A Note On Computing Set Overlap Classes

Pierre Charbit (1) , Michel Habib (1) , Vincent Limouzy (1) , Fabien de Montgolfier (1) , Mathieu Raffinot (1) , Michaël Rao (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier

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.

Mots clés: Overlap class; Set system; spanning tree

Domaines: Algorithme et structure de données [cs.DS]

Pierre Charbit, Michel Habib, Vincent Limouzy, Fabien de Montgolfier, Mathieu Raffinot, et al.. A Note On Computing Set Overlap Classes. 2007. ⟨hal-00192579⟩