Mathematical Analysis of Algorithms

Mathematical Analysis of Non recursive Algorithms

• Identify input size

• Identify Basic Operation

• Identify Case Complexity i.e. (Every Case, Best Case, Worst Case, Average Case)

• Setup summation/recurrence relation

• solve summation/recurrence relation or atleast order of growth of its relation

Finding largest element in array

Algorithm max_element(A,n)

maxval=A[0]

for i=1 to n-1

if A[i]>maxval

maxval=A[i]

Return maxval

• Input size : number of elements in array

• Basic operation : comparison

• Case complexity : Every Case

• Number of times basic operation executed are

• T(n)=∑ 1=n-1є Ө (n)i=1

n-1

Element uniqueness problem

Algorithm element_unique(A,n)

for i=0 to n-2

for j=i+1 to n-1

if A[i]==A[j]

return false

return true

• Input size : number of elements in array

• Basic operation : comparison of element of Array

• Case complexity : Worst Case

• Number of times basic operation executed are

T(n)=∑ ∑ 1 = ∑ [(n-1)-(i+1)+1]

=n(n-1)/2

i=0

n-2

j=i+1

n-1

i=0

n-2

Matrix Multiplication

Algorithm matrix_mul(A,B,C)

for i=1 to n

for j=1 to n

C[i,j]=0

for k=1 to n

C[i,j]=C[i,j]+A[i,k ]*B[k,j]

return C

• Input Size : order of matrix

• Basic operation:multiplication

• Case complexity: every case

• T(n)=∑ ∑ ∑ 1= ∑ ∑ n = ∑ n2 = n3 n

i=1 j=1 k=1

n n

i=1 i=1

nn n

i=1

MATHEMATICAL ANALYSIS OF RECURSIVE ALGORITHMS

Mathematical Analysis Of Recursive Algorithms

• Identify input size

• Identify Basic Operation

• Identify Case Complexity i.e. (Every Case, Best Case, Worst Case, Average Case)

• Setup recurrence relation

• solve recurrence relation or atleast order of growth of its relation

Finding Factorial

Algorithm fact(n)

if n==0 return 1

else

return fact(n-1)*n

• Input size: 4 bytes

• Basic Operation: multiplication

• Case complexity: Every case

• Recurrence Relation: An equation or inequality that describes function in terms of its smaller inputs.

• to determine solution we need initial condition that makes recursion stop.

• F(n)=F(n-1) *n for n>0 (Basic Operation)M(n)=M(n-1) +1 (to multiply F(n-1) by n)

compute(F(n-1))

M(n-1)=(M(n-2)+1)

M(n-2)=(M(n-3)+1)

by substituting values backward

M(n-1)=M(n-3)+1+1

M(n)=M(n-3)+1+1+1

• M(n)=M(n-i)+i

we know the value of initial condition

M(0)=0

by taking i=n

M(n)=M(n-n)+n

=0+n=nєӨ(n)

Tower of hanoi

• Problem:

Algorithm hanoi(n,source,auxiliary,destination)

if n=1 move disk from source to destination

else

hanoi(n-1,source,destination,auxiliary)

move disk from source to destination

hanoi(n-1,auxiliary,source,destination)

• input size:number of diks, n

• Basic operation:move

• Case Complexity: Every caseso recurrence relation would be

• M(n)=M(n-1)+1+M(n-1)

=2M(n-1)+1

or M(n-1)=2M(n-2)+1

M(n-2)=2M(n-3)+1

Initial Condition: M(1)=1

by backward substituting valuesM(n)=2M(n-1)+1

=2(2(M(n-2)+1)+1

=22M(n-2)+2+1

=22(2M(n-3)+1)+2+1

=23M(n-3)+22+2+1

By generalizing

=2iM(n-i)+2i-1+2i-2+…+1

M(n)=2iM(n-i)+2i-1+2i-2+…+1 geometric series so

=2n-1M(n-(n-1))+2n-1 -1

=2n-1M(1)+2n-1 -1

=2n-1+2n-1 -1

=2n -1

• Recurrence Tree

Calculating number of bits to store decimal number

Algorithm bincov(n)

if n=1 return 1

else

return bincov (floor(n/2))+1

• Input size: 4 bytes

• Basic operation: Summation

• complexity:every case

Recurrence Relation:

A(n)=A(floor(n/2))+1

initial condition:A(1)=0

• Smoothness rule: for solving problems of ceiling or flooring assume that n= 2k ;

which claims under broad assumption order of growth observed for n=2k correct results for all values of n.

A(n)=A( n/2)+1 A(n/2)=A(n/4)+1

A(n/4)=A(n/8)+1

by backward substitution

A(n)=A(n/4)+2=A(n/8)+3=A(n/ 2k )+k

by taking 2k =n

A(n)=A(1)+k=k=lg(n)єӨ(lgn)