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Binary Planar Partition

Binary Planar Partition. Lecture 2 Advanced Algorithms II. Slides by: Arjun Dasgupta. Slides by: Arjun Dasgupta. Example 1. l1. 1. 3. l2. l3. 2. 4. Each oval node stores information about the infinite line l i The leaves denote the line segments being partitioned. Example 2. l1.

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Binary Planar Partition

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  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)

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