Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $operatorname{rb-index}(k)$ be the maximum of $operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $operatorname{rb-index}(k) eq k$, and we prove that for $kge 5$, [ frac{40lfloor (k-1)/2 rfloor -8}{19} %Birgit: leqoperatorname{rb-index}(k)leq 10 bigglfloorfrac{k}{7}biggrfloor + 11. ] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 lfloorfrac{k}{7}rfloor + 11$ vertices can be computed in $O(nlog n)$ time.
تحميل البحث