A Note On Computing Set Overlap Classes

Let V be a finite set of n elements and F = {X1, X2, . . . , Xm} a family of m subsets of V. Two sets Xi and Xj of F overlap if Xi∩Xj 6= ∅, Xj \ Xi 6= ∅, and Xi \ Xj 6= ∅. Two sets X, Y ∈ F are in the same overlap class if there is a series X = X1, X2, . . . , Xk = Y of sets of F in which each XiXi+1 overlaps. In this note, we focus on efficiently identifying all overlap classes in O(n + ∑m i=1 |Xi|) time. We thus revisit the clever algorithm of Dahlhaus [2] of which we give a clear presentation and that we simplify to make it practical and implementable in its real worst case complexity. An useful variant of Dahlhaus’s approach is also explained.