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 worst-case 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 P˘ atra¸scuatra¸scu, 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.

Connectivity Oracles for Graphs Subject to Vertex Failures

“…In Section 6.1 we show howÕ(d 3 ) time can be achieved with a 2D colored (aka categorical) range searching structure [58]. Section 6.2 uses a 2D range counting [22] data structure, and Section 6.3 uses a 3D range emptiness data structure [20]. The method of Section 6.3 was suggested to us by Shiri Chechik.…”

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

“…Figure 1 illustrates the process. Our array is A[1, 9] = 3, 1, 2, 4,6,5,7,8,9 , where π 0 is encoded in A [1,3] and π 1 in A [4,6] (with values shifted by ik = 3). Then, sel(1, 3, 3) = 2 tells us that the minimum among the first 3 elements in π 0 (i.e.…”

“…We describe Jørgensen and Larsen's "shallow cuttings" idea [23], and the way Chan and Wilkinson [9] take advantage of it. In general terms, our encoding for sel(·) queries will implement their solution in an encoding scenario.…”

“…Using the properties of shallow cuttings, Chan and Wilkinson [9] reduce the O(lg n/ lg lg n) time of Brodal and Jørgensen [8] as follows. At each node v ∈ T C , they store the structure of Brodal and Jørgensen for the array…”

“…Chan and Wilkinson [9] manage to store them in O(n(lg κ + lg lg n + (lg n)/κ)) bits, which gives O(n lg n) bits when added over a set of suitable κ values (their structure works for every 1 ≤ k ≤ n, so several κ-capped structures are built). Also, their solution requires to access A and thus does not immediately translate into our setting.…”