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John Cook is an applied mathematician working in Houston, Texas. His career has been a blend of research, software development, consulting, and management. John is a DZone MVB and is not an employee of DZone and has posted 172 posts at DZone. You can read more from them at their website. View Full User Profile

Algorithm of the Week: Finding Sum-Free Subsets

06.04.2013
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A set of integers is called sum-free if no element of the set is the sum of any other pair of elements in the set. For example, {1, 10, 100} is sum-free.

Let’s look at pulling out a sum-free subset of a larger set. For example, if we start with {1, 2, 3, …, 10}, then {1, 5, 10} as a sum-free subset. So is {1, 2, 4, 7}. Notice in this case 1 + 2 + 4 = 7, but that’s OK because we’re only concerned with whether an element is the sum of two other elements.

Now let A be a set of integers with n elements. How large of a sum-free subset does A contain? It could be as large as n if the set A were sum-free to begin with, so that’s an upper bound. But what is a lower bound on the size of the largest sum-free subset?

There is a theorem that gives a number k such that every set of n non-zero integers contains a sum-free subset of size at least kn. You could let k be zero, but that’s no fun. Can you find a larger value of k? I’ll tell you later what value of k the theorem has. Until then, maybe you could try to find your own value.

Suppose you want to write a program to explore this empirically. For a given set, how would you find a maximal sum-free subset? Brute force examination of all subsets would take 2n steps, so hopefully you could do better than that.

What are some sets that have relatively small maximal sum-free subsets?

UpdateResults here.

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