Dynamic Orthogonal Range Searching on the RAM, Revisited

Dynamic Orthogonal Range Searching on the RAM, Revisited∗ Timothy M. Chan1 and Konstantinos Tsakalidis2 1 Dept. of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada 2 Abstract We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range   log n reporting. We present a new data structure achieving O log log n + k optimal query time and   O log2/3+o(1) n update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement  in over 10 years of Mortensen’s 7/8+ε previous result [SIAM J. Comput., 2006], which has O log n update time for an arbitrarily small constant ε.   In the case of 3-sided queries, our update time reduces to O log1/2+ε n , improving Wilkin-   son’s previous bound [ESA 2014] of O log2/3+ε n . 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases dynamic data structures, range searching, computational geometry Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.28 1 Introduction Orthogonal range searching is one of the most well-studied and fundamental problems in computational geometry: the goal is to design a data structure to store a set of n points so that we can quickly report all points inside a query axis-aligned rectangle. In the “emptiness” version of the problem, we just want to decide if the rectangle contains any point. (We will not study the counting version of the problem here.) The static 2-d problem has been extensively investigated [15, 6, 25, 13, 11, 22, 1, 21], with the current best results in the word RAM model given by Chan, Larsen, and Pătraşcu [9]. In this paper, we are interested in the dynamic 2-d problem, allowing insertions and deletions of points. A straightforward dynamization of the

standard range tree [27] supports queries in O log2 n + k time and updates in O log2 n time, where k denotes the number of reported points (for the emptiness problem, we can take k = 0). Mehlhorn and Näher [17] improved the query time to O (log n log log n + k) and the update time to O (log n log log n) by dynamic fractional cascading. ∗ This work was done while the first author was at the University of Waterloo, and was partially supported by an NSERC Discovery Grant. © Timothy M. Chan and Konstantinos Tsakalidis; licensed under Creative Commons License CC-BY 33rd International Symposium on Computational Geometry (SoCG 2017). Editors: Boris Aronov and Matthew J. Katz; Article No. 28; pp. 28:1–28:13 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 28:2 Dynamic Orthogonal Range Searching on the RAM, Revisited The first data structure to achieve logarithmic query and update (amortized) time was presented by Mortensen [19].  In fact, heobtained sublogarithmic bounds in the word RAM  model: the query time is O logloglogn n + k and the amortized update time is O log7/8+ε n where ε denotes an arbitrarily small positive constant. On the lower bound side, Alstrup et al.[2] showed  that any data structure with tu update n time for 2-d range emptiness requires Ω log(tlog query time in the cell-probe model. u log n) Thus, Mortensen’s query bound is optimal for any data structure with polylogarithmic update time. However, it is conceivable that the update time  could be improved further  while keeping the same query time. Indeed, the O log7/8+ε n update bound looks too peculiar to be optimal, one would think. Let us remark how intriguing this type of “fractional-power-of-log” bound is, which showed up only on a few occasions in the literature. For example, Chan and Pătraşcu [10] gave a dynamic data structure of elements less than a given for 1-d rank queries (counting number   value) with O log n log log n √ query time and O log1/2+ε n update time. Chan and Pătraşcu also obtained more log n-type results for various offline range counting problems. Another example is Wilkinson’s recent paper [24]: he studied a special case of 2-d orthogonal range   reporting for 2-sided and 3-sided rectangles and obtained a solution with O logloglogn n + k     amortized query time, O log1/2+ε n update time for the 2-sided case, and O log2/3+ε n   update time for 3-sided; the latter improves Mortensen’s O log5/6+ε n update bound for 3-sided [19]. He also showed that in the insertion-only and deletion-only settings, it is possible to get fractional-power-of-log bounds for both the update and the query time. However, he was unable to make progress for general 4-sided rectangles in the insertion-only and deletion-only settings, let alone the fully dynamic setting. New results. Our mainnew result is a fully dynamic data structure   for 2-d orthogonal log n range reporting with O log log n + k optimal query time and O log2/3+o(1) n update   time, greatly improving Mortensen’s O log7/8+ε n bound. In the 3-sided case, we obtain     O log1/2+ε n update time, improving Wilkinson’s O log2/3+ε n bound. (See Table 1 for comparison.) Our update bounds seem to reach a natural limit with this type of approach. √ In particular, it is not unreasonable to conjecture that the near- log n update bound for the 3-sided case is close to optimal, considering prior “fractional-power-of-log” upper-bound results in the literature (although there have been no known lower bounds of this type so far). Like previous methods, our bounds are amortized (this includes query time). Our results are in the word-RAM model, under the standard assumption that the word size w is at least log n bits (in fact, except for an initial predecessor search during each query/update, we only need operations on (log n)-bit words). Even to researchers uncomfortable with sublogarithmic algorithms on the word RAM, such techniques are still relevant. For example, Mortensen  d−1  log n extended his data structure to d ≥ 3 dimensions and obtained O + k query log log n   time and O logd−9/8+ε n update time, even in the real-RAM model (where each word can hold an input real number or a (log n)-bit number). Our result automatically leads to improvements in higher dimensions as well. T. M. Chan and K. Tsakalidis 28:3 Table 1 Dynamic planar orthogonal range reporting: previous and new results. 4-sided Lueker and Willard [27] Mehlhorn and Näher [17] Mortensen [19] New 3-sided McCreight [16] Willard [26] Mortensen [19] Wilkinson [24] Wilkinson [24] New Update time log2 n log n log log n log7/8+ε n log2/3 n logO(1) log n Query time log2 n + k log n log log n + k log n +k log log n log n +k log log n log n log n + k log n +k log log n log n +k log log n log n + k log n +k log log n log n +k log log n log n log log n 5/6+ε log n (log n log log n)2/3 log2/3+ε n log1/2+ε n Overview of techniques: Micro- and macro-structures. Our sol (...truncated)