3,000 research outputs found
Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane
Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for computing a minimum-link shortest s-t path, the previous algorithm runs in O(n log^{3/2} n) time while the time of our new algorithm is O(n + h log^{3/2} h)
Computing a rectilinear shortest path amid splinegons in plane
We reduce the problem of computing a rectilinear shortest path between two
given points s and t in the splinegonal domain \calS to the problem of
computing a rectilinear shortest path between two points in the polygonal
domain. As part of this, we define a polygonal domain \calP from \calS and
transform a rectilinear shortest path computed in \calP to a path between s and
t amid splinegon obstacles in \calS. When \calS comprises of h pairwise
disjoint splinegons with a total of n vertices, excluding the time to compute a
rectilinear shortest path amid polygons in \calP, our reduction algorithm takes
O(n + h \lg{n}) time. For the special case of \calS comprising of concave-in
splinegons, we have devised another algorithm in which the reduction procedure
does not rely on the structures used in the algorithm to compute a rectilinear
shortest path in polygonal domain. As part of these, we have characterized few
of the properties of rectilinear shortest paths amid splinegons which could be
of independent interest
Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain
We study the computation of the diameter and radius under the rectilinear
link distance within a rectilinear polygonal domain of vertices and
holes. We introduce a \emph{graph of oriented distances} to encode the distance
between pairs of points of the domain. This helps us transform the problem so
that we can search through the candidates more efficiently. Our algorithm
computes both the diameter and the radius in time, where denotes the matrix
multiplication exponent and is the number of
edges of the graph of oriented distances. We also provide a faster algorithm
for computing the diameter that runs in time
Routing for analog chip designs at NXP Semiconductors
During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP.
This resulted in an heuristic approach, which we presented at the end of the
week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach
The ‘Dynamics’ of Leibnizian Relationism: Reference Frames and Force in Leibniz’s Plenum
This paper explores various metaphysical aspects of Leibniz’s concepts of space, motion, and matter, with the intention of demonstrating how the distinctive role of force in Leibnizian physics can be used to develop a theory of relational motion using privileged reference frames. Although numerous problems will remain for a consistent Leibnizian relationist account, the version developed within our investigation will advance the work of previous commentators by more accurately reflecting the specific details of Leibniz’s own natural philosophy, especially his handling of the dynamical interactions of plenum bodies
An O(n^{5/2} log n) Algorithm for the Rectilinear Minimum Link-Distance Problem in Three Dimensions (Extended Abstract)
In this paper we consider the Rectilinear Minimum Link-Distance Problem in Three Dimensions. The problem is well studied in two dimensions, but is relatively unexplored in higher dimensions. We solve the problem in O(B n log n) time, where n is the number of corners among all obstacles, and B is the size of a BSP decomposition of the space containing the obstacles. It has been shown that in the worst case B = Theta(n^{3/2}), giving us an overall worst case time of O(n^{5/2} log n). Previously known algorithms have had worst-case running times of Omega(n^3)
Planar rectilinear shortest path computation using corridors
AbstractThe rectilinear shortest path problem can be stated as follows: given a set of m non-intersecting simple polygonal obstacles in the plane, find a shortest L1-metric (rectilinear) path from a point s to a point t that avoids all the obstacles. The path can touch an obstacle but does not cross it. This paper presents an algorithm with time complexity O(n+m(lgn)3/2), which is close to the known lower bound of Ω(n+mlgm) for finding such a path. Here, n is the number of vertices of all the obstacles together
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