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Finding minimum area \(k\)-gons. (English) Zbl 0746.68038

Let \(P\) be a set of \(n\) points in the plane. Assume \(k\geq 3\). The problem considered is to find a convex polygon \(C\) with vertices from \(P\) of minimum area that satisfies one of the following conditions:
(1) \(C\) is a convex \(k\)-gon,
(2) \(C\) is an empty convex \(k\)-gon (i.e., \(P\cap\text{int} C=\emptyset\)),
(3) \(C\) is the convex hull fo exactly \(k\) points of \(P\).
It is shown here that each of these problems can be solved by an algorithm of time complexity \(O(kn^ 3)\) and space complexity \(O(kn^ 2)\) (for \(k=4\) this is only \(O(n)\)). The algorithms are based on dynamic programming. The method extends to several similar extremum problems.

MSC:

68Q25 Analysis of algorithms and problem complexity
52A10 Convex sets in \(2\) dimensions (including convex curves)
90C39 Dynamic programming
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References:

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