Optimal rank and select queries on dictionary-compressed text

We study the problem of supporting queries on a string S of length n within a space bounded by the size \u3b3 of a string attractor for S. In the paper introducing string attractors it was shown that random access on S can be supported in optimal O(log(n/\u3b3)/ log log n) time within O (\u3b3 polylog n) space. In this paper, we extend this result to rank and select queries and provide lower bounds matching our upper bounds on alphabets of polylogarithmic size. Our solutions are given in the form of a space-time trade-off that is more general than the one previously known for grammars and that improves existing bounds on LZ77-compressed text by a log log n time-factor in select queries. We also provide matching lower and upper bounds for partial sum and predecessor queries within attractor-bounded space, and extend our lower bounds to encompass navigation of dictionary-compressed tree representations