We propose measures for compressed data structures, in which space usage is measured in a data-aware manner. In particular, we consider the fundamental dictionary problem on set data, where the task is to construct a data structure to represent a set S of n items out of a universe U = {0, . . . , u - 1} and support various queries on S. We use a well-known data-aware measure for set data called gap to bound the space of our data structures.

We describe a novel dictionary structure taking gap+O(n log(u/n)/ log n)+O(n log log(u/n)) bits. Under the RAM model, our dictionary supports membership, rank, select, and predecessor queries in nearly optimal time, matching the time bound of Andersson and Thorup?s predecessor structure [AT00], while simultaneously improving upon their space usage. Our dictionary structure uses exactly gap bits in the leading term (i.e., the constant factor is 1) and answers queries in near-optimal time. When seen from the worst case perspective, we present the first O(n log(u/n))-bit dictionary structure which supports these queries in nearoptimal time under RAM model. We also build a dictionary which requires the same space and supports membership, select, and partial rank queries even more quickly in O(log log n) time. To the best of our knowledge, this is the first of a kind result which achieves data-aware space usage and retains near-optimal time.