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
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

2. Divide-and-Conquer Technique (cont.)
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
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

3. Divide-and-Conquer Examples
Sorting: mergesort and quicksort
Binary tree traversals
Multiplication of large integers
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

4. General Divide-and-Conquer Recurrence
T(n) = aT(n/b) + f (n) where f(n)  (nd), d  0
Master Theorem: If a < bd, T(n)  (nd)
If a = bd, T(n)  (nd log n)
If a > bd, T(n)  (nlog 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) + n2  T(n)  ?
T(n) = 4T(n/2) + n3  T(n)  ?
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

5. Mergesort
Split array A[0..n-1] in two about equal halves and make copies of each half in arrays B and C
Sort arrays B and C recursively
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.
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

6. Pseudocode of Mergesort
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

7. Pseudocode of Merge
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

8. Mergesort Example
Go over this example in detail, then do another example of merging, something like:
( )
(3 4 6)
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

9. Analysis of Mergesort All cases have same efficiency: Θ(n log n)
Number of comparisons in the worst case is close to theoretical minimum for comparison-based sorting:
log2 n! ≈ n log2 n n
Space requirement: Θ(n) (not in-place)
Can be implemented without recursion (bottom-up)
even if not analyzing in detail, show the recurrence for mergesort in worst case:
T(n) = 2 T(n/2) + (n-1)
worst case comparisons for merge
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

10. Quicksort
Select a pivot (partitioning element) – here, the first element
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)
Exchange the pivot with the last element in the first (i.e., ) subarray — the pivot is now in its final position
Sort the two subarrays recursively p
A[i]p
A[i]p
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

11. Hoare’s Partitioning Algorithm
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

12. Quicksort Example
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

13. Analysis of Quicksort Best case: split in the middle — Θ(n log n)
Worst case: sorted array! — Θ(n2)
Average case: random arrays — Θ(n log n)
Improvements:
better pivot selection: median of three partitioning
switch to insertion sort on small subfiles
elimination of recursion
These combine to 20-25% improvement
Considered the method of choice for internal sorting of large files (n ≥ 10000)
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

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(Tleft) b c b c
print(root of T) d e   d e
Inorder(Tright)    
Efficiency: Θ(n)
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

15. Binary Tree Algorithms (cont.)
Ex. 2: Computing the height of a binary tree
h(T) = max{h(TL), h(TR)} + 1 if T   and h() = -1
Efficiency: Θ(n)
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

16. Multiplication of large integers
a , b are both n-digit integers
If we use the brute-force approach to compute c = a * b, what is the time efficiency?
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

17. 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:
a1 a2 … an b1 b2 … bn (d10) d11d12 … d1n
(d20) d21d22 … d2n
… … … … … … …
(dn0) dn1dn2 … dnn
Efficiency: n2 one-digit multiplications
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

18. Divide-and-Conquer Recursive Algorithm I
a = a1a0 and b = b1b0
c = a * b
= (a110n/2 + a0) * (b110n/2 + b0)
=(a1 * b1)10n + (a1 * b0 + a0 * b1)10n/2 + (a0 * b0)
For instance: a = , b = :
Then c = a * b = (123* )*(117* )
=(123 * 117)106 + (123 * * 117)103 + (456 * 933)
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

19. Divide-and-Conquer Recursive Algorithm I
a = a1a0 and b = b1b0
a, b: n-digit integers
a1, a0, b1, b0: n/2-digit integers
c = a * b
=(a1 * b1)10n + (a1 * b0 + a0 * b1)10n/2 + (a0 * b0)
How many single-digit multiplications are needed to compute a*b, if we ignore the shifting operations required for multiplying 10n and 10n/2?
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

20. Divide-and-Conquer Recursive Algorithm II
a = a1a0 and b = b1b0
c = a * b
=(a1 * b1)10n + (a1 * b0 + a0 * b1)10n/2 + (a0 * b0)
=c210n + c110n/2 + c0,
where
c2 = a1 * b1 is the product of their first halves
c0 = a0 * b0 is the product of their second halves
c1 = (a1 + a0) * (b1 + b0) – (c2 + c0) is the product of the sum of the a’s halves and the sum of the b’s halves minus the sum of c2 and c0.
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

21. Divide-and-Conquer Recursive Algorithm II
c =c210n + c110n/2 + c0,
where
c2 = a1 * b1
c0 = a0 * b0
c1 = (a1 + a0) * (b1 + b0) – (c2 + c0)
How many single-digit multiplications are needed to compute a*b, if we ignore the shifting operations required for multiplying 10n and 10n/2?
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

22. Divide-and-Conquer Recursive Algorithm II
c =c210n + c110n/2 + c0,
where
c2 = a1 * b1
c0 = a0 * b0
c1 = (a1 + a0) * (b1 + b0) – (c2 + c0)
Multiplication of n-digit numbers requires three multiplications of n/2-digit numbers
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

23. Divide-and-Conquer Recursive Algorithm II
M(n) = 3M(n/2) for n>1, M(1) = 1
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

24. Divide-and-Conquer Recursive Algorithm II
M(n) = 3M(n/2) for n>1, M(1) = 1
M(n)  n1.585
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

25. Example of Large-Integer Multiplication
2135  4014
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.