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[Efi Fogel <>]

Given two concave polygons P and Q, the complexity of the Minkowski sum (the structure itself) M = P + Q is O(n^2m^2). This bound is tight.

The complexity of constructing M depends on the alg. and can be done in O(n^2m^2 log nm).

I think that it is possible to come up with a concave polygon P', such that the complexity of the Minkowski sum M' = P' + P' is O(n^4). Take the two concave polygons P and Q the Minkowski sum of which has complexity O(n^2m^2) and somehow unify them into P'. I believe that the same holds for a polygon P'', P'' + -P''... If I'm right, you are, theoretically speaking, out of luck, but typically the complexities are much smaller.

On Thu, Jan 3, 2013 at 12:59 PM, diethelm <> wrote:

Dear all,

I hope this forum is the right place to ask my question. If not, I would
appreciate any pointers to the correct place.

My problem is that I want to compute Minkowski sums of two simple but
non-convex polygons P and Q, say, with m and n vertices, respectively. The
polygons are actually simply connected, i.e. there are no holes. Classical
knowledge tells me that the usual algorithms solve this problem in O((mn)^2)
time. Now my particular interest is in the two special cases where P = Q and
P = -Q. Thus, a straightfoward application of the standard algorithm gives a
method that has an O(n^4) complexity. Is there an algorithm for these two
special cases, or at least for one of them, that has a lower complexity?

Many thanks for your help.

Kai Diethelm



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