Fault-Equivalent Lowest Common Ancestors

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 $T$ be a rooted tree in which a set $M$ of vertices are marked. The lowest common ancestor (LCA) of $M$ is the unique vertex $\ell$ with the following property: after failing (i.e., deleting) any single vertex $x$ from $T$, the root remains connected to $\ell$ if and only if it remains connected to some marked vertex. In this note, we introduce a generalized notion called $f$-fault-equivalent LCAs ($f$-FLCA), obtained by adapting the above view to $f$ failures for arbitrary $f \geq 1$. We show that there is a unique vertex set $M^* = \operatorname{FLCA}(M,f)$ of minimal size such after the failure of any $f$ vertices (or less), the root remains connected to some $v \in M$ iff it remains connected to some $u \in M^*$. Computing $M^*$ takes linear time. A bound of $|M^*| \leq 2^{f-1}$ always holds, regardless of $|M|$, and holds with equality for some choice of $T$ and $M$.