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Topic: distance of points (Read 598 times) |
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birbal
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distance of points
« on: Dec 24th, 2009, 9:34am » |
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Given n points on a 2D coordinate system . What is the most efficient way of finding nearest point for each point? How can we find all the points at a distance k from a given point efficiently? Also does the method followed will depend on the density of points ?
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inexorable
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Re: distance of points
« Reply #1 on: Jan 2nd, 2010, 11:12am » |
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Plane sweep algorithm can be used to find the nearest point to each point in O(nlogn).
For a given specific point you can do a plane sweep from X-K coordinate to X+K coordinate to find all points within K distance.
If many queries, for all points within k distance are expected. then preprocessing can be done by constructing n-ary tree with each node representing a square on plane with all points present in it.
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abhijeet
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Re: distance of points
« Reply #2 on: Jan 7th, 2010, 1:17am » |
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You can also use KD Trees for this
en.wikipedia.org/wiki/Kd-tree
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Hippo
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Re: distance of points
« Reply #3 on: Jan 7th, 2010, 1:31am » |
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I would construct Voronoi diagram/Delaunay triangulation at first (in O(n log n)). The triangulation has O(n) edges so leaving only minimal directed one going from a vertex neads an additional O(n) time.
on Jan 8th, 2010, 3:36am, birbal wrote:@hippo
can you please explain Voronoi Diagram and how to make it ? |
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Google will ... line sweep algorithm.
one of the first hits ... http://www.diku.dk/hjemmesider/studerende/duff/Fortune/
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| « Last Edit: Jan 9th, 2010, 5:45pm by Hippo » |
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birbal
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Re: distance of points
« Reply #4 on: Jan 8th, 2010, 3:36am » |
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@hippo
can you please explain Voronoi Diagram and how to make it ?
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baba
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Re: distance of points
« Reply #5 on: Feb 7th, 2010, 11:31am » |
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Isn't this similar to the Top K selection from N given items. Can't we apply Selection Algorithm to get the k nearest point in O(n). 
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