1. Divide and Conquer Strategy

2. General Concept of Divide & Conquer
Given a function to compute on n inputs, the divide-and-conquer strategy consists of:
splitting the inputs into k distinct subsets, 1kn, yielding k subproblems.
solving these subproblems
combining the subsolutions into solution of the whole.
if the subproblems are relatively large, then divide_Conquer is applied again.
if the subproblems are small, the are solved without splitting.

3. The Divide and Conquer Algorithm
Divide_Conquer(problem P) {
if Small(P) return S(P);
else {
divide P into smaller instances P1, P2, …, Pk, k1;
Apply Divide_Conquer to each of these subproblems;
return Combine(Divide_Conque(P1), Divide_Conque(P2),…, Divide_Conque(Pk)); }

4. Divde_Conquer recurrence relation
The computing time of Divde_Conquer is
T(n) is the time for Divde_Conquer on any input size n.
g(n) is the time to compute the answer directly (for small inputs)
f(n) is the time for dividing P and combining the solutions.
n small
otherwise

5. Three Steps of The Divide and Conquer Approach
The most well known algorithm design strategy:
Divide the problem into two or more smaller subproblems.
Conquer the subproblems by solving them recursively.
Combine the solutions to the subproblems into the solutions for the original problem.

6. A Typical Divide and Conquer Case
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

7. An Example: Calculating a0 + a1 + … + an-1
ALGORITHM RecursiveSum(A[0..n-1])
//Input: An array A[0..n-1] of orderable elements
//Output: the summation of the array elements
if n > 1
return ( RecursiveSum(A[0.. n/2 – 1]) + RecursiveSum(A[n/2 .. n– 1]) )
Efficiency: (for n = 2k)
A(n) = 2A(n/2) + 1, n >1
A(1) = 0;

8. General Divide and Conquer recurrence
The Master Theorem
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)
the time spent on solving a subproblem of size n/b.
the time spent on dividing the problem
into smaller ones and combining their solutions.
This is the same slide from chapter 2 (analysis of recursive algorithms)
It is a good idea to remind students at this point and do some examples especially
if you went over this quickly and did not do examples the previous time.
Here are examples from before:
T(n) = T(n/2) + 1
T(n) = 2T(n/2) + n
T(n) = 3 T(n/2) + n
=> this last example is important to mention at some point:
Θ(nlog23)
NB: cannot “eliminate” or simplify this constant

9. Divide and Conquer Other Examples
Sorting: mergesort and quicksort
Binary search
Matrix multiplication-Strassen’s algorithm

10. The Divide, Conquer and Combine Steps in Quicksort
Divide: Partition array A[l..r] into 2 subarrays, A[l..s-1] and A[s+1..r] such that each element of the first array is ≤A[s] and each element of the second array is ≥ A[s]. (computing the index of s is part of partition.)
Implication: A[s] will be in its final position in the sorted array.
Conquer: Sort the two subarrays A[l..s-1] and A[s+1..r] by recursive calls to quicksort
Combine: No work is needed, because A[s] is already in its correct place after the partition is done, and the two subarrays have been sorted.

11. Quicksort p A[i]≤p A[i] ≥ p
Select a pivot whose value we are going to divide the sublist. (e.g., p = A[l])
Rearrange the list so that it starts with the pivot followed by a ≤ sublist (a sublist whose elements are all smaller than or equal to the pivot) and a ≥ sublist (a sublist whose elements are all greater than or equal to the pivot ) (See algorithm Partition in section 4.2)
Exchange the pivot with the last element in the first sublist(i.e., ≤ sublist) – the pivot is now in its final position
Sort the two sublists recursively using quicksort. p
A[i]≤p
A[i] ≥ p

12. The Quicksort Algorithm
ALGORITHM Quicksort(A[l..r])
//Sorts a subarray by quicksort
//Input: A subarray A[l..r] of A[0..n-1],defined by its left and right indices l and r
//Output: The subarray A[l..r] sorted in nondecreasing order
if l < r
s  Partition (A[l..r]) // s is a split position
Quicksort(A[l..s-1])
Quicksort(A[s+1..r]

13. The Partition Algorithm
ALGORITHM Partition (A[l ..r])
//Partitions a subarray by using its first element as a pivot
//Input: A subarray A[l..r] of A[0..n-1], defined by its left and right indices l and r (l < r)
//Output: A partition of A[l..r], with the split position returned as this function’s value
P A[l]
i l; j  r + 1;
Repeat
repeat i  i + 1 until A[i]>=p //left-right scan
repeat j j – 1 until A[j] <= p//right-left scan
if (i < j) //need to continue with the scan
swap(A[i], a[j])
until i >= j //no need to scan
swap(A[l], A[j])
return j

14. Quicksort Example

15. Efficiency of Quicksort
Based on whether the partitioning is balanced.
Best case: split in the middle — Θ( n log n)
T(n) = 2T(n/2) + Θ(n) //2 subproblems of size n/2 each
Worst case: sorted array! — Θ( n2)
T(n) = T(n-1) + n+1 //2 subproblems of size 0 and n-1 respectively
Average case: random arrays — Θ( n log n)

16. Binary Search – an Iterative Algorithm
ALGORITHM BinarySearch(A[0..n-1], K)
//Implements nonrecursive binary search
//Input: An array A[0..n-1] sorted in ascending order and
// a search key K
//Output: An index of the array’s element that is equal to K
// or –1 if there is no such element
l  0, r  n – 1
while l  r do //l and r crosses over can’t find K.
m (l + r) / 2
if K = A[m] return m //the key is found
else
if K < A[m] r m – 1 //the key is on the left half of the array
else l  m + 1 // the key is on the right half of the array
return -1

17. Strassen’s Matrix Multiplication
Brute-force algorithm
c c a00 a b00 b01
= *
c c a10 a b10 b11
a00 * b00 + a01 * b10 a00 * b01 + a01 * b11 =
a10 * b00 + a11 * b10 a10 * b01 + a11 * b11
8 multiplications
Efficiency class:  (n3)
4 additions

18. Strassen’s Matrix Multiplication
Strassen’s Algorithm (two 2x2 matrices)
c c a00 a b00 b01
= *
c c a10 a b10 b11
m1 + m4 - m5 + m m3 + m5 =
m2 + m m1 + m3 - m2 + m6
m1 = (a00 + a11) * (b00 + b11)
m2 = (a10 + a11) * b00
m3 = a00 * (b01 - b11)
m4 = a11 * (b10 - b00)
m5 = (a00 + a01) * b11
m6 = (a10 - a00) * (b00 + b01)
m7 = (a01 - a11) * (b10 + b11)
7 multiplications
18 additions

19. Strassen’s Matrix Multiplication
Two nxn matrices, where n is a power of two (What about the situations where n is not a power of two?)
C C A00 A B00 B01
= *
C C A10 A B10 B11
M1 + M4 - M5 + M M3 + M5 =
M2 + M M1 + M3 - M2 + M6

20. Submatrices: M1 = (A00 + A11) * (B00 + B11) M2 = (A10 + A11) * B00
M3 = A00 * (B01 - B11)
M4 = A11 * (B10 - B00)
M5 = (A00 + A01) * B11
M6 = (A10 - A00) * (B00 + B01)
M7 = (A01 - A11) * (B10 + B11)

21. Efficiency of Strassen’s Algorithm
If n is not a power of 2, matrices can be padded with rows and columns with zeros
Number of multiplications
Number of additions
Other algorithms have improved this result, but are even more complex

22. Important Recurrence Types
Decrease-by-one recurrences
A decrease-by-one algorithm solves a problem by exploiting a relationship between a given instance of size n and a smaller size n – 1.
Example: n!
The recurrence equation for investigating the time efficiency of such algorithms typically has the form
T(n) = T(n-1) + f(n)
Decrease-by-a-constant-factor recurrences
A decrease-by-a-constant algorithm solves a problem by dividing its given instance of size n into several smaller instances of size n/b, solving each of them recursively, and then, if necessary, combining the solutions to the smaller instances into a solution to the given instance.
Example: binary search.
T(n) = aT(n/b) + f (n)

23. Decrease-by-one Recurrences
One (constant) operation reduces problem size by one.
T(n) = T(n-1) + c T(1) = d
Solution:
A pass through input reduces problem size by one.
T(n) = T(n-1) + c n T(1) = d
T(n) = (n-1)c + d linear
T(n) = [n(n+1)/2 – 1] c + d quadratic

24. Decrease-by-a-constant-factor recurrences – The Master Theorem
T(n) = aT(n/b) + f (n), where f (n) ∈ Θ(nk) , k>=0
a < bk T(n) ∈ Θ(nk)
a = bk T(n) ∈ Θ(nk log n )
a > bk T(n) ∈ Θ(nlog b a)
Examples:
T(n) = T(n/2) + 1
T(n) = 2T(n/2) + n
T(n) = 3 T(n/2) + n
Θ(logn)
Θ(nlogn)
Θ(nlog23)