Divide and Conquer

1. 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

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

3. Divide and Conquer Examples • Sorting: mergesort and quicksort • Tree traversals • Binary search • Matrix multiplication-Strassen’s algorithm • Convex hull-QuickHull algorithm

4. General Divide and Conquer recurrence: T(n) = aT(n/b) + f (n)where f (n)=Θ(nk) • a < bk T(n) =Θ(nk) • a = bk T(n) =Θ(nk lg n ) • a > bk T(n) =Θ(nlog b a)

5. Mergesort Algorithm: • Split array A[1..n] in two and make copies of each half in arrays B[1.. n/2 ] and C[1.. n/2 ] • Sort arrays B and C • Merge sorted arrays B and C into array A

6. Mergesort Algorithm: • Merge sorted arrays B and C into array A as follows: • Repeat the following until no elements remain in one of the arrays: • compare the first elements in the remaining unprocessed portions of the arrays • copy the smaller of the two into A, while incrementing the index indicating the unprocessed portion of that array • Once all elements in one of the arrays are processed, copy the remaining unprocessed elements from the other array into A.

7. How Merging Works A: 7 2 1 6 4 9 5 B: 7 2 1 6 C: 4 9 5 Split The List B: 1 2 6 7 C: 4 5 9 Sort Each List Merge the Lists B: 1 2 4 5 6 7 9

8. Putting it Together A: 7 2 1 6 4 9 5 B: 7 2 1 6 C: 4 9 5 Split The List B: 1 2 6 7 C: 4 5 9 Sort Each List Each List is sorted by recursively applying mergesort to the sub-lists D: 7 2 E: 1 6 F: 4 9 G: 5 Split Again H: 7 I: 2 J: 1 K: 6 L: 4 M: 9 N: 5

9. Efficiency of mergesort • All cases have same time efficiency: Θ( n log n) • Number of comparisons is close to theoretical minimum for comparison-based sorting: log (n !) ≈ n log n - 1.44 n • Space requirement: Θ( n ) (NOT in-place) • Can be implemented without recursion (bottom-up)

10. p A[i]≤p A[i]>p Quicksort • Select a pivot (partitioning element) • Rearrange the list so that all the elements in the positions before the pivot are smaller than or equal to the pivot and those after the pivot are larger than the pivot (See algorithm Partition in section 4.2) • Exchange the pivot with the last element in the first (i.e., ≤ sublist) – the pivot is now in its final position • Sort the two sublists

11. The partition algorithm Example: 8 1 12 2 6 10 14 15 4 13 9 11 3 7 5

12. Efficiency of quicksort • Best case: split in the middleΘ( n log n) • Worst case: sorted array!Θ( n2) • Average case: random arraysΘ( n log n)

13. Efficiency of quicksort • Improvements: • better pivot selection: median of three partitioning avoids worst case in sorted files • switch to insertion sort on small subfiles • elimination of recursion these combine to 20-25% improvement • Considered the method of choice for internal sorting for large files (n ≥ 10000)

14. QuickHull Algorithm Inspired by Quicksort compute Convex Hull: • Assume points are sorted by x-coordinate values • Identify extreme points P1 and P2 (part of hull) P2 P1

15. QuickHull Algorithm • Compute upper hull: • find point Pmax that is farthest away from line P1P2 • compute the hull to the left of line P1Pmax • compute the hull to the right of line P2Pmax • Compute lower hull in a similar manner Pmax P2 P1

16. 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: Θ( n2) • Average case: Θ( n log n)

17. Efficiency of QuickHull algorithm • If points are not initially sorted by x-coordinate value, this can be accomplished in Θ( n log n) — no increase in asymptotic efficiency class • Other algorithms for convex hull: • Graham’s scan • DCHull also in Θ( n log n)

18. Closest-Pair Problem: Divide and Conquer • Brute force approach requires comparing every point with every other point • Given n points, we must perform 1 + 2 + 3 + … + n-2 + n-1 comparisons. • Brute force  O(n2) • The Divide and Conquer algorithm yields  O(n log n) • Reminder: if n = 1,000,000 then • n2 = 1,000,000,000,000 whereas • n log n = 20,000,000

19. Closest-Pair Algorithm Given: A set of points in 2-D

20. Closest-Pair Algorithm Step 1: Sort the points in one D

21. Closest-Pair Algorithm Lets sort based on the X-axis O(n log n) using quicksort or mergesort 4 9 13 6 2 14 11 7 3 10 5 1 8 12

22. Closest-Pair Algorithm Step 2: Split the points, i.e., Draw a line at the mid-point between 7 and 8 4 9 13 6 2 14 11 7 3 10 5 1 8 12 Sub-Problem 1 Sub-Problem 2

23. Closest-Pair Algorithm Advantage: Normally, we’d have to compare each of the 14 points with every other point. (n-1)n/2 = 13*14/2 = 91 comparisons 4 9 13 6 2 14 11 7 3 10 5 1 8 12 Sub-Problem 1 Sub-Problem 2

24. Closest-Pair Algorithm Advantage: Now, we have two sub-problems of half the size. Thus, we have to do 6*7/2 comparisons twice, which is 42 comparisons solution d = min(d1, d2) 4 9 13 d1 6 2 d2 14 11 7 3 10 5 1 8 12 Sub-Problem 1 Sub-Problem 2

25. Closest-Pair Algorithm Advantage: With just one split we cut the number of comparisons in half. Obviously, we gain an even greater advantage if we split the sub-problems. d = min(d1, d2) 4 9 13 d1 6 2 d2 14 11 7 3 10 5 1 8 12 Sub-Problem 1 Sub-Problem 2

26. Closest-Pair Algorithm Problem: However, what if the closest two points are each from different sub-problems? 4 9 13 d1 6 2 d2 14 11 7 3 10 5 1 8 12 Sub-Problem 1 Sub-Problem 2

27. Closest-Pair Algorithm Here is an example where we have to compare points from sub-problem 1 to the points in sub-problem 2. 4 13 d1 6 2 d2 9 14 11 7 3 10 8 5 1 12 Sub-Problem 1 Sub-Problem 2

28. Closest-Pair Algorithm However, we only have to compare points inside the following “strip.” d = min(d1, d2) 4 9 d d 13 d1 6 2 d2 14 11 7 3 10 5 1 8 12 Sub-Problem 1 Sub-Problem 2

29. Closest-Pair Algorithm Step 3: But, we can continue the advantage by splitting the sub-problems. 4 9 13 6 2 14 11 7 3 10 5 1 8 12

30. Closest-Pair Algorithm Step 3: In fact we can continue to split until each sub-problem is trivial, i.e., takes one comparison. 4 9 13 6 2 14 11 7 3 10 5 1 8 12

31. Closest-Pair Algorithm Finally: The solution to each sub-problem is combined until the final solution is obtained 4 9 13 6 2 14 11 7 3 10 5 1 8 12

32. Closest-Pair Algorithm Finally: On the last step the ‘strip’ will likely be very small. Thus, combining the two largest sub-problems won’t require much work. 4 9 13 6 2 14 11 7 3 10 5 1 8 12

33. Closest-Pair Algorithm • In this example, it takes 22 comparisons to find the closets-pair. • The brute force algorithm would have taken 91 comparisons. • But, the real advantage occurs when there are millions of points. 4 9 13 6 2 14 11 7 3 10 5 1 8 12

34. Closest-Pair Problem: Divide and Conquer • Here is another animation: • http://www.cs.mcgill.ca/~cs251/ClosestPair/ClosestPairApplet/ClosestPairApplet.html

35. Remember • Homework is due on Friday • In class and on paper • There is a talk today: • 4PM RB 340 or 328 • Darren Lim (faculty candidate) • Bioinformatics: Secondary Structure Prediction in Proteins

36. Long Term • HW7 will be given on Friday. • It’ll be due on Wed. • It’ll be returned on Friday. • Exam2 will be on Monday the 22nd