1. Lecture 30
CSE 331
Nov 12, 2012

2. Another feedback
Writing up the solutions takes disproportionately long!
Yes, writing well is very hard to do

3. Mergesort algorithm Input: a1, a2, …, an
Output: Numbers in sorted order
MergeSort( a, n )
If n = 1 return the order a1
aL = a1,…, a[n/2]
aR = a[n/2]+1,…, an
return MERGE ( MergeSort(aL, [n/2]), MergeSort(aR, n-[n/2]) )
If n = 2 return the order min(a1,a2); max(a1,a2)

4. Inductive step follows from correctness of MERGE
Input: a1, a2, …, an
Output: Numbers in sorted order
By induction on n
MergeSort( a, n )
If n = 2 return the order min(a1,a2); max(a1,a2)
aL = a1,…, a[n/2]
aR = a[n/2]+1,…, an
return MERGE ( MergeSort(aL, [n/2]), MergeSort(aR, n-[n/2]) )
If n = 1 return the order a1
Inductive step follows from correctness of MERGE

5. Run time recurrence
T(n) ≤ c if n ≤ 2
2*T(n/2) + c*n otherwise

6. Today’s agenda
Solve the recurrence
Multiplying two integers

7. Divide and Conquer
Divide up the problem into at least two sub-problems
Recursively solve the sub-problems
“Patch up” the solutions to the sub-problems for the final solution

8. Improvements on a smaller scale
Greedy algorithms: exponential  poly time
(Typical) Divide and Conquer: O(n2)  asymptotically smaller running time

9. Multiplying two numbers
Given two numbers a and b in binary
a=(an-1,..,a0) and b = (bn-1,…,b0)
Running time of primary school algorithm?
Compute c = a x b

10. The current algorithm scheme
Shift by O(n) bits
Adding O(n) bit numbers
Mult over n bits
a  b = a1b1 22[n/2] + (a1b0+a0b1)2[n/2] + a0b0
Multiplication over n/2 bit inputs
T(n) is O(n2)
T(n) ≤ 4T(n/2) + cn
T(1) ≤ c