analysis of algorithms

1. Divide and Conquer
Analysis of Algorithms

2. Master Theorem
The solution to the equation
T(n) = aT(n/b) + θ(nk
), where a >= 1 and b >1 is
Ө(nlog
b
a
) if a > bk
T(n) = Ө(nk
logn) if a = bk
Ө(nk
) if a < bk

3. Example
Solve the following recurrence using Master Theorem
T(n) = 8T(n/2) + O(n2
)
a = 8, b = 2, k = 2
⇒a > bk
, 8 > 22
)()()( 38lg2
nOnOnT ==⇒

4. Matrix Multiplication
procedure stmul(n,A,B,C)
if n>=1
partition A into A11,A12,A21,A22
partition B into B11,B12,B21,B22
c=AxB
stmul(n/2,A11,B11,m1)
stmul(n/2,A12,B21,m2)
stmul(n/2,A11,B12,m3)
stmul(n/2,A12,B22,m4)
stmul(n/2,A21,B11,m5)
stmul(n/2,A22,B21,m6)
stmul(n/2,A21,B12,m7)
stmul(n/2,A22,B22,m8)

5. c11=m1+m2, c12=m5+m6
c21=m3+m4, c22=m7+m8
Function Calls
m1=A11B11, m2=A12B21, m3=A11B12
m4=A12B22, m5=A21B11, m6=A22B21
m7=A21B12, m8=A22B22

6. Analysis, Multiplications
T(n)=8T(n/2)
T(n/2)=8T(n/4)
T(n/4)=8T(n/8)
By backward substitution
T(n)=82
T(n/4)
=83
T(n/8)
=8i
T(n/2i
)
=8 i
T(n/2i
)
let 2i
=n
T(n)=8lgn
=nlg8
=n^3

7. Additions
t(n)=8t(n/2)+4(n/2)2
a=8, b=2, k=2
by applying master theorem
T(n)= n log
b
a
= nlg8
=n3
єӨ(n3
)

8. Strassen’s Multiplication
C11 C12 A11 A12 B11 B12
= *
C21 C22 A21 A22 B21 B22
m1=A11(B12-B22)
m2=(A11+A12)B22
m3=(A21+A22)B11
m4=A22(B21-B11)
m5=(A11+A22)(B11+B22)
m6=(A12-A22)(B21+B22)
m7=(A11-A21)(B11+B12)
C11=m5+m4-m2+m6, C21=m3+m4
C12=m1+m2 C22=m1+m5-m3-m7

9. Complexity(multiplication, addition&
Subtraction)
t(n)=7t(n/2)+kn 2
a=7,b=2,d=2
byapplyingmastertheorem
T(n)= n log
b
a
= nlg7
=n2.8
єӨ(n2.8
)

10. Complexity(multiplication, addition&
Subtraction)
t(n)=7t(n/2)+kn 2
a=7,b=2,d=2
byapplyingmastertheorem
T(n)= n log
b
a
= nlg7
=n2.8
єӨ(n2.8
)