1. Binary Planar Partition Lecture 2 Advanced Algorithms II Slides by: Arjun Dasgupta Slides by: Arjun Dasgupta

2. Example 1 l1 1 3 l2 l3 2 4 • Each oval node stores information about the infinite line li • The leaves denote the line segments being partitioned

3. Example 2 l1 1 l2 l5 3 2 l3 4 l4 • Smallest Tree that can be created from the partitions is O(n)

4. Auto-Partition Algorithm • Index(u , v) = # of cuts that u makes when extended to v • Algorithm: • Input: S = {S1,S2, …. Sn} • Generate a random permutation of S • U = {u1,u2,…..un} • Start constructing the tree by using the segments in this order as partitioning lines • Upper Bound of the size of tree created by Auto-Partition -> O(n)

5. Analysis • Our objective is to calculate ∑ni=1∑nj≠i,j=1Prob(i cuts j) • Now, ∑nj≠iProb(i cuts j) ≤ (1/2 + 1/3 + …..) ≤ 2 ln n And, ∑ni=1∑nj≠i,j=1Prob(i cuts j) ≤ 2 n ln n • Thus, E[# of cuts] ≤ 2 n ln n and, E[Tree Size] = O(nlogn)