Is the complement of a Max Prefix Sum a Min Suffix Sum?
It depends on whether the maximum prefix and minimum suffix are unique. Consider for example the array $0$ &mdash; you can easily find a counterexample to your claim. Suppose therefore that $\sum_{i \leq j \leq k} A[j] \neq 0$ for all $i \leq k$. Let now $j$ maximize $\sum_{i=1}^j A[i]$, and let $k$ minimize $\sum_{i=k+1}^n A[i]$. We want to show that $j=k$. Otherwise, there are two cases:


If $j &lt; k$, let $S = \sum_{i=j}^{k-1} A[i]$. By assumption $S \neq 0$. If $S &gt; 0$ then $\sum_{i=1}^{k-1} A[i] &gt; \sum_{i=1}^j A[i]$, and otherwise $\sum_{i=j+1}^n A[i] &lt; \sum_{i=k+1}^n A[i]$.
If $j &gt; k$, let $S = \sum_{i=k}^{j-1} A[i]$. By assumption $S \neq 0$. If $S &lt; 0$ then $\sum_{i=1}^{k-1} A[i] &gt; \sum_{i=1}^j A[i]$, and otherwise $\sum_{i=j+1}^n A[i] &lt; \sum_{i=k+1}^n A[i]$.