The document outlines a methodical approach for analyzing recursive algorithms, particularly through the example of the factorial function. It details steps for determining input size, identifying basic operations, establishing recursive relations, and solving them using substitution methods. The analysis concludes with the time complexity of the recursive algorithm being O(n).

What's hot

Similar to Mathematical Analysis of Recursive Algorithm.

More from mohanrathod18

Recently uploaded

Mathematical Analysis of Recursive Algorithm.

• 1.
  Mathematical analysis ofrecursivealgorithmBharati. CM.tech
• 2.
  General plans foranalysis of recursivealgorithm1) decide the input size.2) identify the basic operation.3) check how many time the basic operationperformed and determine the best worst andaverage case.4) set up a recursive relation with some initialcondition expressing the basic operation.5)solve recursion with using forward and backwardsubstitution and correct formula using mathematicalinduction.
• 3.
  Algorithm: factorial(n)if(n==0)return 1elsereturnfact(n-1)*n
• 4.
  Mathematical analysis:Step 1:input size is nstep 2 :basic operation is multiplicationstep 3 :recursive function isF(n)=F(n-1)*n where n>0Step 4: multiplication is given asM(n)=M(n-1)+1
• 5.
  Step 4: Forwordssubstitution:M(n)=M(n-1)+1M(0)=0 n=0M(1)=M(1-1)+1=1.............. n=1M(2)=M(2-1)+1=2.................n=2M(3)=M(3-1)+1=3
• 6.
  Backward substitution:M(n)=M(n-1)+1M(n)=[M(n-2)+1]+1=M(n-2)+2M(n)=[M(n-3)+1]+2=M(n-3)+3M(n)=M(n-i)+i
• 7.
  Now we correctthe formulaProve: M(n)=n by using mathematical inductionLet n=0M(n)=0.M(0)=0=nInduction: M(n-1)=n-1M(n)=M(n-1)+1=n-1+1=nM(n)=nTime complexity. (n)
• 8.
  Thank you