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Commite24c674

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Update knuth-optimization.md
In time complexity proof cancellation explanation, j=N -> j=N-1
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‎src/dynamic_programming/knuth-optimization.md

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\sum\limits_{i=1}^N \sum\limits_{j=i}^{N-1} [opt(i+1,j+1)-opt(i,j)].
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As you see, most of the terms in this expression cancel each other out, except for positive terms with $j=N$ and negative terms with $i=1$. Thus, the whole sum can be estimated as
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As you see, most of the terms in this expression cancel each other out, except for positive terms with $j=N-1$ and negative terms with $i=1$. Thus, the whole sum can be estimated as
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$$
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\sum\limits_{k=1}^N[opt(k,N)-opt(1,k)] = O(n^2),

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