Abstract
A wellstudied problem in fault diagnosis is to identify the set of all good processors in a given set $\{p_1,p_2,\ldots,p_n\}$
of processors via asking some processors $p_i$ to test whether processor $p_j$ is good or faulty. Mathematically, the set $C$ of the indices of good processors forms an isolated clique in the graph with the edges $E = \{(i,j):$ if you ask $p_i$
to test $p_j$ then $p_i$ states that ``$p_j$ is good''$\}$; where $C$ is an isolated clique iff it holds for every $i \in C$ and $j \neq i$ that $(i,j) \in E$ iff $j \in C$.
In the present work, the classical setting of fault diagnosis is modified by no longer requiring that $C$ contains at least $\frac{n+1}{2}$ of the $n$ nodes of the graph. Instead, one is given a lower bound $a$ on the size of $C$ and the number
$n$ of nodes and one has to find a list of up to $n/a$ candidates containing all isolated cliques of size $a$ or more where the number of queries whether a given edge is in $E$ is as small as possible.
It is shown that the number of queries necessary differs at most by $n$ for the case of directed and undirected graphs. Furthermore,
for directed graphs the lower bound $n^2/(2a2)3n$ and the upper
bound $2n^2/a$ are established. For some constant values of $a$, better bounds are given. In the case of parallel queries, the number of rounds is at least $n/(a1)6$ and at most $O(\log(a)n/a)$.
BibTeX  Entry
@InProceedings{gasarch_et_al:DagSemProc.04421.3,
author = {Gasarch, William and Stephan, Frank},
title = {{Finding Isolated Cliques by Queries – An Approach to Fault Diagnosis with Many Faults}},
booktitle = {Algebraic Methods in Computational Complexity},
pages = {116},
series = {Dagstuhl Seminar Proceedings (DagSemProc)},
ISSN = {18624405},
year = {2005},
volume = {4421},
editor = {Harry Buhrman and Lance Fortnow and Thomas Thierauf},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2005/106},
URN = {urn:nbn:de:0030drops1066},
doi = {10.4230/DagSemProc.04421.3},
annote = {Keywords: Isolated Cliques , QueryComplexity , Fault Diagnosis}
}
Keywords: 

Isolated Cliques , QueryComplexity , Fault Diagnosis 
Collection: 

04421  Algebraic Methods in Computational Complexity 
Issue Date: 

2005 
Date of publication: 

24.03.2005 