A note on computing set overlap classes

Authors:
Pierre Charbit;Michel Habib;Vincent Limouzy;Fabien de Montgolfier;Mathieu Raffinot;Michaël Rao
Affiliations:
LIAFA, Univ. Paris Diderot, Paris 7, 75205 Paris Cedex 13, France;LIAFA, Univ. Paris Diderot, Paris 7, 75205 Paris Cedex 13, France;LIAFA, Univ. Paris Diderot, Paris 7, 75205 Paris Cedex 13, France;LIAFA, Univ. Paris Diderot, Paris 7, 75205 Paris Cedex 13, France;LIAFA, Univ. Paris Diderot, Paris 7, 75205 Paris Cedex 13, France;LIRMM, 161 rue Ada, 34392 Montpellier, France
Venue:
Information Processing Letters
Year:
2008

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Abstract

Let V be a finite set of n elements and F={X"1,X"2,...,X"m} a family of m subsets of V. Two sets X"i and X"j of F overlap if X"i@?X"j@A, X"j@?X"i@A, and X"i@?X"j@A. Two sets X,Y@?F are in the same overlap class if there is a series X=X"1,X"2,...,X"k=Y of sets of F in which each X"iX"i"+"1 overlaps. In this note, we focus on efficiently identifying all overlap classes in O(n+@?"i"="1^m|X"i|) time. We thus revisit the clever algorithm of Dahlhaus [E. Dahlhaus, Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition, J. Algorithms 36 (2) (2000) 205-240] 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.