Binary Planar Partition

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 -gt O(n)

5
Analysis

• Our objective is to calculate
• ?ni1?nj?i,j1 Prob(i cuts j)
• Now,
• ?nj?i Prob(i cuts j) (1/2 1/3 ..)
• 2 ln n
• And, ?ni1?nj?i,j1 Prob(i cuts j) 2 n ln n
• Thus,
• E of cuts 2 n ln n and,
• ETree Size O(nlogn)