Equal Sum Subarrays

Note: The time limit for this problem is 3s, 1.5x the default.

FJ gave Bessie an array $a$ of length $N$ ($2\le N\le 500, -10^{15}\le a_i\le 10^{15}$) with all $\frac{N(N+1)}{2}$ contiguous subarray sums distinct. For each index $i\in [1,N]$, help Bessie compute the minimum amount it suffices to change $a_i$ by so that there are two different contiguous subarrays of $a$ with equal sum.

The first line contains $N$.

The next line contains $a_1,\dots, a_N$ (the elements of $a$, in order).

• Input 3: $N\le 40$
• Input 4: $N \le 80$
• Inputs 5-7: $N \le 200$
• Inputs 8-16: No additional constraints.

One line for each index $i\in [1,N]$.

Decreasing $a_1$ by $2$ would result in $a_1+a_2=a_2$. Similarly, increasing $a_2$ by $3$ would result in $a_1+a_2=a_1$.

Increasing $a_1$ or decreasing $a_3$ by $1$ would result in $a_1=a_3$. Increasing $a_2$ by $6$ would result in $a_1=a_1+a_2+a_3$.