Lecture 9

Dynamic Programming III

⇒ Problem: Minimum Length Triangulation

• Input:
  • $n$ points $q_1,q_2,...,q_n$ in 2D space that form a convex $n$-gon $P$
    • Assume that these points are sorted clockwise around the centre of $P$
• Find:
  • given the $n$ points, triangulation is the process of decomposing the $n$-gon into a set of non-overlapping triangles
  • we want to find a triangulation of $P$ such that the sum of the perimeters of the $n-2$ triangles is minimized
• Output:

  • the sum of the perimeters of the triangles in $P$

So, given this $n$-gon, how many possible triangulations are there — there are (n-2) Catalan number many, which is: $C_{n-2}=\frac{1}{n-1}\left(\genfrac{}{}{0ex}{}{2n-4}{n-2}\right)$. We don’t need to know how, but $C_{n-2}\in \Theta (\frac{4^n}{(n-2{)}^{\frac{3}{2}}})$

come back to this