What is the quickest way to find the shortest cart

Question

I have 1 red polygon say and 50 randomly placed blue polygons - they are situated in geographical 2D space. What is the quickest/speediest algorithim to find the the shortest distance between a red polygon and its nearest blue polygon?

Bear in mind that it is not a simple case of taking the points that make up the vertices of the polygon as values to test for distance as they may not necessarily be the closest points.

So in the end - the answer should give back the closest blue polygon to the singular red one.

This is harder than it sounds!

Answer 1

You might want to look at Voronoi Culling. Paper and video here:

http://www.cs.unc.edu/~geom/DVD/

Answer 2

You might be able to reduce the problem, and then do an intensive search on a small set.

Process each polygon first by finding:

• Center of polygon
• Maximum radius of polygon (i.e., point on edge/surface/vertex of the polygon furthest from the defined center)

Now you can collect, say, the 5-10 closest polygons to the red one (find the distance center to center, subtract the radius, sort the list and take the top 5) and then do a much more exhaustive routine.

Answer 3

The naive approach is to find the distance between the red and 50 blue objects -- so you're looking at 50 3d Pythagorean calculations + sorting to find the answer. That would only really work for finding the distance between center points though.

If you want arbitrary polygons, maybe your best best is a raytracing solution that emits rays from the surface of the red polygon with respect to the normal, and reports when another polygon is hit.

A hybrid might work -- we could find the distance from the center points, assuming we had some notion of the relative size of the blue polygons, we could cull the result set to the closest among those, then use raytracing to narrow down the truly closest polygon(s).

Answer 4

As others have mentioned using bounding areas (boxes, circles) may allow you to discard some polygon-polygon interactions. There are several strategies for this, e.g.

1. Pick any blue polygon and find the distance from the red one. Now pick any other polygon. If the minimum distance between the bounding areas is greater than the already found distance you can ignore this polygon. Continue for all polygons.
2. Find the minimum distance/centroid distance between the red polygon and all the blue polygons. Sort the distances and consider the smallest distance first. Calculate the actual minimum distance and continue through the sorted list until the maximum distance between the polygons is greater than the minimum distance found so far.

Your choice of circles/axially aligned boxes, or oriented boxes can have a great affect on performance of the algorithm, dependent on the actual layout of the input polygons.

For the actual minimum distance calculation you could use Yang et al's 'A new fast algorithm for computing the distance between two disjoint convex polygons based on Voronoi diagram' which is O(log n + log m).

Answer 5

You could start by comparing the distance between the bounding boxes. Testing the distance between rectangles is easier than testing the distance between polygons, and you can immediately eliminate any polygons that are more than nearest_rect + its_diagonal away (possibly you can refine that even more). Then, you can test the remaining polygons to find the closest polygon.

There are algorithms for finding polygon proximity - I'm sure Wikipedia has a good review of them. If I recall correctly, those that only allow convex polygons are substantially faster.

Answer 6

This screening technique is intended to reduce the number of distance computations you need to perform in the average case, without compromising the accuracy of the result. It works on convex and concave polygons.

Find the the minimum distance between each pair of vertexes such that one is a red vertex and one is a blue. Call it r. The distance between the polygons is at most r. Construct a new region from the red polygon where each line segment is moved outward by r and is joined to its neighbors by an arc of radius r is centered at the vertex. Find the distance from each vertex inside this region to every line segment of the opposite color that intersects this region.

Of course you could add an approximate method such as bounding boxes to quickly determine which of the blue polygons can't possibly intersect with the red region.