ClosestPair Problem by Divide and Conquer

1
Closest-Pair Problem by Divide and Conquer

• Brute-force algorithm (in Chapter 3) takes
  time because the distance between all
  possible pair of points are computed and compared
• Divide-and-conquer strategy can be used to
  achieve more efficient algorithm
• Given n points
• Assume that they have been sorted in ascending
  order of their x coordinate. Sorting complexity
  (mergesort)
• Then we divide n points into two subsets S1 and
  S2 of n/2 points by a vertical line xc. All the
  points in S1 are located to the left of or on
  this line and all the points in S2 are located to
  the right or on this line. Complexity in finding
  this line

2
Recursive Algorithm

• Find the closest pair in S1 and S2 separately,
  denoting the closest distance in S1 and S2 are d1
  and d2.
• However, mind1,d2 is not the desired distance
  between the closest pair among all n points, why?
• Closest pair may be located in different side of
  the partitioning line!
• How to solve this problem?

3
Solution

• Suppose these n points are also sorted according
  to their y-coordinate. This can be done in
  if S1 and S2 were y-sorted.
• Then we can construct the y-sorted version of
  points in C1 and C2 (two strips shown in the
  above figure) in
• Merge y-sorted version of points in C1 and C2
  into a single sorted list, check each elements
  distance to constant of its neighboring
  elements in
• If there is any such distance is less than d,
  update the shortest distance and the closest pair
• Recursive equation is

4
Convex-Hull Problem

• Find the smallest convex polygon that contains n
  points in a plane. Brute-force convex-hull
  algorithm
• Divide and Conquer quickhull
• Given n points
  sorted in
  increasing order of their x-coordinates with ties
  resolved by increasing order of y coordinates.
• First, leftmost point and rightmost point
  must belong to the convex hull
• Connecting and , we have a vector
  , which partitions the plane into two
  semi-planes and the remaining points into left
  point sets and right point sets.

5
Quickhull

• The upper/lower parts of the convex hull are
  called upper/lower hull, which can be constructed
  in the similar fashion (recursive algorithm)

6
QuickHull

• Compute upper hull
• find point Pmax that is farthest away from line
  P1P2
• compute the hull of the points to the left of
  line P1Pmax
• compute the hull of the points to the left of
  line PmaxP2
• Compute lower hull in a similar manner

Pmax
P2
P1
7
Efficiency of QuickHull algorithm

• Finding point farthest away from line P1P2 can be
  done in linear time
• This gives same efficiency as quicksort
• Worst case T( n2)
• Average case T( n log n)
• If points are not initially sorted by
  x-coordinate value, this can be accomplished in
  T( n log n) no increase in asymptotic
  efficiency class
• Other algorithms for convex hull
• Grahams scan
• DCHull
• also in T( n log n)