Adaptive and Approximate Orthogonal Range Counting

Abstract

We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model.• It is well known that there are linear-space data structures for 2-D orthogonal range counting with worstcase optimal query time O(log w n).We give an O(n log log n)-space adaptive data structure that improves the query time to O(log log n + log w k), where k is the output count. When k = O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan, Larsen, and Pȃtraşcu, SoCG 2011].• We give an O(n log log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1 + δ)-factor approximation to the count in O(log log n) time for any fixed constant δ > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem.• Lastly, we consider the 1-D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jørgensen and Larsen [SODA 2011] presented a linear-space adaptive data structure with query time O(log log n + log w k). We give a new linear-space structure that improves the query time to O(1 + log w k), exactly matching the lower bound proved by Jørgensen and Larsen.

Range Selection Queries in Data Aware Space and Time

“…Observe that this timing is regardless of the number of integers, n, in the original sequence, and actually solely depends on the answer of the query. Previously, an adaptive time complexity had been achieved in [5] as O(log k/ log log n), and in [2] O(log σ) was reported, where σ is the number of distinct elements in X. In both cases the space usage was linear as O(n) words, assuming the word size is u bits.…”

“…Space complexity (in bits) Gagie et al (2009) [2] O(log σ) S(X) + n log σ + o(n log σ) Brodal et al (2011) [3] O(log n/ log log n) S(X) + O(n log n) Jørgensen & Larsen (2011) [4] O(log k/ log log n + log log n) S(X) + O(n log n) Chan & Wilkinson (2013) [5] O(log k/ log log n) S(X) + O(n log n) This study O(log u + log x ) ( n log u + ∀i log x i )(1 + o(1)) Table 1: Time and space complexities of the solutions proposed to answer general range selection query (i, j, k) on a given integer sequence X = x 1 , x 2 , . .…”

“…However, the time-complexity was improved as O(log k/ log log n + log log n). Finally Chan and Wilkinson [5] presented a linear space data structure with optimal O(log k/ log log n) query time. On a related node, a generalization of this problem called tree path selection queries is studied in [10][11][12].…”

“…The query is to select the 4th smallest element between positions 8 and 12 of the given the sequence X = 16,17,3,6,2,11,5,2,3,15,16,9,13 . The elements of the X in this range are 2, 3, 15, 16, 9 . We begin from the root node of the combined wavelet tree and consider the corresponding bits from position 8 to 12, which is 00111.…”

Efficient Data Structures

“…We can assume that A is a permutation of [n], since replacing each element A[i] by its rank in A yields correct answers to those queries. The range selection problem has received a lot of interest in recent years [4,3,13,5]. Following a series of earlier papers, Brodal and Jørgensen [4] presented a structure using linear space and O(lg n/ lg lg n) time, for any k given at query time.…”