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I am attempting to find the maximum value in a matrix (or 2d array) and want to find it in less than O(n) time. The easiest way, which results in O(n) run time, is an element wise search. If a better run time is possible, I would also like to find any values over a specific threshold in less than O(n) time. Is any of this possible?

I cannot re-sort and do a binary search or something along those lines as the ordering is important.

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    $\begingroup$ If you can't sort or if you don't have any information on the organization of the data, you cannot do better than O(n). $\endgroup$
    – ryuu9187
    Dec 1, 2015 at 19:34
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    $\begingroup$ consider this problem at its simplest. You have value A = randomly 4 (fair dice roll). You want to know if value B is bigger. You have no information on value B. Can you figure that out without looking at value B? $\endgroup$
    – njzk2
    Dec 1, 2015 at 20:53

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If you don't know anything about the contents of the matrix (such as some kind of monotonicity property), linear time is the best you can do for a one-off search with a deterministic algorithm by a simple adversary argument: if you don't look at everything, then you can't distinguish between the cases where the maximum component is/isn't one of the ones you didn't look at. If you want to maintain max-information for a dynamic matrix, then there might be suitable preprocessing and maintenance (for example, a search tree) that can speed things up.

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