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
