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\title{CS 141: Advanced Topics in Algorithms \\ Spring 2006 \\ Homework Cris and Chris}
\author{}
\date{Due:  Thursday, April 6}
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\begin{enumerate}

  \item \Points{10} \Name{Delaunay Triangulations and Minimum Spanning Trees!}
  Delaunay Triangulations have many interesting properties.  One is this:   Given a set of points $P$
  in the plane and the Delaunay Triangulation of these points, two points $p_1, p_2 \in P$
  are connected by an edge in the Delaunay Triangulation if and only if there exists some
  circle through $p_1$ and $p_2$ that contains no other points of $P$.  
  
  Use this property to prove the following claim:  Consider a set of points $P$ in the plane.  Let $G_P$ be
  the complete graph induced by these points such that the weight on each edge is the Euclidean
  distance between its endpoints.  Let $D_P$ be the Delaunay Triangulation of $P$.  Then any
  minimum spanning tree of $G_P$ is a subgraph of $D_P$.  
\end{enumerate}

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