So I watched a couple of videos regarding this and the divide and conquer made sense to me. But I am still not convinced as to how recursing on the side with the larger number guarantees us the solution.
Please can someone explain it Mathematically.
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1$\begingroup$ Do you mean a local peak (item greater than its neighbors) or a global peak (item greater than all others)? $\endgroup$– Reinstate MonicaCommented Nov 5, 2018 at 21:43
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$\begingroup$ Yes, I meant a local peek $\endgroup$– Shivangi SinghCommented Nov 6, 2018 at 1:11
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$\begingroup$ related: cs.stackexchange.com/questions/96457/… $\endgroup$– Reinstate MonicaCommented Nov 6, 2018 at 18:31
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$\begingroup$ It looks like you are not aware that you can accept an answer if you are the asker. It is part of protocol and etiquette to try accepting the best answer that has answered your question by clicking on the check mark beside the answer. "Thanks" without upvote or acceptance might be the antithesis of appreciation on Stack exchange. A low acceptance rate might drive some experience users away from answering. You can also check how important is accepting an answer. $\endgroup$– John L.Commented Dec 1, 2018 at 2:54
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2 Answers
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To reformulate the question, there is the following problem: given an array of numbers, find an index in the array that is a local maximum, meaning the value at that index $\ge$ the values at adjacent indices.
The algorithm suggested works as follows. Perform a binary search on the array. At each iteration choose a middle element of the array; examine its neighbors; if it is a local maximum or the binary search has narrowed down to just one item, output its index; otherwise recurse on a side with a greater neighbor. Clearly this algorithm runs in $O(\log n)$.
Your question is why this algorithm works.
The first observation is that every array has a local maximum. That is because an array of length 1 has a local maximum and if every array of length $n$ has a local maximum then given an array of length $n+1$, either its first element is a local maximum or each local maximum in the sub-array starting at the second element is a local maximum of the full array.
Now suppose there are two consecutive elements of the array $a, b$ and $a < b$. Every local maximum of the sub-array starting at $b$ is a local maximum of the full array and as shown in the previous paragraph, there must be at least one. Similarly if $a > b$, every local maximum of the sub-array ending at $a$ is a local maximum of the full array and there must be at least one. The algorithm always recurses to one of those mentioned sub-arrays and therefore when it gets to a single-element sub-array, whose only element is a local maximum of itself, that element is also a local maximum of the sub-array in the previous step, which must be a local maximum of the sub-array two steps prior, etc., all the way back to the full array.
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No, it is not possible to find the max (peak) element in an unsorted array better than $\mathcal{O}(n)$.
When you executed your algorithm $\mathcal{A}$ that has $c \log n$ compare operations, that means; $\mathcal{A}$ only compared $c \log n$ elements. The remaining are not compared by $\mathcal{A}$.
We can easily construct an adversary argument for any $\mathcal{O}(\log n)$ algorithm by just placing the peak, within the elements for $\mathcal{A}$ didn't touch.
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$\begingroup$ The question asks for finding the local peak, not the global peak. $\endgroup$– SOFeCommented May 6, 2019 at 11:07
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1$\begingroup$ @SOFe If you look at the original question, it was not local. The OP changed the question after the my answer, so I've kept it. $\endgroup$– kelalakaCommented May 6, 2019 at 11:48