Minimize the Difference

Chef has an array $A$ consisting of $N$ integers. Consider an array $S$ of length $N$, such that each $S_i$ is either $+1$ or $-1$. For such an array, we define $f(S)$ as follows: - Consider a $0$-indexed array $B$ of length $N+1$, where $B_0 = 0$ and for $1 \leq i \leq N$, $B_i = \sum_{j=1}^i S_j\cdot A_j$ Informally, $B$ is the *signed prefix sum* array of $A$, with $S$ giving the signs.