03 mathematical anaylsis
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The document discusses mathematical analysis of algorithms, including both non-recursive and recursive algorithms. For non-recursive algorithms, it describes identifying the input size, basic operation, complexity case, and setting up a summation or recurrence relation to solve. For recursive algorithms, it similarly identifies these elements and sets up a recurrence relation to solve. It provides examples of analyzing algorithms for finding the largest array element, element uniqueness, matrix multiplication, factorial, Tower of Hanoi, and calculating number of bits to store a decimal number.
![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](https://image.slidesharecdn.com/03-mathematicalanaylsis-150601060826-lva1-app6891/85/03-mathematical-anaylsis-3-320.jpg)
![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](https://image.slidesharecdn.com/03-mathematicalanaylsis-150601060826-lva1-app6891/85/03-mathematical-anaylsis-5-320.jpg)
![• 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](https://image.slidesharecdn.com/03-mathematicalanaylsis-150601060826-lva1-app6891/85/03-mathematical-anaylsis-6-320.jpg)
![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](https://image.slidesharecdn.com/03-mathematicalanaylsis-150601060826-lva1-app6891/85/03-mathematical-anaylsis-7-320.jpg)
03 mathematical anaylsis
- 2. 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
- 3. 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
- 4. • 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
- 5. 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
- 6. • 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
- 7. 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
- 8. • 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
- 9. MATHEMATICAL ANALYSIS OF RECURSIVE ALGORITHMS
- 10. 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
- 11. Finding Factorial Algorithm fact(n) if n==0 return 1 else return fact(n-1)*n
- 12. • 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.
- 13. • 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
- 14. • 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)
- 15. Tower of hanoi • Problem:
- 16. 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)
- 17. • input size:number of diks, n • Basic operation:move • Case Complexity: Every case so 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
- 18. Initial Condition: M(1)=1 by backward substituting values M(n)=2M(n-1)+1 =2(2(M(n-2)+1)+1 =22 M(n-2)+2+1 =22 (2M(n-3)+1)+2+1 =23 M(n-3)+22 +2+1 By generalizing =2i M(n-i)+2i-1 +2i-2 +…+1
- 19. M(n)=2i M(n-i)+2i-1 +2i-2 +…+1 geometric series so =2n-1 M(n-(n-1))+2n-1 -1 =2n-1 M(1)+2n-1 -1 =2n-1 +2n-1 -1 =2n -1 • Recurrence Tree
- 20. Calculating number of bits to store decimal number Algorithm bincov(n) if n=1 return 1 else return bincov (floor(n/2))+1
- 21. • Input size: 4 bytes • Basic operation: Summation • complexity:every case Recurrence Relation: A(n)=A(floor(n/2))+1 initial condition:A(1)=0
- 22. • 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)