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I am looking for an algorithm in $O(N^2)$ that finds the maximum value that be obtained from a sequence of real numbers greater than 0 (e.g. $\{ 1, 2, 3 , 4\}$) by inserting a plus ($+$) or multiplication ($\times$) between the elements.

Really not having much luck and wondering if anyone has any insight on this?

It must follow normal order of precedence.

For example with $\{1,2,3,4\}$, the maximum is $1+2\times3\times4 = 25$ not $1\times2\times3\times4 = 24$ and not $1\times2+3+4 = 9$.

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    $\begingroup$ Do you mean integers by "real numbers"? Examples are only integers here. Is it possible to take numbers like 0.1, 0.2 etc.? $\endgroup$ – Evil Sep 17 at 3:52
  • $\begingroup$ cs.stackexchange.com/tags/dynamic-programming/info $\endgroup$ – D.W. Sep 17 at 21:52
  • $\begingroup$ Hint: For given $i, j \le N$, see if you can compute the maximum value that can be obtained from just the first $j$ elements by inserting $+$ or $\times$ between them, with the additional restriction that the last $i$ operations consist of a $+$ followed by $i-1$ $\times$s. $\endgroup$ – j_random_hacker Sep 17 at 22:09
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Let's say all numbers are positive (i.e. $\geq0$). Before and after $0$s you should always use $+$. This could also solved in $O(n^2)$ by first finding all zeros and then find maximum value for each group separated by zeros. Inside each group, for positive numbers, you should always use $\times$ except in $1$ cases, if ones are in the middle of a group, still you should choose $\times$. But when ones are at the very beginning or at the very end of the group, you should choose $+$.

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    $\begingroup$ $2+1+2>2\times 1\times 2$. $\endgroup$ – xskxzr Sep 17 at 4:08
  • $\begingroup$ There are no 0 in the question. $\endgroup$ – Evil Sep 17 at 15:03

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