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A Note on Computing Set Overlap Classes

Abstract : Let ${\cal V}$ be a finite set of $n$ elements and ${\cal F}=\{X_1,X_2, \ldots , X_m\}$ a family of $m$ subsets of ${\cal V}.$ Two sets $X_i$ and $X_j$ of ${\cal F}$ overlap if $X_i \cap X_j \neq \emptyset,$ $X_j \setminus X_i \neq \emptyset,$ and $X_i \setminus X_j \neq \emptyset.$ Two sets $X,Y\in {\cal F}$ are in the same overlap class if there is a series $X=X_1,X_2, \ldots, X_k=Y$ of sets of ${\cal F}$ in which each $X_iX_{i+1}$ overlaps. In this note, we focus on efficiently identifying all overlap classes in $O(n+\sum_{i=1}^m |X_i|)$ time. We thus revisit the clever algorithm of Dahlhaus 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.
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Submitted on : Monday, September 29, 2008 - 10:03:49 AM
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Pierre Charbit, Michel Habib, Vincent Limouzy, Fabien de Montgolfier, Mathieu Raffinot, et al.. A Note on Computing Set Overlap Classes. Information Processing Letters, Elsevier, 2008, 108 (4), pp.186-191. ⟨10.1016/j.ipl.2008.05.005⟩. ⟨lirmm-00325371⟩



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