1 Divide-and-Conquer The most-well known algorithm design strategy: 1. Divide instance of problem into two or more smaller instances 2. Solve smaller instances recursively 3. Obtain solution to original (larger) instance by combining these solutions

2 Divide-and-Conquer Technique (cont.) subproblem 2 of size n/2 subproblem 1 of size n/2 a solution to subproblem 1 a solution to the original problem a solution to subproblem 2 a problem of size n

3 Divide-and-Conquer Examples b Sorting: mergesort and quicksort b Binary tree traversals b Multiplication of large integers b Matrix multiplication: Strassen’s algorithm b Closest-pair and convex-hull algorithms b Binary search: decrease-by-half (or degenerate divide&conq.)

4 General Divide-and-Conquer Recurrence T(n) = aT(n/b) + f (n) where f(n)   (n d ), d  0 Master Theorem: If a < b d, T(n)   (n d ) If a = b d, T(n)   (n d log n) If a = b d, T(n)   (n d log n) If a > b d, T(n)   ( n log b a ) If a > b d, T(n)   ( n log b a ) Note: The same results hold with O instead of . Examples: T(n) = 4T(n/2) + n  T(n)  ? T(n) = 4T(n/2) + n 2  T(n)  ? T(n) = 4T(n/2) + n 2  T(n)  ? T(n) = 4T(n/2) + n 3  T(n)  ? T(n) = 4T(n/2) + n 3  T(n)  ?

5 Mergesort b Split array A[0..n-1] in two about equal halves and make copies of each half in arrays B and C b Sort arrays B and C recursively b Merge sorted arrays B and C into array A as follows: Repeat the following until no elements remain in one of the arrays: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.Once all elements in one of the arrays are processed, copy the remaining unprocessed elements from the other array into A.

6 Pseudocode of Mergesort

7 Pseudocode of Merge

8 Mergesort Example

9 Analysis of Mergesort b All cases have same efficiency: Θ(n log n) b Number of comparisons in the worst case is close to theoretical minimum for comparison-based sorting:  log 2 n!  ≈ n log 2 n n  log 2 n!  ≈ n log 2 n n b Space requirement: Θ(n) (not in-place) b Can be implemented without recursion (bottom-up)

10 Quicksort b Select a pivot (partitioning element) – here, the first element b Rearrange the list so that all the elements in the first s positions are smaller than or equal to the pivot and all the elements in the remaining n-s positions are larger than or equal to the pivot (see next slide for an algorithm) b Exchange the pivot with the last element in the first (i.e.,  ) subarray — the pivot is now in its final position b Sort the two subarrays recursively p A[i]  p A[i]  p

11 Hoare’s Partitioning Algorithm

12 Quicksort Example

13 Analysis of Quicksort b Best case: split in the middle — Θ(n log n) b Worst case: sorted array! — Θ(n 2 ) b Average case: random arrays — Θ(n log n) b Improvements: better pivot selection: median of three partitioningbetter pivot selection: median of three partitioning switch to insertion sort on small subfilesswitch to insertion sort on small subfiles elimination of recursionelimination of recursion These combine to 20-25% improvement b Considered the method of choice for internal sorting of large files (n ≥ 10000)

14 Binary Tree Algorithms Binary tree is a divide-and-conquer ready structure! Ex. 1: Classic traversals (preorder, inorder, postorder) Algorithm Inorder(T) if T   a a Inorder(T left ) b c b c Inorder(T left ) b c b c print(root of T) d e d e print(root of T) d e d e Inorder(T right ) Inorder(T right ) Efficiency: Θ(n)

15 Binary Tree Algorithms (cont.) Ex. 2: Computing the height of a binary tree h(T) = max{h(T L ), h(T R )} + 1 if T  and h(  ) = -1 h(T) = max{h(T L ), h(T R )} + 1 if T   and h(  ) = -1 Efficiency: Θ(n)

16 Multiplication of Large Integers Consider the problem of multiplying two (large) n-digit integers represented by arrays of their digits such as: A = B = The grade-school algorithm: a 1 a 2 … a n b 1 b 2 … b n (d 10 ) d 11 d 12 … d 1n (d 20 ) d 21 d 22 … d 2n (d 20 ) d 21 d 22 … d 2n … … … … … … … … … … … … … … (d n0 ) d n1 d n2 … d nn Efficiency: n 2 one-digit multiplications Efficiency: n 2 one-digit multiplications

17 First Divide-and-Conquer Algorithm A small example: A  B where A = 2135 and B = 4014 A = (21· ), B = (40 · ) So, A  B = (21 · )  (40 · ) = 21  40 · (21   40) ·  14 = 21  40 · (21   40) ·  14 In general, if A = A 1 A 2 and B = B 1 B 2 (where A and B are n-digit, A 1, A 2, B 1, B 2 are n/2-digit numbers), A  B = A 1  B 1 ·10 n + (A 1  B 2 + A 2  B 1 ) ·10 n/2 + A 2  B 2 Recurrence for the number of one-digit multiplications M(n): M(n) = 4M(n/2), M(1) = 1 Solution: M(n) = n 2 M(n) = 4M(n/2), M(1) = 1 Solution: M(n) = n 2

18 Second Divide-and-Conquer Algorithm A  B = A 1  B 1 ·10 n + (A 1  B 2 + A 2  B 1 ) ·10 n/2 + A 2  B 2 The idea is to decrease the number of multiplications from 4 to 3: (A 1 + A 2 )  (B 1 + B 2 ) = A 1  B 1 + (A 1  B 2 + A 2  B 1 ) + A 2  B 2, I.e., (A 1  B 2 + A 2  B 1 ) = (A 1 + A 2 )  (B 1 + B 2 ) - A 1  B 1 - A 2  B 2, which requires only 3 multiplications at the expense of (4-1) extra add/sub. (A 1 + A 2 )  (B 1 + B 2 ) = A 1  B 1 + (A 1  B 2 + A 2  B 1 ) + A 2  B 2, I.e., (A 1  B 2 + A 2  B 1 ) = (A 1 + A 2 )  (B 1 + B 2 ) - A 1  B 1 - A 2  B 2, which requires only 3 multiplications at the expense of (4-1) extra add/sub. Recurrence for the number of multiplications M(n): M(n) = 3M(n/2), M(1) = 1 Solution: M(n) = 3 log 2 n = n log 2 3 ≈ n 1.585

19 Example of Large-Integer Multiplication 2135  4014

20 Strassen’s Matrix Multiplication Strassen observed [1969] that the product of two matrices can be computed as follows: Strassen observed [1969] that the product of two matrices can be computed as follows: C 00 C 01 A 00 A 01 B 00 B 01 = * = * C 10 C 11 A 10 A 11 B 10 B 11 M 1 + M 4 - M 5 + M 7 M 3 + M 5 M 1 + M 4 - M 5 + M 7 M 3 + M 5 = M 2 + M 4 M 1 + M 3 - M 2 + M 6 M 2 + M 4 M 1 + M 3 - M 2 + M 6

21 Formulas for Strassen’s Algorithm M 1 = (A 00 + A 11 )  (B 00 + B 11 ) M 2 = (A 10 + A 11 )  B 00 M 3 = A 00  (B 01 - B 11 ) M 4 = A 11  (B 10 - B 00 ) M 5 = (A 00 + A 01 )  B 11 M 6 = (A 10 - A 00 )  (B 00 + B 01 ) M 7 = (A 01 - A 11 )  (B 10 + B 11 )

22 Analysis of Strassen’s Algorithm If n is not a power of 2, matrices can be padded with zeros. Number of multiplications: M(n) = 7M(n/2), M(1) = 1 M(n) = 7M(n/2), M(1) = 1 Solution: M(n) = 7 log 2 n = n log 2 7 ≈ n vs. n 3 of brute-force alg. Algorithms with better asymptotic efficiency are known but they are even more complex.

23 Closest-Pair Problem by Divide-and-Conquer Step 1 Divide the points given into two subsets P l and P r by a vertical line x = m so that half the points lie to the left or on the line and half the points lie to the right or on the line.

24 Closest Pair by Divide-and-Conquer (cont.) Step 2 Find recursively the closest pairs for the left and right subsets. Step 3 Set d = min{d l, d r } We can limit our attention to the points in the symmetric vertical strip S of width 2d as possible closest pair. (The points are stored and processed in increasing order of their y coordinates.) We can limit our attention to the points in the symmetric vertical strip S of width 2d as possible closest pair. (The points are stored and processed in increasing order of their y coordinates.) Step 4 Scan the points in the vertical strip S from the lowest up. For every point p(x,y) in the strip, inspect points in in the strip that may be closer to p than d. There can be no more than 5 such points following p on the strip list!

25 Efficiency of the Closest-Pair Algorithm Running time of the algorithm is described by T(n) = 2T(n/2) + M(n), where M(n)  O(n) T(n) = 2T(n/2) + M(n), where M(n)  O(n) By the Master Theorem (with a = 2, b = 2, d = 1) T(n)  O(n log n) T(n)  O(n log n)

26 Quickhull Algorithm Convex hull: smallest convex set that includes given points b Assume points are sorted by x-coordinate values b Identify extreme points P 1 and P 2 (leftmost and rightmost) b Compute upper hull recursively: find point P max that is farthest away from line P 1 P 2find point P max that is farthest away from line P 1 P 2 compute the upper hull of the points to the left of line P 1 P maxcompute the upper hull of the points to the left of line P 1 P max compute the upper hull of the points to the left of line P max P 2compute the upper hull of the points to the left of line P max P 2 b Compute lower hull in a similar manner P1P1 P2P2 P max

27 Efficiency of Quickhull Algorithm b Finding point farthest away from line P 1 P 2 can be done in linear time b Time efficiency: worst case: Θ(n 2 ) (as quicksort)worst case: Θ(n 2 ) (as quicksort) average case: Θ(n) (under reasonable assumptions about distribution of points given)average case: Θ(n) (under reasonable assumptions about distribution of points given) b If points are not initially sorted by x-coordinate value, this can be accomplished in O(n log n) time b Several O(n log n) algorithms for convex hull are known