Optimal Encodings for Range Majority Queries

We study the problem of designing a data structure that reports the positions of the distinct τ-majorities within any range of an array A[1, n], without storing A. A τ-majority in a range A[i, j], for 0 < τ < 1, is an element that occurs more than τ(j−i+1) times in A[i, j]. We show that Ω(n log(1/τ) ) bits are necessary for any data structure just able to count the number of distinct τ-majorities in any range. Then, we design a structure using O(n log(1/τ) ) bits that returns one position of each τ-majority of A[i, j] in O((1/τ) log logw(1/τ) log n) time, on a RAM machine with word size w (it can output any further position where each τ-majority occurs in O(1) additional time). Finally, we show how to remove a log n factor from the time by adding O(n log log n) bits of space to the structure.

• Type: Journal
• Year: 2016
• Where: ALGORITHMICA
• Authors: Gonzalo Navarro, Sharma V. Thankachan
• Keywords: ALGORITHMICA 2016, Algorithms