Shortest Paths in an Arrangement with k Line Orientations (1999) [5 citations — 0 self]
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author = {David Eppstein and David W. Hart},
title = {Shortest Paths in an Arrangement with k Line Orientations},
booktitle = {Proceedings 10th Annual ACM-SIAM Symposium on Discrete Algorithms},
year = {1999},
pages = {310--316}
}
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Abstract:
Suppose one has a line arrangement in which many lines are parallel, so that the number of different line orientations is k, and one wants to find a shortest path from one point on a line in the arrangement to another such point. Using known techniques one can find the shortest path in time and space O(n 2 ). We present an algorithm that can find the shortest path in time and space O(n + k 2 ). 1 Introduction Given a line arrangement and two points located on lines of the arrangement, it is natural to ask how to compute a shortest path from one point to the other, traveling only on lines of the arrangement. Of course, one can simply construct the arrangement, interpret its vertices and edges as forming a planar graph, and find a shortest path in this graph. It is well known how to compute arrangements in time O(n 2 ), and the resulting planar graph has O(n 2 ) vertices and edges. Klein et al. [4] show how to compute shortest paths in planar graphs in linear time, hence the sh...
Citations
| 99 | Faster shortest-path algorithms for planar graphs – Henzinger, Klein, et al. - 1997 |
| 8 | Approximating shortest paths in arrangements of lines – Bose, Evans, et al. - 1996 |
| 8 | An efficient algorithm for shortest paths in vertical and horizontal segments – Eppstein, Hart - 1997 |
| 4 | Shortest paths in line arrangements (entry in open algorithmic problems). http://compgeom.cs.uiuc.edu/ jeffe/open/ algo.html#shortpath – Erickson |
| 2 | Open problem – Kreveld - 1995 |
