# Bichromatic Line Segment Intersection Counting in O(n log n) Time

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1 Bichromatic Line Segment Intersection Counting in O(n log n) Time Timothy M. Chan Bryan T. Wilkinson Abstract We give an algorithm for bichromatic line segment intersection counting that runs in O(n 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. Introduction We consider bichromatic line segment intersection counting: given a set of disjoint blue line segments and a set of disjoint red line segments in the plane, output the number of intersections of blue segments with red segments. Bichromatic line segment intersection problems arise in applications such as map overlay in GIS. We give an algorithm that runs in O(n log n) time, where n is the total number of segments, under the standard word RAM model with w-bit words. The input segments are specified by their endpoints, which are given as O(w)- bit integer or rational coordinates. Our result answers an open question of Chan and Pǎtraşcu [6] by showing that bichromatic line segment intersection counting can be solved in o(n log n) time. Thus, the problem finally joins the ranks of many other problems in computational geometry that have Ω(n log n) lower bounds under the comparison model but o(n log n) upper bounds under the word RAM model. Recent examples include -d Voronoi diagrams [], 3-d convex hulls [6], and 3-d layers-of-maxima []. The word RAM model is important because its power corresponds very closely to that of actual programming languages (our algorithm uses only standard operations, such as arithmetic and bitwise logical operations) and it includes only the reasonable assumption that each input value is an integer (or rational) that fits in a word. Chazelle et al. [7] give algorithms based on segment trees for bichromatic segment intersection reporting in O(n log n + k) time, where k is the number of intersections reported, and for counting in O(n log n) time. Chan s trapezoid sweep algorithm [3] for the reporting problem also achieves O(n log n + k) time, but has David R. Cheriton School of Computer Science, University of Waterloo, David R. Cheriton School of Computer Science, University of Waterloo, Supported by NSERC. smaller constant factors and can be extended to curve segments. Mantler and Snoeyink [] modify the trapezoid sweep to use operations of algebraic degree at most and also to support counting in O(n log n) time. Our algorithm follows the high-level idea of Mantler and Snoeyink [] but reduces the low-level computations to dynamic predecessor search, offline point location, and offline dynamic ranking. Note that we cannot base similar reductions on just any high-level algorithm; the algorithm of Mantler and Snoeyink [] gives particularly exploitable structure to the bichromatic line segment intersection counting problem. The algorithm of Chazelle et al. [7] inherently requires O(n log n) time due to the use of segment trees, and Chan s trapezoid sweep does not address the counting problem. Recently, efficient algorithms have arisen for both offline point location and offline dynamic ranking under the word RAM model. Chan and Pǎtraşcu [6] give an algorithm for offline point location that locates O(n) points in a subdivision of the plane defined by O(n) segments in n O( log log n) time. Chan and Pǎtraşcu [5] also give an algorithm for offline dynamic ranking that processes O(n) queries and updates in O(n log n) time. We use both of these algorithms as subroutines. In Section we give an overview of Mantler and Snoeyink s algorithm [] and describe a -d data structure problem to which it reduces. In Section 3 we discuss a rank space reduction which is integral to achieving speed ups under the word RAM model. In Section 4 we describe our data structure, which we analyse in Sections 5 and 6. High-Level Algorithm We assume that the endpoints of all of the given segments are in general position. Otherwise, perturbation techniques can be used to handle any degeneracies [0]. We begin with an overview of the algorithm of Mantler and Snoeyink [], simplified under our non-degeneracy assumption. The algorithm is a plane sweep that keeps track of all of the segments that intersect the vertical sweep line. The efficiency of the algorithm is achieved by storing the segments in maximal monochromatic bundles of segments. The algorithm keeps an alternating list of red and blue bundles. The order of the bundles in this list may not be consistent with the order of the segments along the sweep line. However, the order of the

3 Size(b) Calculates the number of segments in bundle b. The analysis presented by Mantler and Snoeyink [] reveals that each of these operations is invoked at most O(n) times. The main idea of their analysis is that during the processing of each endpoint, there is a constant upper bound on the number of bundle splits, and thus there are a linear number of bundles over the course of the algorithm. All of the other operations can be charged to bundles. 3 Rank Space Reduction Before we describe our data structure, we discuss a preprocessing step in which we perform rank space reduction. Rank space reduction is important under the word RAM model to, for example, reduce the cost of predecessor search with van Emde Boas trees [4] from O(log log U), where U is the size of the universe, to O(log log n). In particular, we want to assign ids to segments such that segment s s id is greater than segment s s id if and only if s is above s. Since red and blue segments can intersect, there may be no mapping that is consistent for all segments throughout the entire plane sweep. However, Palazzi and Snoeyink [3] give a topological ordering that is consistent for a set of disjoint segments. We can thus assign ids that are consistent with aboveness to all red segments and all blue segments separately, based on their own separate topological orderings. The topological sort involves a plane sweep that runs in O(n log n) time, since it uses a balanced search tree in order to find the segment above the current endpoint. Instead of using this balanced search tree, we can find the segment above every endpoint in advance by computing the trapezoidal decomposition of the segments. Chan and Pǎtraşcu [4] reduce general offline -d point location to offline -d point location in a vertical slab that is divided into regions by disjoint segments that cut across the slab. We call this latter problem the slab problem. The same techniques (persistence and exponential search trees) can be used to reduce trapezoidal decomposition of disjoint segments to the slab problem. Chan and Pǎtraşcu [6] give an algorithm for the slab problem that yields an algorithm for trapezoidal decomposition of disjoint segments that runs in n O( log log n) deterministic time. The topological sort thus has the same runtime. 4 Data Structure All segments have ids, as assigned by the rank space reduction described in Section 3. We keep a doubly linked list of bundles. A bundle stores pointers to its top and bottom segments. Only segments which are top or bottom segments of a bundle have pointers back to their bundle. In addition to this doubly linked list, we keep two predecessor search data structures that use segment ids as keys. These trees, T and B, contain only those red segments that are at the tops and bottoms of their bundles, respectively. If we were to use van Emde Boas trees [4], updates would run in O(log log U) expected time. Since we want to avoid randomization, we use instead a data structure of Andersson and Thorup [], which supports queries and updates log log U in O(log log n log log log U ) deterministic time. Due to the rank space reduction, these operations actually run in ) time. Finally, we keep dynamic ranking data structures R r and R b, also using segment ids as keys, for all red segments and blue segments, respectively, that intersect the sweep line. We defer our selection of a particular dynamic ranking data structure to Section 5. O( log log n log log log n Lemma After a preprocessing step that runs in n O( log log n) time, we can find the segment of a given colour above or below endpoint e along the sweep line in O() time. Proof. Assume without loss of generality that e is red. The red segments above and below e can be found in O() time by navigating the red trapezoidal decomposition that was computed in n O( log log n) time in Section 3. Finding the blue segments above and below e is equivalent to locating e within the blue trapezoidal decomposition. So, in the preprocessing step, we perform offline planar point location of all red endpoints in the blue trapezoidal decomposition. The blue trapezoidal decomposition has linear complexity and has vertices with O(w)-bit rational coordinates. We can perform offline planar point location of the red endpoints in such a subdivision of the plane in n O( log log n) time using another algorithm of Chan and Pǎtraşcu [6]. We now describe how we implement all of the operations of the data structure, using the ability to find the segments above and below an endpoint in constant time via Lemma as a primitive. Insert(s, b l, b h ) We create a new bundle b in which s is both the top and bottom segment and rewire the next/previous bundle pointers between b, b l, and b h. We insert s into R c, where c is the colour of s. If s is red, we also insert it into both T and B. Delete(s) If s is both the top and bottom segment of its bundle b, s has a pointer to b. We rewire the next/previous bundle pointers around b to exclude b. If s is only the top (bottom) segment of b, we can find the new boundary of b by finding the

5 [5] T. M. Chan and M. Pătraşcu. Counting inversions, offline orthogonal range counting, and related problems. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 0, pages 6 73, Philadelphia, PA, USA, 00. Society for Industrial and Applied Mathematics. [6] T. M. Chan and M. Pǎtraşcu. Transdichotomous results in computational geometry, II: Offline search. CoRR, abs/00.948, 00. Also in Proceedings of the Thirty- Ninth Annual ACM Symposium on Theory of Computing, STOC 07, pages 3 39, New York, NY, USA, 007. ACM. [7] B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir. Algorithms for bichromatic line-segment problems and polyhedral terrains. Algorithmica, :6 3, /BF0877. [8] P. F. Dietz. Optimal algorithms for list indexing and subset rank. In Proceedings of the Workshop on Algorithms and Data Structures, WADS 89, pages 39 46, London, UK, 989. Springer-Verlag. [9] Y. Han. Deterministic sorting in O(n log log n) time and linear space. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC 0, pages , New York, NY, USA, 00. ACM. [0] H. Mairson and J. Stolfi. Reporting and counting intersections between two sets of line segments. In Theoretical Foundations of Computer Graphics and CAD, volume 40 of Proceedings of the NATO Advanced Science Institute, Series F, pages Springer-Verlag, 988. [] A. Mantler and J. Snoeyink. Intersecting red and blue line segments in optimal time and precision. In J. Akiyama, M. Kano, and M. Urabe, editors, Discrete and Computational Geometry, volume 098 of Lecture Notes in Computer Science, pages Springer Berlin / Heidelberg, / [] Y. Nekrich. A fast algorithm for three-dimensional layers of maxima problem. CoRR, abs/ , 00. [3] L. Palazzi and J. Snoeyink. Counting and reporting red/blue segment intersections. CVGIP: Graph. Models Image Process., 56:304 30, July 994. [4] P. van Emde Boas. Preserving order in a forest in less than logarithmic time. In Proceedings of the 6th Annual Symposium on Foundations of Computer Science, pages 75 84, Washington, DC, USA, 975. IEEE Computer Society.

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