Find maximum element in sorted arrays in logarithmic time

Answered By : Aryabhata

If the elements need not be distinct, then you cannot have an $O(\log n)$ time algorithm.

Consider the sorted array $[0,0, \dots, 1]$ which has been cyclic shifted $k$ (unknown) times and you need to find where the $1$ appears. This needs $\Omega(n)$ time, as you need to examine at least $n-1$ elements.

However, if you assume the elements are distinct, then you can indeed give an $O(\log n)$ time algorithm.

Assume the array was sorted ascending. Once it is cyclic shifted, we will have that, in the rotated array (say $a[1,2, \dots n]$), that $a[1] \gt a[n]$. (It might help to draw a figure here, plotting $i$ on x-axis and $a[i]$ on the y-axis).

Now if you pick a $j$, you compare $a[j]$ with $a[1]$ and move right or left, depending on whether it is greater or lesser, like binary search.

Problem Detail: 

Been stuck on this for a while, would really appreciate some help:

Suppose you are given an array A[1...n] of sorted integers that has been circularly shifted k positions to the right. For example, [35,42,5,15,27,29] is a sorted array that has been circularly shifted k = 2 positions, while [27,29,35,42,5,15] has been shifted k = 4 positions. Give an algorithm for finding the maximum element in A that runs in O(log n) time.

The elements in A are distinct.

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I understand that to achieve O(log n) time I'll probably have to search through the list by starting at the middle, and then going left or right, then splitting the list in half over and over, but I'm not sure how to attack it beyond that.

Asked By : user2317714

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Question Source : http://cs.stackexchange.com/questions/11545