On Approximate Range Mode and Range Selection

For any epsilon in (0,1), a (1+epsilon)-approximate range mode query asks for the position of an element whose frequency in the query range is at most a factor (1+epsilon) smaller than the true mode. For this problem, we design a data structure occupying O(n/epsilon) bits of space to answer queries in O(lg(1/epsilon)) time. This is an encoding data structure which does not require access to the input sequence; the space cost of this structure is asymptotically optimal for constant epsilon as we also prove a matching lower bound. Furthermore, our solution improves the previous best result of Greve et al. (Cell Probe Lower Bounds and Approximations for Range Mode, ICALP\u2710) by saving the space cost by a factor of lg n while achieving the same query time. In dynamic settings, we design an O(n)-word data structure that answers queries in O(lg n /lg lg n) time and supports insertions and deletions in O(lg n) time, for any constant epsilon in (0,1); the bounds for non-constant epsilon = o(1) are also given in the paper. This is the first result on dynamic approximate range mode; it can also be used to obtain the first static data structure for approximate 3-sided range mode queries in two dimensions. Another problem we consider is approximate range selection. For any alpha in (0,1/2), an alpha-approximate range selection query asks for the position of an element whose rank in the query range is in [k - alpha s, k + alpha s], where k is a rank given by the query and s is the size of the query range. When alpha is a constant, we design an O(n)-bit encoding data structure that can answer queries in constant time and prove this space cost is asymptotically optimal. The previous best result by Krizanc et al. (Range Mode and Range Median Queries on Lists and Trees, Nordic Journal of Computing, 2005) uses O(n lg n) bits, or O(n) words, to achieve constant approximation for range median only. Thus we not only improve the space cost, but also provide support for any arbitrary k given at query time. We also analyse our solutions for non-constant alpha

Range Quantile Queries: Another Virtue of Wavelet Trees

We show how to use a balanced wavelet tree as a data structure that stores a list of numbers and supports efficient {\em range quantile queries}. A range quantile query takes a rank and the endpoints of a sublist and returns the number with that rank in that sublist. For example, if the rank is half the sublist's length, then the query returns the sublist's median. We also show how these queries can be used to support space-efficient {\em coloured range reporting} and {\em document listing}.Comment: Added note about generalization to any constant number of dimensions

Linear-Space Data Structures for Range Mode Query in Arrays

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.Comment: 13 pages, 2 figure