Asymptotically Optimal Encodings of Range Data Structures for Selection and Top- k Queries

Abstract

Given an array A[1, n] of elements with a total order, we consider the problem of building a data structure that solves two queries: (a) selection queries receive a range [i, j] and an integer k and return the position of the kth largest element in A[i, j]; (b) top-k queries receive [i, j] and k and return the positions of the k largest elements in A [i, j]. These problems can be solved in optimal time, O(1 + lg k/ lg lg n) and O(k), respectively, using linear-space data structures.We provide the first study of the encoding data structures for the above problems, where A cannot be accessed at query time. Several applications are interested in the relative order of the entries of A, and their positions, rather their actual values, and thus we do not need to keep A at query time. In those cases, encodings save storage space: we first show that any encoding answering such queries requires n lg k − O(n + k lg k) bits of space; then, we design encodings using O(n lg k) bits, that is, asymptotically optimal up to constant factors, while preserving optimal query time.

Encoding Two-Dimensional Range Top-k Queries

“…This improves upon the trivial ⌈mn log (mn)⌉-bit encoding when m = o(lg n) , and also generalizes the mn(m + 3)∕2-bit encoding [10] for answering queries. The trivial encoding of the input array takes O(nm lg n) bits, whereas one can easily show a lower bound of Ω(nm lg (max (m, k))) bits for any encoding of an m × n array that supports -k queries since at least O(nm lg m) bits are necessary for answering queries [3], and at least n lg k bits are necessary for answering -k queries for each row [11]. Thus, there is only a small range of parameters where a strict improvement over the trivial encoding is possible.…”

“…For general k, on a 1D array of size n, Grossi et al [11] proposed an O(n lg k)-bit encoding which supports sorted -k queries in O(k) time, and showed that at least n lg k − O(n) bits are necessary for answering 1-sided -k queries; Gawrychowski and Nicholson [9] proposed a (k + 1)nH(1∕(k + 1)) + o(n)-bit 2 encoding for -k queries (although the queries are not supported efficiently), and showed that at least (k + 1)nH(1∕(k + 1))(1 − o(1)) bits are required to encode -k queries. They also proposed a (k + 1.5)nH(1.5∕(k + 1.5)) + o(n lg k)-bit data structure for answering -k queries in O(k 6 lg 2 nf (n)) time, for any strictly increasing function f. For a 2D array A of size m × n , one can answer -k queries using O(nm lg n) bits, by storing the rank of all elements in A.…”

“…Next we propose an optimal encoding for sorted 1-sided -k queries on A. For a 1D array [3,11] Then one can answer the -k(1, i, A � ) by finding the rightmost occurrence of every element 1 … k in X[1 … i] . By representing X (along with some additional auxiliary structures) using n lg k + O(n) bits, Grossi et al [12] obtained an encoding which supports 1-sided -k queries on A ′ in O(k) time.…”

“…We now show that the query time can be improved to O(k 2 + kT(n, k)) time if we use (4k + 7)n + ko(n) additional bits. Note that if we simply use the data structure of Grossi et al [11] (which takes 44n lg k + O(n lg lg k) bits to encode a 1D array of length n to support -k queries in O(k) time) on the 1D array of size 2n obtained by writing the values of A in column-major order, we can answer -k queries on A in O(k) time using 88n lg k + O(n lg lg k) additional bits. Although our data structure takes more query time and takes asymptotically more space, it uses less space for small values of k (note that 4k + 7 < 88 lg k for all integers 2 ≤ k < 160 ) when n is sufficiently large.…”

Encoding two-dimensional range top-k queries

“…For general k, on a 1D array of size n, Grossi et al [10] proposed an O(n lg k)-bit 1 encoding which supports sorted Top-k queries in O(k) time, and showed that at least n lg k − O(n) bits are necessary for answering 1-sided Top-k queries; Gawrychowski and Nicholson [7] proposed a (k + 1)nH(1/(k + 1)) + o(n)bit 2 encoding for Top-k queries (although the queries are not supported efficiently), and showed that at least (k + 1)nH(1/(k + 1))(1 − o(1)) bits are required to encode Top-k queries. They also proposed a (k + 1.5)nH(1.5/(k + 1.5)) + o(n lg k)-bit data structure for answering Top-k queries in O(k 6 lg 2 nf (n)) time, for any strictly increasing function f .…”