8,068 research outputs found

    New results on stabbing segments with a polygon

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    We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loftier and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard. (C) 2014 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author's final draft

    Optimal Constrained Wireless Emergency Network Antennae Placement

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    With increasing number of mobile devices, newly introduced smart devices, and the Internet of things (IoT) sensors, the current microwave frequency spectrum is getting rapidly congested. The obvious solution to this frequency spectrum congestion is to use millimeter wave spectrum ranging from 6 GHz to 300 GHz. With the use of millimeter waves, we can enjoy very high communication speeds and very low latency. But, this technology also introduces some challenges that we hardly faced before. The most important one among these challenges is the Line of Sight (LOS) requirement. In the emergent concept of smart cities, the wireless emergency network is set to use millimeter waves. We have worked on the problem of efficiently finding a line of sight for such wireless emergency network antennae in minimal time. We devised two algorithms, Sequential Line of Sight (SLOS) and Tiled Line of Sight (TLOS), both perform better than traditional algorithms in terms of execution time. The tiled line of sight algorithm reduces the time required for a single line of sight query from 200 ms for traditional algorithms to mere 1.7 ms on average

    Approximating the Maximum Overlap of Polygons under Translation

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    Let PP and QQ be two simple polygons in the plane of total complexity nn, each of which can be decomposed into at most kk convex parts. We present an (1−Δ)(1-\varepsilon)-approximation algorithm, for finding the translation of QQ, which maximizes its area of overlap with PP. Our algorithm runs in O(cn)O(c n) time, where cc is a constant that depends only on kk and Δ\varepsilon. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time
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