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 able just 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 $O((1/τ)\log\log_w(1/τ)\log n)$ 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.

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