Lecture 31
CSE 331
Nov 16, 2009
Jeff is out of town this week
No regular recitation or Jeff’s normal office hours
I’ll hold extra Question sessions
Mon, Tue 4-6pm
Bell 242
Email me if you
plan to come in
after 5:30pm
Divide and Conquer
Divide up the problem into at least two sub-problems
Solve all sub-problems: Mergesort
Recursively solve the sub-problems
Solve some sub-problems: Multiplication
Solve stronger sub-problems: Inversions
“Patch up” the solutions to the sub-problems for the final solution
Integer Multiplication
Input: a = (an-1,..,a0) and b = (bn-1,…,b0)
Output: c = a x b
a = 1101 b = 1001
c =1110101
a = a12n/2 + a0
a1 = 11 and a0 = 01
b = b12n/2 + b0
b1 = 10 and b0 = 01
c=
2n + (
+
)2n/2 +
First attempt
Shift by O(n) bits
Adding O(n) bit numbers
Mult over n
bits
ab=
2n + (
+
)2n/2 +
Multiplication over n/2 bit inputs
T(n) = 4T(n/2) + cn
T(1) = c
T(n) is O(n2)
The key identity
+
= (a1+a0)(b1+b0) -
-
The final algorithm
Input: a = (an-1,..,a0) and b = (bn-1,…,b0)
T(1) = c
Mult (a, b)
If n = 1 return a0b0
T(n) = 3T(n/2) + cn
a1 = an-1,…,an/2 and a0 = an/2-1,…, a0
Compute b1 and b0 from b
x = a1 a0 and y = b1 b0
O(nlog 3) = O(n1.59)
run time
Let p = Mult (x, y), D = Mult (a1, b1), E = Mult (a0, b0)
All
operations
are O(n) time
F=p D E
return D 2n F 2n/2 E
ab=
2n +( (a1+a0)(b1+b0) -
-
2n/2 +
(Old) Reading Assignment
Sec 5.2 of [KT]
Rankings
How close are two rankings?
Today’s agenda
Formal problem: Counting inversions
Divide and Conquer algorithm