A Note On Computing Set Overlap Classes

TL;DR

This paper revisits and simplifies Dahlhaus's algorithm to efficiently identify all overlap classes among a family of sets, achieving optimal time complexity and improving practical implementability.

Contribution

The authors provide a clearer, simplified presentation of Dahlhaus's algorithm for overlap class computation, making it more practical and easier to implement.

Findings

Achieves optimal O(n + sum of set sizes) time complexity.

Provides a simplified, clear version of Dahlhaus's algorithm.

Includes a practical variant of the original approach.

Abstract

Let ${\cal V}$ be a finite set of $n$ elements and ${\cal F}=\{X_1,X_2, >..., X_m\}$ 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 $X=X_1,X_2, ..., X_k=Y$ 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.