I want to find the minimum distance between two polygons with million number of vertices(not the minimum distance between their vertices). I have to find the minimum of shortest distance between each vertex of first shape with all of the vertices of the other one. Something like the Hausdorff Distance, but I need the minimum instead of the maximum.
18
votes
Could you update your question to mention that polygons may have millions of points, and whether they are convex/concave, and whether they can overlap.
– brainjam
Yeah, the polygons have millions of points and they are non-convex. Actually an algorithm which works for two sets of points is preferred more.
– Pouya BCD
the minimum distance between two polyogns is not always the minimum distance between the verticies in each polygon. which do you really want
– jk.
You are right. Sometimes, the minimum distance is between a vertex and a segment. But it cannot be between two segments. By the way, the most naive algorithm to find this is checking each vertex with the other polygon and doing it for both polygons.
– Pouya BCD
2 Answers
24
votes
Perhaps you should check out (PDF warning! Also note that, for some reason, the order of the pages is reversed) "Optimal Algorithms for Computing the Minimum Distance Between Two Finite Planar Sets" by Toussaint and Bhattacharya:
It is shown in this paper that the minimum distance between two finite planar sets if [sic] n points can be computed in O(n log n) worst-case running time and that this is optimal to within a constant factor. Furthermore, when the sets form a convex polygon this complexity can be reduced to O(n).
If the two polygons are crossing convex ones, perhaps you should also check out (PDF warning! Again, the order of the pages is reversed) "An Optimal Algorithm for Computing the Minimum Vertex Distance Between Two Crossing Convex Polygons" by Toussaint:
Let P = {p1, p2,..., pm} and Q = {q1, q2,..., qn} be two intersecting polygons whose vertices are specified by their cartesian coordinates in order. An optimal O(m + n) algorithm is presented for computing the minimum euclidean distance between a vertex pi in P and a vertex qj in Q.
0
votes
There is a simple algorithm that uses Minkowski Addition that allows calculating min-distance of two convex polygonal shapes and runs in O(n + m).
Links:
algoWiki, boost.org, neerc.ifmo.ru (in russian).
If Minkowski subtraction of two convex polygons covers (0, 0), then they intersect
