Divide and Conquer

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Divide and Conquer

• The most well known algorithm design strategy
• Divide instance of problem into two or more
  smaller instances
• Solve smaller instances recursively
• Obtain solution to original (larger) instance by
  combining these solutions

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Divide-and-conquer technique
a problem of size n
subproblem 2 of size n/2
subproblem 1 of size n/2
a solution to subproblem 1
a solution to subproblem 2
a solution to the original problem
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Divide and Conquer - Closest pair

• Divide the set into two equal sized parts by the
  line l, and recursively compute the minimal
  distance in each part.
• Let d be the minimal of the two minimal
  distances.
• Eliminate points that lie farther than d apart
  from l
• Sort the remaining points according to their
  y-coordinates
• Scan the remaining points in the y order and
  compute the distances of each point to its five
  neighbors.
• If any of these distances is less than d then
  update d.
• Step 2 takes O(1) time
• Step 3 takes O(n) time
• Step 4 is a sort that takes O(nlogn) time
• Step 5 takes O(n) time (as we saw in the previous
  section)
• Step 6 takes O(1) time
• O((logn)nlogn)

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Closest pair - improved algorithm

• Step 1 Divide the set into..., and recursively
  compute the distance in each part, returning the
  points in each set in sorted order by
  y-coordinate.
• ....
• Step 4 Merge the two sorted lists into one
  sorted list in O(n) time.
• O(nlogn)