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Asked: 2026-05-28T08:12:52+00:00

I have a list of formulae for combining elements: A + B + C

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I have a list of formulae for combining elements:

A + B + C = X
D + E + F = Y
G + H + I = Z

I want to ensure that given any random 4 elements, there will never be more than 1 applicable formula. For example the formulae below should not be allowed as if I get elements A, B, C and D then both are applicable:

A + B + C = X
B + C + D = Y

Every formula will consist of 3 elements on the LHS and it’s the LHS that I want to enforce the rule on.  
The elements are sortable, if that helps.

An alternative, equivalent problem:

I have a list of an array of 3 elements: List<Element[3]> How do I ensure that no 2 elements appear in more than one array.

What would be a reasonably efficient (runtime speed) way of doing this for a large number of elements and a large number for formulae (beyond brute forcing)?

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1 Answer

  1. Editorial Team

    You can represent the constraints on the set of equations as a graph. The vertices are the elements, with n[i] elements in equation i. For equation i, there are thus n[i]*(n[i]-1)/2 pairs of elements; these become edges. Iterate through the equations, adding the edges to the graph. Anytime you’d want to add an edge that is already present, you’ve found a conflict.

    For each edge, you could store a set of equation numbers that would generate the edge; this allows the specific conflicts to be identified, rather than just the presence of conflicts.

    Let N be the number of equations and M the number of elements in the equation with the most elements. The time complexity is O(M^2*N), as is the space complexity. If all equations have a fixed number of elements, time and space usage will thus be O(N).

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