Approximate Range Mode and Range Median Queries

We consider data structures and algorithms for preprocessing a labelled list of length n so that, for any given indices i and j we can answer queries of the form: What is the mode or median label in the sequence of labels between indices i and j. Our results are on approximate versions of this problem. Using O( n 1−α ) space, our data structure can find in O(log log 1 α n) time an element whose number of occurrences is at least α times that of the mode, for some user-specified parameter 0 < α < 1. Data structures are proposed to achieve constant query time for α = 1/2, 1/3 and 1/4, using storage space of O(n logn), O(n log logn) and O(n), respectively. Finally, if the elements are comparable, we construct an O( n 1−α ) space data structure that answers approximate range median queries. Specifically, given indices i and j, in O(1) time, an element whose rank is at least α× |j − i+ 1|/2 and at most (2− α)× |j − i+ 1|/2 is returned for 0 < α < 1.