Optimal Polygon Triangulation

Given:

Find T that minimizes the sum of weights

Example: $w (\triangle v_i v_j v_k) \;=\; \mid v_i v_j \mid \;+\; \mid v_j v_k \mid \;+\; \mid v_k v_i \mid$ (Euclidean distance)

Looks a bit like matrix chaining

Optimal substructure

T contains $\triangle v_0 v_k v_n$ .
$w(T) \;=\; w(\triangle v_0 v_k v_n) \;+\; m[0,k] \;+\; m[k+1,n]$ .
The two subproblem solutions must be optimal or w(T) would be less.

This algorithm requires $\Theta(n^3)$ time.

This algorithm requires $\Theta(n^2)$ memory.