CS 361 – Chapter 11

1. CS 361 – Chapter 11 • Divide and Conquer ! • Examples: • Merge sort √ • Ranking inversions • Matrix multiplication • Closest pair of points • Master theorem • A shortcut way of solving many, but not all, recurrences.

2. Finding inversions • Two judges have ranked n movies. We want to measure how similar their rankings are. • Approach: • First, arrange the movies in order of one of the judge’s rankings. • Look for numerical “inversions” in the second judge’s list. • Example

3. How to do it • We’re given (3, 1, 5, 2, 4). Compare with (1, 2, 3, 4, 5). • One obvious solution is O(n2). Look at all possible pairs of rankings and count those that are out of order. • For each # in list, see if subsequent values are smaller. This would be an inversion. • 3 > 1, 2 • 1 > [nothing] • 5 > 2, 4 • 2 > [nothing] • 4 > [nothing] • The answer is 4 inversions. • But we can devise an O(n log2 n) solution that operates like merge sort!

4. Algorithm • Split up the list in halves until you have singleton sets. • Our merge procedure will • Take as input 2 sorted lists, along with the number of inversions that appeared in each • Return a (sorted) merged list and the number of inversions • Singleton set has no inversions. Return 0. • How to combine: • Let i point to front/smallest value in A. • Let j point to front/smallest value in B. • Which is smaller, A[i] or B[j]? Remove smaller to C. • If it came from B, increment count by |A|. • When one list empty, you are essentially done. • Return # of inversions = count + A.count + B.count

5. Example • Count inversions in: 5, 4, 7, 3, 2, 6, 1, 8 • Along the way, we find that: • (5, 4) has 1 inversion • (7, 3) has 1 inversion • ((4, 5), (3, 7)) has 2 additional inversions, subtotal 1+1+2 = 4 • ((2, 6), (1, 8)) has 2 additional inversions, subtotal 0+0+2 = 2 • ((3, 4, 5, 7), (1, 2, 6, 8)) has 9 additional inversions, bringing our total to 4 + 2 + 9 = 15. • We can verify that there are in fact 15 inversions. • Interesting that we have an O(n log2 n) algorithm to find inversions, even though our answer can be as high as C(n, 2) which is O(n2). Sorting helped!

6. Matrix operations • Many numerical problems in science & engineering involve manipulating values in a 2-d arrays called matrices. • Let’s assume dimensions are n x n. • Adding and subtracting matrices is simple: just add corresponding values. O(n2). • Multiplication is more complicated. Here is example: A B C=AB • C[1,2] = A[1,1]*B[1,2] + A[1,2]*B[2,2], etc. • C[i,j] = sum from k=1 to n of A[i,k]*B[k,j]

7. Multiplication • Matrix multiplication can thus be done in O(n3) time, compared to O(n2) for addition and subtraction. • Actually, we are overstating the complexity, because usually “n” is size of input. The complexities of n2 and n3 here assume we have n2 of input. • But in the realm of linear algebra, “n” means how many rows/columns of data. • How can we devise a divide-and-conquer algorithm for matrix multiplication? • Partition the matrices into 4 quadrants each. • Multiply each quadrant independently.

8. Example Multiply quadrants this way: Compute the 8 products AE, BG, etc. and then combine results. 

9. Complexity • The divide-and-conquer approach for matrix multiplication gives us: T(n) = 8 T(n/2) + O(n2). • This works out to a total of O(n3). • No better than classic definition of matrix multiplication. • But, very useful in parallel computation! • Strassen’s algorithm: By doing some messy matrix algebra, it’s possible to compute the n*n product with only 7 quadrant multiplications instead of 8. • T(n) = 7 T(n/2) + O(n2) implies T(n)  O(n log 2 7). • Further optimizations exist. Conjecture: matrix multiplication is only slightly over O(n2).

10. Closest pair • A divide-and-conquer problem. • Given a list of (x,y) points, find which 2 points are closest to each other. • (First, think about how you’d do it in 1 dimension.) • Divide & conquer: repeatedly divide the points into left and right halves. Once you only have a set of 2 or 3 points, finding the closest is easy. • Convenient to have list of points sorted by x & by y. • Combine is a little tricky because it’s possible that the closest pair may straddle the dividing line between left and right halves.

11. Combining solutions • Given a list of points P, we divided this into a “left” half Q and a right half R. • Thru divide and conquer we now know the closest pair in Q [q1,q2] and closest pair in R [r1,r2], and we can determine which is better. • Let  = min(dist(q1,q2),dist(r1,r2)) • But, there might be a pair of points [q,r] with q  Q and r  R whose dist(q,r) < . How would we find this mysterious pair of points? • These 2 points must be within  distance of the vertical line passing thru the rightmost point in Q. Do you see why?

12. Boundary case • Let’s call S the list of points within  horizontal (i.e. “x”) distance of the dividing line L. In effect, a thin vertical strip of territory. • We can restate our earlier observation as follows: There are two points q  Q and r  R with dist(q,r) <   there are two points s1, s2  S with dist(s1,s2) < . • We can restrict our search for the boundary case even more. • If we sort S by y-coordinate, then s1,s2 are within 15 positions of each other on the list. • This means that searching for the 2 closest points in S can be done in O(n) time, even though it’s a nested loop. One loop is bounded by 15 iterations.

13. Why 15? • Take a closer look at the boundary region. • Let Z be the portion of the plane within  x-distance from L. Partition Z into squares of side length /2. We will have many rows of squares, with 4 squares per row. • Each square contains at most 1 point from S. Why? • Suppose one square contains 2 points from S. Both are either in Q or in R, so their distance is at least . But the diagonal of the square is not this long. Contradiction. • s1 and s2 are within 15 positions in S sorted by y. Why? • Suppose separated by 16 or more. Then, there have to be at least 3 rows of squares separating them, so their distance apart must be at least 3/2. However, we had chosen s1 and s2 because their distance was < . Contradiction.

15. Algorithm closestPair(list of points P): px and py are P sorted by x and y respectively if |px| <= 3, return the simple base case solution Create Q and R: the left & right halves of P. closestQ = closestPair(Q) closestR = closestPair(R) delta = min(dist(closestQ), dist(closestR)) S = points in P within delta of rightmost Q sy = S sorted by y for each s in sy: find dist from s to each of next 15 in sy let [s,s’] be this shortest pair in sy return closest of: closestQ, closestR, or [s,s’]

16. Master theorem • What is the complexity of an algorithm if we have this type of recurrence: T(n) = a T(n / b) + f(n) • For example, splitting a problem into “a” instances of the problem, each with size 1/b of the original. • 3 cases, basically comparing n log b a with f(n) to see which is “larger”:

17. How to use • If we are given T(n) = a T(n / b) + f(n) • Compare n log b a with f(n) • Case 1: If n log b a is larger, then T(n) = O(n log b a ). • Case 2: If equal, then T(n) = O(f(n) log2 n). Incidentally, if f(n) is exactly (log2 n)ktimes n log b a T(n) = O(f(n) (log2 n)k+1). • Case 3: If f(n) is larger, then T(n) = O(f(n)). But need to verify that a f(n/b) < f(n) or else master theorem can’t guarantee answer.

18. Examples • T(n) = 9 T(n/3) + n • T(n) = T((2/3) n) + 1 • T(n) = 3 T(n/4) + n log2 n • T(n) = 4 T(n/2) + n • T(n) = 4 T(n/2) + n2 • T(n) = 4 T(n/2) + n3 • T(n) = 2 T(n/2) + n3 • T(n) = T(9n/10) + n • T(n) = 16 T(n/4) + n2 • T(n) = 7 T(n/3) + n2 • T(n) = 7 T(n/2) + n2 • T(n) = 2 T(n/4) + n0.5 • T(n) = 3 T(n/2) + n log2 n • T(n) = 2 T(n/2) + n/ log2n • T(n) = 2 T(n/2) + n log2n • T(n) = 8 T(n/2) + n3(log2n)5

19. Var substitution • Sometimes the recurrence is not expressed as a fraction of n. (See bottom of page 311.) • Example: T(n) = T(n0.5) + 1 • Let x = log2 n. This means that n = 2x and n0.5 = 2 x/2 • T(n) formula becomes T(2x) = T(2x/2) + 1 • If you don’t like powers of 2 inside T, temporarily define similar function S(x) = T(2x). • Now it is: S(x) = S(x/2) + 1. We know how to solve it. • S(x) = O(log2 x). • Thus, T(2 x) = O(log2 x). Replace x with log2 n. • Final answer: T(n) = O(log2 (log2 n)).