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P.K.: Dynamic range majority data structures
 In: Proceedings of the 22nd International Symposium on Algorithms and Computation
, 2011
"... Abstract. Given a set P of n coloured points on the real line, we study the problem of answering range αmajority (or “heavy hitter”) queries on P. More specifically, for a query range Q, we want to return each colour that is assigned to more than an αfraction of the points contained in Q. We prese ..."
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Abstract. Given a set P of n coloured points on the real line, we study the problem of answering range αmajority (or “heavy hitter”) queries on P. More specifically, for a query range Q, we want to return each colour that is assigned to more than an αfraction of the points contained in Q. We present a new data structure for answering range αmajority queries on a dynamic set of points, where α ∈ (0, 1). Our data structure uses O(n) space, supports queries in O((lgn)/α) time, and updates in O((lgn)/α) amortized time. If the coordinates of the points are integers, then the query time can be improved to O(lgn/(α lg lgn)). For constant values of α, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in ddimensions, for d ≥ 2, as well as dynamic arrays, in which each entry is a colour. 1
A faster algorithm for computing motorcycle graphs
 PROC. 29TH SYMP. ON COMPUTATIONAL GEOMETRY, SOCG ’13, ACM
, 2013
"... We present a new algorithm for computing motorcycle graphs that runs in O(n4/3+ε) time for any ε> 0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a nondegener ..."
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We present a new algorithm for computing motorcycle graphs that runs in O(n4/3+ε) time for any ε> 0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a nondegenerate polygon with h holes in O(n h+ 1 log2 n + n4/3+ε) expected time. If all input coordinates are O(log n)bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with h holes in O(n h+ 1 log3 n) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in O(n log3 n) expected time if all input coordinates are O(log n)bit rationals, while all previously known algorithms have worstcase running time ω(n3/2).
Light Orthogonal Networks with Constant Geometric Dilation
, 2008
"... An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axisaligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length of at m ..."
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Cited by 3 (1 self)
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An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axisaligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length of at most a constant times the length of a Euclidean minimum spanning tree for the point set; (ii) is small having O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most a constant times longer than the (Euclidean) distance between u and v. Such a network can be constructed in O(n log n) time.
A SpaceEfficient Framework for Dynamic Point Location
"... Abstract. Let G be a planar subdivision with n vertices. A succinct geometric index for G is a data structure that occupies o(n) bits beyond the space required to store the coordinates of the vertices of G, while supporting efficient queries. We describe a general framework for converting dynamic d ..."
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Abstract. Let G be a planar subdivision with n vertices. A succinct geometric index for G is a data structure that occupies o(n) bits beyond the space required to store the coordinates of the vertices of G, while supporting efficient queries. We describe a general framework for converting dynamic data structures for planar point location into succinct geometric indexes, provided that the subdivision G to be maintained has bounded face size. Using this framework, we obtain several succinct geometric indexes for dynamic planar point location on G with query times matching the currently best (nonsuccinct) data structures and polylogarithmic update times. 1
Fullfledged RealTime Indexing for Constant Size Alphabets
"... Abstract. In this paper we describe a data structure that supports pattern matching queries on a dynamically arriving text over an alphabet of constant size. Each new symbol can be prepended to T in O(1) expected worstcase time. At any moment, we can report all occurrences of a pattern P in the c ..."
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Abstract. In this paper we describe a data structure that supports pattern matching queries on a dynamically arriving text over an alphabet of constant size. Each new symbol can be prepended to T in O(1) expected worstcase time. At any moment, we can report all occurrences of a pattern P in the current text in O(P + k) time, where P  is the length of P and k is the number of occurrences. This resolves, under assumption of constant size alphabet, a longstanding open problem of existence of a realtime indexing method for string matching (see [2]). 1
unknown title
, 2012
"... In the last lecture we saw the concepts of persistence and retroactivity as well as several data structures implementing these ideas. In this lecture we are looking at data structures to solve the geometric problems of point location and orthogonal range queries. These problems encompass application ..."
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In the last lecture we saw the concepts of persistence and retroactivity as well as several data structures implementing these ideas. In this lecture we are looking at data structures to solve the geometric problems of point location and orthogonal range queries. These problems encompass applications such as determining which GUI element a user clicked on, what city a set of GPS coordinates is in, and certain types of database queries. 2 Planar Point Location Planar point location is a problem in which we are given a planar graph (with no crossings) defining a map, such as the boundary of GUI elements. Then we wish to support a query that takes a point given by its coordinates (x, y) and returns the face that contains it (see Figure 1 for an example). As is often the case with these problems, there is both a static and dynamic version of this problem. In the static version, we are given the entire map beforehand and just want to be able to answer queries. For the dynamic version, we also want to allow the addition and removal of edges. Figure 1: An example of a planar map and some query points
Watchman Routes for Lines and Line Segments
, 2013
"... Given a set L of nonparallel lines in the plane, a watchman route (tour) for L is a closed curve contained in the union of the lines in L such that every line is visited (intersected) by the route; we similarly define a watchman route (tour) for a connected set S of line segments. The watchman rout ..."
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Given a set L of nonparallel lines in the plane, a watchman route (tour) for L is a closed curve contained in the union of the lines in L such that every line is visited (intersected) by the route; we similarly define a watchman route (tour) for a connected set S of line segments. The watchman route problem for a given set of lines or line segments is to find a shortest watchman route for the input set, and these problems are natural special cases of the watchman route problem in a polygon with holes (a polygonal domain). In this paper, we show that the problem of computing a shortest watchman route for a set of n nonparallel lines in the plane is polynomially tractable, while it becomes NPhard in 3D. We give an alternative NPhardness proof of this problem for line segmentsintheplaneandobtainapolynomialtimeapproximationalgorithmwithratio O(log 3 n). Additionally, we consider some special cases of the watchman route problem on line segments, for which we provide exact algorithms or improved approximations.
Orienting Parts with Shape Variation
, 2014
"... Industrial parts are manufactured to tolerances as no production process is capable of delivering perfectly identical parts. It is unacceptable that a plan for a manipulation task that was determined on the basis of a CAD model of a part fails on some manufactured instance of that part, and therefor ..."
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Industrial parts are manufactured to tolerances as no production process is capable of delivering perfectly identical parts. It is unacceptable that a plan for a manipulation task that was determined on the basis of a CAD model of a part fails on some manufactured instance of that part, and therefore it is crucial that the admitted shape variations are systematically taken into account during the planning of the task. We study the problem of orienting a part with given admitted shape variations by means of pushing with a single frictionless jaw. We use a very general model for admitted shape variations that only requires that any valid instance must contain a given convex polygon PI while it must be contained in another convex polygon PE. The problem that we solve is to determine, for a given h, the sequence of h push actions that puts all valid instances of a part with given shape variation into the smallest possible interval of final orientations. The resulting algorithm runs in O(hn) time, where n = PI  + PE . 1