that would be a good place to start.
Design and implement a modified version of the $\mathtt{find(x)}$ operation in an XFastTrie that runs in $O(1+\log\log d(\mathtt{x}))$ expected time.

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