Question

(20 points) You are given an array A of distinct integers of size n. The sequence A[1], A[2], ..., A[n] is unimodal if for some index k between 1 and n the values increase up to position k and then decrease the reminder of the way until position n. (example 1, 4, 5, 7, 9, 10, 13, 14, 8, 6, 4, 3, 2 where the values increase until 14 and then decrease until 1). (a) Propose a recursive algorithm to find the peak entry index k of a unimodal array by reading at most Olg n) entries of A. (b) Write the recurrence relation and solve it to show that the complexity of your algorithm is indeed QAg n).

Answer 1

a)

Algorithm(A , start, end, n)

mid = start + (end - start)/2

    if ((mid == 0 || A[mid-1] <= A[mid]) && (mid == n-1 || A[mid+1] <= A[mid]))

        return mid

    else if (mid > 0 && A[mid-1] > A[mid])

        return Algorithm(A, start, (mid -1), n)

    else

return Algorithm(A, (mid + 1), end, n)

  

b) T(n) = T(n/2) +1

We can solve using Masters theorem
=> a = 1, b = 2 and f(n) = 1
=> n^{log ₂1} = n⁰ = 1

f(n) = g(n) => n^{log_b a} log n

=> O(log n)