In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log^{1+o(1)}n + k log^{1/2+o(1)}n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log^{2+o(1)}n) time (amortized) in the standard RAM model. This is the first dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near sqrt{log n}). We also give an alternative data structure with O(log^{1+o(1)} n + k log^{3/4+o(1)}n) query time and O(log^{3/2+o(1)}n) update time (amortized). We also mention extensions to higher constant dimensions.
We present a number of new results about range searching for colored (or "categorical") data:
Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.
Range searching on categorical, or "colored", data has been studied extensively for over two decades. In this paper, we obtain the current best results for perhaps the most basic, and most often studied, version of the geometric problem: colored orthogonal range reporting.
Given n colored points in two-dimensional space [U]^2, we present a data structure with O(n log^{3/4+epsilon} n) space, for an arbitrarily small constant epsilon > 0, so that all k distinct colors in any axis-aligned query rectangle can be reported in (optimal) O(loglog U + k) time; this is the first method to break the O(n log n) space barrier.
In three dimensions, we present a data structure with O(n log^{9/5+epsilon} n) space and O(log n/loglog n + k) time; this improves the previous space bound of O(n log^4 n).
In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing.
Dynamic planar orthogonal point location in sublogarithmic time
(with Konstantinos Tsakalidis)
We study a longstanding problem in computational geometry: dynamic 2-d orthogonal point location, i.e., vertical ray shooting among n horizontal line segments. We present a data structure achieving O(log n / loglog n) optimal expected query time and O(log^{1/2+epsilon} n) update time (amortized) in the word-RAM model for any constant epsilon > 0, under the assumption that the x-coordinates are integers bounded polynomially in n. This substantially improves previous results of Giyora and Kaplan [SODA 2007] and Blelloch [SODA 2008] with O(log n) query and update time, and of Nekrich (2010) with O(log n / loglog n) query time and O(log^{1+epsilon} n) update time. Our result matches the best known upper bound for simpler problems such as dynamic 2-d dominance range searching.
We also obtain similar bounds for orthogonal line segment intersection reporting queries, vertical ray stabbing, and vertical stabbing-max, improving previous bounds, respectively, of Blelloch [SODA 2008] and Mortensen [SODA 2003], of Tao (2014), and of Agarwal, Arge, and Yi [SODA 2005] and Nekrich [ISAAC 2011].
We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(n log n + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((n log n + k) loglog n) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Patrascu [SoCG'11] that attains the same time complexity but in expectation.
The 2D rectangle enclosure problem is related to the offline dominance range reporting problem in 4D, and our result leads to the currently fastest deterministic algorithm for offline dominance reporting in any constant dimension d >= 4.
A key tool behind Chan et al.'s previous randomized algorithm is shallow cuttings for 3D dominance ranges. Recently, Afshani and Tsakalidis [SODA'14] obtained a deterministic O(n log n)-time algorithm to construct such cuttings. We first present an improved deterministic construction algorithm that runs in O(n loglog n) time in the word-RAM; this result is of independent interest. Many additional ideas are then incorporated, including a linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree point location problem.
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.
We give an algorithm for bichromatic line segment intersection counting that runs in O(n sqrt{log n}) time under the word RAM model via a reduction to dynamic predecessor search, offline point location, and offline dynamic ranking. This algorithm is the first to solve bichromatic line segment intersection counting in o(n log n) time.
We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model:
The most recent previous development on (a) was reported back in SoCG'95 by Gupta, Janardan, Smid, and Dasgupta, whose main result was an O([n lg n + k] lglg n) algorithm. The best previous result on (b) was an O(n lg n lglg n) algorithm due to Gabow, Bentley, and Tarjan---from STOC'84! As a consequence, we also obtain the current-record time bound for the maxima problem in all constant dimensions above 4.
We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,...,U} under an arbitrary sequence of n insertions and deletions, with O(loglog U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linear-space structures and improves previous near-O((loglog U)^2) methods.
The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(loglog U) worst-case query time and linear space. This result is again optimal in U for linear-space structures and improves the previous O((loglog U)^2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higher-dimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions >= 3 on the RAM.
Our key technique is an interesting new van-Emde-Boas-style recursion that alternates between two strategies, both quite simple.
We give an O(n sqrt{lg n})-time algorithm for counting the number of inversions in a permutation on n elements. This improves a long-standing previous bound of O(n lg n/lg lg n) that followed from Dietz's data structure [WADS'89], and answers a question of Andersson and Petersson [SODA'95]. As Dietz's result is known to be optimal for the related dynamic rank problem, our result demonstrates a significant improvement in the offline setting. Our new technique is quite simple: we perform a "vertical partitioning" of a trie (akin to van Emde Boas trees), and use ideas from external memory. However, the technique finds numerous applications: for example, we obtain
As a bonus, we also give a simple (1+epsilon)-approximation algorithm for counting inversions that runs in linear time, improving the previous O(n lg lg n) bound by Andersson and Petersson.
Given a point set in a fixed dimension, we note that a well-separated pair decomposition can be found in linear time if we assume that the ratio of the farthest pair distance to the closest pair distance is polynomially bounded. Many consequences follow; for example, we can construct spanners or solve the all-nearest-neighbors problem in linear time (under the same assumption), and we compute an approximate Euclidean minimum spanning tree in linear time (without any assumption).
We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a two-dimensional analog of sorting where one needs to order points with respect to segments. This result implies, for example, that the Voronoi diagram of n points in the plane can be constructed in (randomized) time n . 2^{O(\sqrt{lg lg n})}. Similar bounds hold for numerous other geometric problems, such as three-dimensional convex hulls, planar Euclidean minimum spanning trees, line segment intersection, and triangulation of non-simple polygons.
In FOCS'06, we developed a data structure for online point location, which implied a bound of O(n lg n / lg lg n) for Voronoi diagrams and the other problems. Our current bounds are dramatically better, and a convincing improvement over the classic O(n lg n) algorithms. As in the field of integer sorting, the main challenge is to find ways to manipulate information, while avoiding the online problem (in that case, predecessor search).
Given a planar subdivision whose coordinates are integers bounded by U <= 2^w, we present a linear-space data structure that can answer point location queries in O(min{ lg n/lglg n, sqrt{lg U/lglg U} }) time on the unit-cost RAM with word size w. This is the first result to beat the standard Theta(lg n) bound for infinite precision models.
As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed.
Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA'92).
In this note, we point out that much simpler algorithms with the same performance are possible using standard, though nonalgebraic, RAM operations. This is interesting, considering that nonalgebraic operations have been used before in the literature...
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