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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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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.
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.