Advanced rank/select data structures: succinctness, bounds and applications.

The thesis explores new theoretical results and applications of rank and select data structures. Given a string, select(c, i) gives the position of the ith occurrence of character c in the string, while rank(c, p) counts the number of instances of character c on the left of position p. Succinct rank/select data structures are space-efficient versions of standard ones, designed to keep data compressed and at the same time answer to queries rapidly. They are at the basis of more involved compressed and succinct data structures which in turn are motivated by the nowadays need to analyze and operate on massive data sets quickly, where space efficiency is crucial. The thesis builds up on the state of the art left by years of study and produces results on multiple fronts. Analyzing binary succinct data structures and their link with predecessor data structures, we integrate data structures for the latter problem in the former. The result is a data structure which outperforms the one of Patrascu 08 in a range of cases which were not studied before, namely when the lower bound for predecessor do not apply and constant-time rank is not feasible. Further, we propose the first lower bound for succinct data structures on generic strings, achieving a linear trade-off between time for rank/select execution and additional space (w.r.t. to the plain data) needed by the data structure. The proposal addresses systematic data structures, namely those that only access the underlying string through ADT calls and do not encode it directly. Also, we propose a matching upper bound that proves the tightness of our lower bound. Finally, we apply rank/select data structures to the substring counting problem, where we seek to preprocess a text and generate a summary data structure which is stored in lieu of the text and answers to substring counting queries with additive error. The results include a theory-proven optimal data structure with generic additive error and a data structure that errs only on infrequent patterns with significative practical space gains

Succinct Indexable Dictionaries with Applications to Encoding kk-ary Trees, Prefix Sums and Multisets

We consider the {\it indexable dictionary} problem, which consists of storing a set $S \subseteq \{0,...,m-1\}$ for some integer $m$, while supporting the operations of \Rank(x), which returns the number of elements in $S$ that are less than $x$ if $x \in S$, and -1 otherwise; and \Select(i) which returns the $i$-th smallest element in $S$. We give a data structure that supports both operations in O(1) time on the RAM model and requires ${\cal B}(n,m) + o(n) + O(\lg \lg m)$ bits to store a set of size $n$, where {\cal B}(n,m) = \ceil{\lg {m \choose n}} is the minimum number of bits required to store any $n$-element subset from a universe of size $m$. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the $O(\lg \lg m)$ additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: An information-theoretically optimal representation of a $k$-ary cardinal tree that supports standard operations in constant time, A representation of a multiset of size $n$ from $\{0,...,m-1\}$ in ${\cal B}(n,m+n) + o(n)$ bits that supports (appropriate generalizations of) \Rank and \Select operations in constant time, and A representation of a sequence of $n$ non-negative integers summing up to $m$ in ${\cal B}(n,m+n) + o(n)$ bits that supports prefix sum queries in constant time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report 2002/1