Question

Design and analysis of algorithms

Problem 5. Given a sorted array of distinct integers A[1- -n], you want to find out whether there is an index I for which Ai-i. Give a divide-and-conquer algorithm that runs in time O(log n)

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Answer 1

Hey,

Note: Brother in case of any queries, just comment in box I would be very happy to assist all your queries

• Function func(a,lo,high)
  • len = hi - lo + 1
  • mid = len / 2 + lo
  • if(len==1)
    • if(a[mid]==mid)
      • RETURN mid;
    • Else
      • RETURN -1
    • EndIf
  • Else
    • if(a[mid]>mid)
      • RETURN func(a, lo, mid - 1);
    • Else
      • RETURN func(a, mid, hi);
    • EndIf
  • EndIf
• EndFunction

The above algo works in O(logn) since it is the modification of binary search

Kindly revert for any queries

Thanks.

Answer 2

Algo(A[1,.......,n-1)

1. if n==1, return (A[n]==n)

2. m=(n/2)

3. if A[m] =m , return (true)

4. else if A[m]>m, return (Algo(A[1,.....,m-1]))

5. else return (Algo(A[m+1,....,n]))

Each division reduce the problem size by half, and we do a constant

amount of work in each function call.  This gives us the following re- currence relation:-

T(n) =T(n/2) +O(1)  

which solves to   T(n) =O(log n).