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up vote 15 down vote favorite

Introduction

Consider a nonempty list L of integers. A zero-sum slice of L is a contiguous subsequence of L whose sum equals 0. For example, [1, -3, 2] is a zero-sum slice of [-2, 4, 1, -3, 2, 2, -1, -1], but [2, 2] is not (because it doesn't sum to 0), and neither is [4, -3, -1] (because it's not contiguous).

A collection of zero-sum slices of L is a zero-sum cover of L if every element belongs to at least one of the slices. For example:

L = [-2, 4, 1, -3, 2, 2, -1, -1]
A = [-2, 4, 1, -3]
B =        [1, -3, 2]
C =                  [2, -1, -1]

The three zero-sum slices A, B and C form a zero-sum cover of L. Multiple copies of the same slice can appear in a zero-sum cover, like this:

L = [2, -1, -1, -1, 2, -1, -1]
A = [2, -1, -1]
B =        [-1, -1, 2]
C =                [2, -1, -1]

Of course, not all lists have a zero-sum cover; some examples are [2, -1] (every slice has nonzero sum) and [2, 2, -1, -1, 0, 1] (the leftmost 2 is not part of a zero-sum slice).

The task

Your input is a nonempty integer list L, taken in any reasonable format. Your output shall be a truthy value if L has a zero-sum cover, and a falsy value if not.

You can write a full program or a function, and the lowest byte count wins.

Test cases

[-1] -> False
[2,-1] -> False
[2,2,-1,-1,0,1] -> False
[2,-2,1,2,-2,-2,4] -> False
[3,-5,-2,0,-3,-2,-1,-2,0,-2] -> False
[-2,6,3,-3,-3,-3,1,2,2,-2,-5,1] -> False
[5,-8,2,-1,-7,-4,4,1,-8,2,-1,-3,-3,-3,5,1] -> False
[-8,-8,4,1,3,10,9,-11,4,4,10,-2,-3,4,-10,-3,-5,0,6,9,7,-5,-3,-3] -> False
[10,8,6,-4,-2,-10,1,1,-5,-11,-3,4,11,6,-3,-4,-3,-9,-11,-12,-4,7,-10,-4] -> False
[0] -> True
[4,-2,-2] -> True
[2,2,-3,1,-2,3,1] -> True
[5,-3,-1,-2,1,5,-4] -> True
[2,-1,-1,-1,2,-1,-1] -> True
[-2,4,1,-3,2,2,-1,-1] -> True
[-4,-1,-1,6,3,6,-5,1,-5,-4,5,3] -> True
[-11,8,-2,-6,2,-12,5,3,-7,4,-7,7,12,-1,-1,6,-7,-4,-5,-12,9,5,6,-3] -> True
[4,-9,12,12,-11,-11,9,-4,8,5,-10,-6,2,-9,10,-11,-9,-2,8,4,-11,7,12,-5] -> True
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up vote 4 down vote

Jelly, 13 12 bytes

JẆịS¥ÐḟċþJḄẠ

Try it online!

How it works

JẆịS¥ÐḟċþJḄẠ  Main link. Argument: A (array)

J             Yield all indices of A.
 Ẇ            Window; yield all slices of indices.
     Ðḟ       Filter; keep slices for which the link to the left returns 0.
    ¥           Combine the two atoms to the left into a dyadic chain.
  ị               Retrieve the elements of A at the slice's indices.
   S              Take the sum.
         J    Yield the indices of A.
       ċþ     Count table; count how many times each index appears in each table.
          Ḅ   Unbinary; convery the array of counts of each index from base 2 to 
              integer. This yields 0 iff an index does not appear in any slice.
           Ạ  All; return 1 iff all integers are non-zero.
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up vote 3 down vote

Mathematica, 66 bytes

Two equally long alternatives, both of which are unnamed functions taking a list of integers as input and returning True or False:

And@@Table[0==Product[Tr@#[[i;;j]],{i,k},{j,k,l}],{k,l=Length@#}]&

0==Norm@Table[Product[Tr@#[[i;;j]],{i,k},{j,k,l}],{k,l=Length@#}]&

In both cases, Tr@#[[i;;j]] computes the sum of the slice of the input from position i to position j (1-indexed). Product[...,{i,k},{j,k,l}] multiples together all these slice-sums, as i ranges over indices that are at most k and j ranges over indices that are at least k. (Note that l=Length@# defines l to be the length of the input list.) In other words, this product equals 0 if and only if the kth element belongs to a zero-sum slice.

In the first version, each of those products is compared to 0, and And@@ returns True precisely when every single product equals 0. In the second version, the list of products is acted upon by the function Norm (the length of the l-dimensional vector), which equals 0 if and only if every entry equals 0.

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up vote 2 down vote

Ruby, 81 bytes

Try it online

Simplistic brute-force solution; for every element of the array, try to find a zero-sum slice that contains it.

->a{(0..l=a.size).all?{|i|(0..i).any?{|j|(i..l).any?{|k|a[j..k].inject(:+)==0}}}}
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