Synergistic Sorting, MultiSelection and Deferred Data Structures on MultiSets

Karp et al. (1988) described Deferred Data Structures for Multisets as "lazy" data structures which partially sort data to support online rank and select queries, with the minimum amount of work in the worst case over instances of size $n$ and number of queries $q$ fixed (i.e., the query size). Barbay et al. (2016) refined this approach to take advantage of the gaps between the positions hit by the queries (i.e., the structure in the queries). We develop new techniques in order to further refine this approach and to take advantage all at once of the structure (i.e., the multiplicities of the elements), the local order (i.e., the number and sizes of runs) and the global order (i.e., the number and positions of existing pivots) in the input; and of the structure and order in the sequence of queries. Our main result is a synergistic deferred data structure which performs much better on large classes of instances, while performing always asymptotically as good as previous solutions. As intermediate results, we describe two new synergistic sorting algorithms, which take advantage of the structure and order (local and global) in the input, improving upon previous results which take advantage only of the structure (Munro and Spira 1979) or of the local order (Takaoka 1997) in the input; and one new multiselection algorithm which takes advantage of not only the order and structure in the input, but also of the structure in the queries. We described two compressed data structures to represent a multiset taking advantage of both the local order and structure, while supporting the operators rank and select on the multiset.