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Fault-Equivalent Lowest Common Ancestors

Asaf Petruschka

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 $M$.

Fault-Equivalent Lowest Common Ancestors

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 .

Paper Structure

This paper contains 7 sections, 2 theorems, 5 equations, 1 algorithm.

Table of Contents

  1. Introduction and Results
  2. Relation to Lowest Common Ancestors.
  3. Results.
  4. Edge Faults.
  5. Aggregation.
  6. Potential Applications.
  7. Proof of \ref{['thm:FLCA']}

Key Result

Theorem 1

Let $T$ be an $n$-vertex tree rooted at vertex $s$, $M \subseteq V(T)$ be a non-empty set of marked vertices, and $f \geq 1$. The following hold:

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 3
  • proof
  • Claim 4: $f$-Fault Equivalence
  • proof
  • Claim 5: Minimality and Uniqueness
  • proof
  • Claim 6: Size Bound
  • ...and 3 more