Convex Hull Part II

• A divide and conquer plan.
• Sort the points by the x coordinate, breaking ties by the y coordinate.
• The leftmost point and the rightmost point must be on the convex hull. (P₁,P_n).
• Use this line to divide the data into two sets, the set below the line and the set above the line.
• If we are careful, we can partition the two sets to maintain order.
• Points on the line can be discarded.
• We then construct the "Upper" hull and "Lower" hull.
• To construct the "Upper" hull
  • Find the point P_max in the upper set furthest from the line (P₁,P_n).
  • P_max is on the convex hull.
  • Any points below P₁,P_max or P_maxP_n are not on the convex hull.
  • Repeat the process for points above P₁P_max and points above P_maxP_n
• We can sort in O(n lg (n))
• We can split in O(n) , using the cross product
  • |u × v| = |u||v| sin(θ)
• Calculating the furthest point is a constant (but expensive) time operation (See this page).
• So finding P_max is O(n) maximum. Probably O(n/2) or better.
• T(n)= 2T(n/2) + O(n) (Yes/No?)