# What constitutes a minimal-sum section of an integer array

I'm having trouble understanding what constitutes a "minimal sum section" of an integer array. My book defines it as the following:

Let $$a[0],\dots, a[n-1]$$ be the integer values of an array $$a$$.
A section of $$a$$ is a continuous piece $$a[i],\dots,a[j]$$, where $$0\le i \le j < n$$. We write $$S_i,_j$$ for the sum of that section: $$a[i] + a[i+1]+\dots+a[j]$$.
A minimal-sum section is a section $$a[i],\dots,a[j]$$ of $$a$$ such that the sum $$S_{i,j}$$ is less than or equal to the sum $$S_{i',j'}$$ of any other section $$a[i'],\dots,a[j']$$ of $$a$$.

My confusion comes with one of the examples that follow this definition:

The array [1,-1,3,-1,1] has two minimal-sum sections [1,-1] and [-1,1] with minimal sum 0.

But, wouldn't the minimal sum section be $$[-1]$$ ?

In a later example they give:

array $$[-1,3,-2]$$

Minimal sum

section $$[-2]$$

So, in the last example they definitely counted one element as the minimal sum section, but not in the first one. Any clarification on why this is so would be greatly appreciated.

• If your book is Huth, M; Ryan, M: "Logic in Computer Science: Modelling and Reasoning about Systems" (part in question 4.3.3 A case study: minimal-sum section), I take it to be a simple error on behalf of the authors and lectors in an example intended to show something else: that minimal-sum sections do not need to be unique. (9780521543101, no errata found at Cambridge University Press.) – greybeard Nov 11 '19 at 6:52
• Looks like a mistake. According to the definition, the minimal sum section in your first example should indeed be $-1$. – Yuval Filmus Nov 11 '19 at 9:54
• @greybeard yes that's the book. – Papaya Automata Nov 11 '19 at 16:39