Fault-Equivalent Lowest Common Ancestors
Asaf Petruschka

TL;DR
This paper introduces a generalized concept of lowest common ancestors in rooted trees that remains connected after multiple vertex failures, providing an efficient linear-time computation method and bounds on the minimal set size.
Contribution
It defines $f$-fault-equivalent LCAs ($f$-FLCA), proves their uniqueness and minimality, and establishes bounds on their size, extending classical LCA concepts to multiple failures.
Findings
Linear-time computation of $M^*$ set.
Bound of $|M^*| \, \leq \, 2^{f-1}$ for all $f \geq 1$.
Exact bounds are achievable for certain trees and sets.
Abstract
Let be a rooted tree in which a set of vertices are marked. The lowest common ancestor (LCA) of is the unique vertex with the following property: after failing (i.e., deleting) any single vertex from , the root remains connected to if and only if it remains connected to some marked vertex. In this note, we introduce a generalized notion called -fault-equivalent LCAs (-FLCA), obtained by adapting the above view to failures for arbitrary . We show that there is a unique vertex set of minimal size such after the failure of any vertices (or less), the root remains connected to some iff it remains connected to some . Computing takes linear time. A bound of always holds, regardless of , and holds with equality for some choice of and .
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Taxonomy
TopicsForensic and Genetic Research · Machine Learning in Bioinformatics
