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06 - 20 Jan - Divide and Conquer

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Neeldhara Misra

The document discusses algorithms for finding the closest pair of points from a set of n points given as (x,y) coordinates. It presents three approaches: computing all pairwise distances which is O(n^2); sorting the points and comparing adjacent pairs which is the time to sort plus O(n); and divide and conquer which splits the points, solves subproblems recursively, and merges in O(n log n) time.

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CS 321. Algorithm Analysis & Design Lecture 6 Divide and Conquer
Closest Pair
06 - 20 Jan - Divide and Conquer
Input: A set of n points given in terms of (x,y) coordinates.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
Compute distances between all pairs. Approach #0 O(n2)
Sort the points. Sweep through adjacent pairs. Approach 1 Time to sort + O(n)
Divide and Conquer. Split the point set, use recursive solutions. Approach 2 Cost of Recursion + Time to merge
Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x) coordinates. Output: The closest pair. Closest Pair on Left - d1 Closest Pair on Right - d2
Input: A set of n points given in terms of (x) coordinates. Output: The closest pair. Closest Pair on Left - d1 Closest Pair on Right - d2
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. 1 2 3 4 5 6 7 8 9 10 11 12 13
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. 13 1 7 5 3 9 8 11 10 4 12 2 6
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. d/2 d/2
Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. d/2 d/2
06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and Conquer
T(n) ≤ 2T(n/2) + O(n) T(n) = O(n log n)

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06 - 20 Jan - Divide and Conquer

  • 1. CS 321. Algorithm Analysis & Design Lecture 6 Divide and Conquer
  • 4. Input: A set of n points given in terms of (x,y) coordinates.
  • 5. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 6. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
  • 7. Compute distances between all pairs. Approach #0 O(n2)
  • 8. Sort the points. Sweep through adjacent pairs. Approach 1 Time to sort + O(n)
  • 9. Divide and Conquer. Split the point set, use recursive solutions. Approach 2 Cost of Recursion + Time to merge
  • 10. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
  • 11. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
  • 12. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair. Closest Pair on Left - d1 Closest Pair on Right - d2
  • 13. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair. Closest Pair on Left - d1 Closest Pair on Right - d2
  • 14. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 15. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. 1 2 3 4 5 6 7 8 9 10 11 12 13
  • 16. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 17. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. 13 1 7 5 3 9 8 11 10 4 12 2 6
  • 18. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 19. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 20. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 21. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 22. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
  • 23. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. d/2 d/2
  • 24. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. d/2 d/2
  • 32. T(n) ≤ 2T(n/2) + O(n) T(n) = O(n log n)