Conditional Diagnosability of Cayley Graphs Generated by
Transposition Trees under the Comparison Diagnosis Model
∗
Cheng-Kuan Lin
Department of Computer Science
National Chiao Tung University
[email protected]
Jimmy J. M. Tan
Department of Computer Science
National Chiao Tung University
[email protected]
Lih-Hsing Hsu
Department of Computer Science and Information Engineering
Providence University
[email protected]
Eddie Cheng
Department of Mathematics and Statistics
Oakland University
[email protected]
L´aszl´o Lipt´ak
Department of Mathematics and Statistics
Oakland University
[email protected]
Abstract
The diagnosis of faulty processors plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. Zheng et al. showed that the diagnosability of then-dimensional star graph Snisn − 1. Lai et al. intro-duced a restricted diagnosability of multiprocessor sys-tems called conditional diagnosability. They consider the situation when no faulty set can contain all the neighbors of any vertex in the system. In this paper, we study the
∗This work was supported in part by the National Science Council of
the Republic of China under Contract NSC 95-2221-E-009-134-MY3.
conditional diagnosability of Cayley graphs generated by transposition trees (which include the star graphs) under the comparison model, and show that it is 3n − 8 for n ≥ 4, except for the n-dimensional star graph, for which
it is3n − 7.
1 Introduction
With the continuous increase in the size of multipro-cessor systems, working in multipromultipro-cessor systems with faults has become unavoidable. Therefore, the problem of fault diagnosis in multiprocessor systems has gained increasing importance and has been widely studied, for example [9–11, 20, 21, 38, 39]. The process of identifying
faulty processors in a system is known as system-level diagnosis. Several different approaches have been devel-oped to diagnose faulty processors, among which there are two fundamental approaches on system-level diagno-sis. One major approach is called the comparison model, proposed by Malek and Maeng [28, 29]. In this model, each processor performs a diagnosis by sending the same inputs to each pair of its distinct neighbors and then com-pares their responses. The result of a comparison is either that the two responses agree or the two responses dis-agree. Based on the results of all the comparisons, one needs to decide the faulty or non-faulty (fault-free) status of the processors in the system. Another major approach is the PMC model established by Preparata, Metze, and Chien [33]. In this model, it is assumed that a processor can test the faulty or fault-free status of another adjacent processor. Under the PMC model, only processors with a direct link are allowed to test each other. It is assumed that if a processor is fault-free, it always gives correct and reliable testing results, and if a processor is faulty, then its testing results may be correct or incorrect. By analyz-ing the collection of all testanalyz-ing results, all of the faulty processors need to be identified.
An interconnection network connects the processors of parallel computers. Its architecture can be represented as a graph in which the vertices correspond to proces-sors and the edges correspond to connections. Hence we use graphs and networks interchangeably. There are many mutually conflicting requirements in designing the topology for computer networks. Then-cube is one of
the most popular topologies [23, 35]. Then-dimensional
star network Sn was proposed in [1] as “an attractive alternative to the n-cube” topology for interconnecting
processors in parallel computers. Since its introduc-tion, the networkSn has received considerable attention.
The star graphs are bipartite, vertex transitive, and edge transitive, and several classes of graphs can be embed-ded into them, e.g. grids [19], trees [3, 5, 13], and hy-percubes [30]. Cycle embeddings and path embeddings are studied in [15–19, 24, 32]. The diameter and fault diameters of star graphs were computed in [1, 22, 34]. Some other interesting properties of star graphs are stud-ied in [12, 14, 25–27].
Reviewing some previous papers (see [10, 11, 21, 38]), the n-dimensional hypercube Qn, the n-dimensional crossed cubeCQn, then-dimensional twisted cube T Qn, and then-dimensional m¨obius cube M Qn, all have di-agnosability n under the comparison model. Zheng et al. [39] showed that the diagnosability of the
n-dimensional star graphSn is n − 1. In classical mea-sures of system-level diagnosability for multiprocessor systems, if all the neighbors of some processorv are faulty
simultaneously, it is not possible to determine whether processorv is fault-free or faulty. As a consequence, the
diagnosability of a system is limited by its minimum de-gree. Hence Lai et al. introduced a restricted diagnos-ability of multiprocessor systems called conditional
diag-nosability in [20]. Lai et al. considered this measure by
requiring that for each processorv in a system, all the
processors that are directly connected tov do not fail at
the same time. Under this condition, the conditional diag-nosability of then-dimensional hypercube Qnis4n − 7 under the PMC model [20].
In this paper, we study the conditional diagnosability of the star graphSn and a class of graphs that arise as a generalization of the star graph. These graphs are Cay-ley graphs generated by transposition trees. We consider the comparison model and show that the conditional di-agnosability of these graphs is3n − 8 for n ≥ 4, except
Hence the conditional diagnosability of these graphs is about three times larger than their classical diagnosabil-ity. Section 2 provides preliminaries and previous results for diagnosing a system. In Section 3 we study the condi-tional diagnosability of Cayley graphs generated by trans-position trees under the comparison model. Our conclu-sions are given in Section 4.
2 Preliminaries
A multiprocessor system can be represented by a graph
G(V, E), where the set of vertices V (G) represents
pro-cessors and the set of edgesE(G) represents
communica-tion links between processors. Throughout this paper, we focus on undirected graphs without loops and follow [4] for graph theoretical definitions and notations.
LetG be a graph. The neighborhood NG(v) of vertex
v in G is the set of all vertices that are adjacent to v. The
cardinality|NG(v)| is called the degree of v, denoted by
degG(v). A graph H is a subgraph of G if V (H) ⊆ V (G) andE(H) ⊆ E(G). Let S be a subset of V (G) ∪ E(G).
The subgraph of G induced by S, denoted by G[S], is
the graph with the vertex setS ∩ V (G) and the edge set {(u, v) | (u, v) ∈ E(G) and u, v ∈ S}. For a set of
ver-tices (respectively, edges)S, we use the notation G − S
to denote the graph obtained fromG by removing all the
vertices (respectively, edges) inS. The components of G are its maximal connected subgraphs. A component is trivial if it has no edges; otherwise, it is nontrivial. The connectivityκ(G) of G is the minimum number of
ver-tices whose removal results in a disconnected or a trivial graph. A graphG is k-regular if degG(u) = k for every vertexu in G. A path P between vertices v1 andvk is a sequence of adjacent vertices,hv1, v2, . . . , vki, in which the verticesv1, v2, . . . , vk are distinct. The length ofP , denoted byl(P ), is the number of edges in P . The
dis-tance of two verticesu and v of G, denoted by dG(u, v), is the length of the shortest path ofG between u and v.
The comparison diagnosis model [28,29] was proposed by Malek and Maeng. In this model, a self-diagnosable system is often represented by a multigraph M (V, C),
whereV is the same vertex set defined in G, and C is a
la-beled edge set. If(u, v) is an edge labeled by w, then the
labeled edge(u, v)wis said to belong toC, which implies that verticesu and v are being compared by vertex w. The
same pair of vertices may be compared by different com-parators, soM can be a multigraph. For (u, v)w ∈ C, we user((u, v)w) to denote the result of comparing ver-ticesu and v by w such that r((u, v)w) = 0 if the outputs ofu and v agree, and r((u, v)w) = 1 if the outputs dis-agree. In this model, ifr((u, v)w) = 0 and w is fault-free, then bothu and v are fault-free. If r((u, v)w) = 1, then at least one of the three verticesu, v, w must be faulty.
If the comparatorw is faulty, then the result of
compari-son is unreliable. The collection of all comparicompari-son results, given by the functionr : C → {0, 1}, is called the syn-drome of the diagnosis. A subsetF ⊂ V is said to be compatible with a syndromer if r can arise from the
cir-cumstance that all vertices inF are faulty and all vertices
inV −F are fault-free. A system is said to be diagnosable
if, for every syndromer, there is a unique F ⊂ V that is
compatible withr.
In our comparison model, we have (u, v)w ∈ C if and only if u and v are both adjacent to w, hence the
original graph determines the multigraphM (V, C).
No-tice that in this model for every set F ⊂ V there is
always a syndrome that is compatible for both F and V − F . Thus in general there is no diagnosable system.
Thus [36] introduced the concept of at-diagnosable
sys-tem, in which the system is diagnosable as long as the number of faulty vertices does not exceedt. The
max-imum number of faulty vertices that the system G can
guarantee to identify is called the diagnosability of G,
written ast(G). A faulty comparator can lead to
unreli-able results, so a set of faulty vertices may produce differ-ent syndromes. LetσF = {σ | σ is compatible with F }. Two distinct subsets F1 andF2 ofV are said to be in-distinguishable if and only if σF1 ∩ σF2 6= ∅; other-wise, F1 andF2 are said to be distinguishable. There are several different ways to verify whether a system is
t-diagnosable under the comparison approach. The sym-metric difference of the two setsS1 andS2is defined as the setS1△ S2= (S1− S2) ∪ (S2− S1). The following theorem given by Sengupta and Dahbura [36] is a neces-sary and sufficient condition for ensuring distinguishabil-ity. F1 F2 1 ( ) 1( ) 2 ( ) 3( )
Figure 1: Description of distinguishability for Theorem 1
Theorem 1. [36] LetG be a graph. For any two distinct subsetsF1andF2ofV (G), (F1, F2) is a distinguishable pair if and only if at least one of the following conditions is satisfied (see Figure1):
(1) there are two distinct verticesu and w in V (G) − (F1∪ F2) and there is a vertex v in F1△F2such that
(u, v)w∈ C,
(2) there are two distinct verticesu and v in F1−F2and there is a vertexw in V (G) − (F1∪ F2) such that
(u, v)w∈ C, or
(3) there are two distinct verticesu and v in F2−F1and there is a vertexw in V (G) − (F1∪ F2) such that
(u, v)w∈ C.
3 Transposition trees graphs
In this section we summarize the connectivity prop-erties of Cayley graphs generated by transposition trees. These graphs arise naturally as a common generalization of star graphs and bubble-sort graphs. Some papers study-ing these graphs include [2, 6–8, 37].
LetΓ be a finite group and S be a set of elements of Γ such that the identity of the group does not belong to S. The Cayley graph Γ(S) is the directed graph whose
vertex set isΓ, and there is an arc from u to v if and only
if there is ans ∈ S such that u = vs. The graph Γ(S)
is connected if and only ifS is a generating set for Γ.
A Cayley graph is always vertex transitive, so it is maxi-mally arc-connected if it is connected; however, its vertex connectivity may be low.
In this paper, we choose the finite group to beΓn, the symmetric group on{1, 2, . . . , n}, and the generating set S to be a set of transpositions. The vertices of the
corre-sponding Cayley graph are permutations, and sinceS only
has transpositions, there is an arc from vertexu to vertex v if and only if there is an arc from v to u. Hence we
can regard these Cayley graphs as undirected graphs by replacing every pair of arcs between two vertices with an edge; let the resulting graph beΓn(S). A simple way to depictS is via a graph G(S) with vertex set {1, 2, . . . , n},
where there is an edge betweeni and j if and only if the
transposition(ij) belongs to S. This graph is called the transposition generating graph ofΓn(S) or simply trans-position (generating) graph if it is clear from the context.
In fact, the star graphSn was introduced via the gener-ating graphK1,n−1, where the center is 1 and the leaves
are2, 3, . . . , n. Notice, that if we change the label of the
center, we still get a graph isomorphic to the star graph
Sn, hence with a slight abuse of terminology we will call all these graphs star graphs. The star graphsS2,S3, and
S4are shown in Figure 2 for illustration.
1423 2413 4213 3214 2134 1234 1342 3142 4132 c d f g 2314 1324 3124 a b e 4123 2143 1243 4231 3241 2431 3412 4312 1432 b c e f 2341 4321 3421 a d g 123 213 321 312 231 132 12 21
S
2S
3S
4Figure 2: The star graphsS2,S3, andS4
Note that the Cayley graphΓn(S) is |S|-regular, and it is connected if and only if the generating graphG(S) is
connected. Since an interconnection network needs to be connected, we require the transposition graph to be con-nected. Here we will only consider the fundamental case, when G(S) is a tree, and call the corresponding
trans-position generating graph a transtrans-position tree. Thus the Cayley graphs obtained by these transposition trees are
(n−1)-regular and have n! vertices. In addition to the star
graph mentioned above, these Cayley graphs also include
the bubble-sort graph whose transposition tree is a path. Figure 3 shows the bubble-sort graph forn = 4.
1423 2413 4213 3214 2134 1234 1342 3142 4132 2314 1324 3124 4123 2143 1243 4231 3241 2431 3412 4312 1432 2341 4321 3421
B
4Figure 3: The bubble-sort graph
LetΓn(S) be a Cayley graph generated by a transpo-sition tree S. To help us describe the structure of the
Cayley graphΓn(S) when G(S) is a tree, without loss of generality we may assume that a leaf of the transposi-tion tree isn. We use boldface letters to denote vertices in Γn(S). Hence, u1, u2, . . . , unis a sequence ofn vertices inΓn(S). It is known that the connectivity of Γn(S) is
n − 1. Clearly Γn(S) is a bipartite graph with one partite set containing the vertices corresponding to odd permuta-tions and the other partite set containing the vertices cor-responding to even permutations. Let u= u1u2. . . unbe any vertex of the Cayley graphΓn(S). We say that uiis thei-th coordinate of u, denoted by (u)i, for1 ≤ i ≤ n. For1 ≤ i ≤ n, let Γ{i}n denote the subgraph ofΓn(S) induced by those vertices u with(u)n= i.
Sincen is a leaf in the generating tree, it is easy to see
that the Cayley graphΓn(S) has the following properties: (I) Γn(S) consists of n vertex-disjoint subgraphs:
Γ{1}n , Γ{2}n , . . . , Γ{n}n ; each isomorphic to another Cayley graphΓn−1(S′) with S′ = S \ {π} where
π is the transposition corresponding to the edge
inci-dent to the leafn.
(II) Γ{i}n has(n − 1)! vertices, and it is (n − 2)-regular for alli.
(III) For all i, each vertex in Γ{i}n has a unique neighbor outsideΓ{i}n , and these outside neighbors are all dif-ferent. There are exactly(n − 2)! independent edges
betweenΓ{i}n andΓ{j}n for alli 6= j.
These properties are illustrated in Figures 2 and 3, as e.g.S4and the bubble-sort graph contain four copies of a smaller Cayley graph, the 6-cycle. Note that the 6-cycle is the shortest cycle in star graphs, whereas in other Cay-ley graphs we also have 4-cycles.
Cayley graphs generated by transposition trees have strong connectivity properties. Roughly speaking, delet-ing a large number of vertices from it, they will still con-tain a large connected component as shown by the follow-ing theorem:
Theorem 2. [8] LetΓn(S) be a Cayley graph obtained from a transposition generating treeS on {1, 2, . . . , n} withn ≥ 4, and let T be a set of vertices of G such that |T | ≤ 3n − 8. Then Γn(S) − T satisfies one of the fol-lowing conditions:
(i) Γn(S) − T is connected.
(ii) Γn(S) − T has two components, one of which is K1 orK2.
(iii) Γn(S) − T has three components, two of which are singletons.
(iv) Γn(S) − T has two components, one of which is a path of length3, and T is the union of the neighbor
sets of the vertices on the path except the vertices of the path itself with|T | = 3n − 8.
(v) Γn(S) − T has four components, three of which are singletons, andT is the union of the neighbor sets of the singletons with|T | = 3n − 8.
(vi) Γn(S) − T has two components, one of which is a 4-cycle,n = 4 and |T | = 4.
Note: Cases (iv), (v), and (vi) can only occur whenΓn(S) is not a star graph, because each require a 4-cycle in the graph.
4 The conditional diagnosability
In classical measures of system-level diagnosability for multiprocessor systems, if all the neighbors of some pro-cessorv are faulty simultaneously, it is not possible to
de-termine whether processorv is fault-free or faulty. So the
diagnosability of a system is limited by its minimum ver-tex degree. In particular, as we mentioned before, the star graphSnhas diagnosabilityn − 1 (see [39]). The same result can be proven easily for Cayley graphs generated by transposition trees as well, whose proof we omit:
Theorem 3. LetΓn(S) be a Cayley graph obtained from a transposition generating treeS on {1, 2, . . . , n} with n ≥ 4. Then t(Γn(S)) = n − 1.
A Cayley graphΓn(S) has n−1n! vertex subsets of size
n−1, among which there are only n! vertex subsets which
contain all the neighbors of some vertex. Since the ra-tion!/ n−1n! is very small for large n, in case of
inde-pendent failures the probability of a faulty set containing all the neighbors of any vertex is very low. For this rea-son, Lai et al. introduced a new restricted diagnosability of multiprocessor systems called conditional diagnosabil-ity in [20]. They considered the situation that no faulty
set can contain all the neighbors of any vertex in a sys-tem. We need some terms to define the conditional di-agnosability formally. A faulty setF ⊂ V (G) is called
a conditional faulty set ifNG(v) * F for every vertex
v ∈ V (G). A system described by the graph G(V, E) is
said to be conditionallyt-diagnosable if F1 andF2 are distinguishable for each pair of distinct conditional faulty sets F1 and F2 ofV (G) with |F1| ≤ t and |F2| ≤ t. The maximum value oft such that G is conditionally
t-diagnosable is called the conditional diagnosability ofG,
denoted bytc(G). It is trivial that tc(G) ≥ t(G).
Now we give an example in the Cayley graphΓn(S) to get a bound on the conditional diagnosability. As shown in Figure 4, we take a path of length two in
u
2u
1u
3F
1F
2n-3 n-2 n-2
Figure 4: An indistinguishable conditional pair(F1, F2)
Γn(S). Let hu1, u2, u3i be a path with length two. We set A = NΓn(S)(u1) ∪ NΓn(S)(u2) ∪ NΓn(S)(u3), F1 = A − {u2, u3} and F2 = A − {u1, u2}. It is straightforward to check that F1 andF2 are two condi-tional faulty sets, and F1 and F2 are indistinguishable by Theorem 1. When Γn(S) is a star graph, it has no cycles with length less than 6, hence the vertices in
NΓn(S)(u1), NΓn(S)(u2), and NΓn(S)(u3) are all differ-ent, thus|F1| = |F2| = 3n − 6. On the other hand, if
Γn(S) is not a star graph, it contains 4-cycles, so some of those neighbors may be the same. However, it is easy
to see that any two vertices inΓn(S) can have at most two common neighbors. Thus when the pathhu1, u2, u3i is part of a 4-cycle, we get|F1| = |F2| = 3n − 7. In both cases we have|F1− F2| = |F2− F1| = 1, there-fore whenΓn(S) is a star graph, it is not conditionally
(3n − 6)-diagnosable, otherwise Γn(S) is not condition-ally(3n − 7)-diagnosable. Hence we have the following
result:
Proposition 4. Forn ≥ 4, tc(Γn(S)) ≤ 3n − 7 when
Γn(S) is a star graph, otherwise tc(Γn(S)) ≤ 3n − 8. The following two lemmas will be needed to show our result on the conditional diagnosability ofΓn(S) for n ≥
4.
Lemma 5. Forn ≥ 4, let F1andF2be any two distinct conditional faulty subsets ofV (Γn(S)) with |F1| ≤ 3n−7 and|F2| ≤ 3n − 7 if Γn(S) is a star graph, and |F1| ≤
3n − 8 and |F2| ≤ 3n − 8 otherwise. Denote by H the maximum component of Γn(S) − (F1∩ F2). Then for every vertex u inF1△F2, u is inH.
Proof. Without loss of generality, we assume that u is
inF1− F2. SinceF2is a conditional faulty set, there is vertex v in(V (Γn(S)) − F2) − {u} such that (u, v) ∈
E(Γn(S)). Suppose that u is not a vertex of H. Then v is not inV (H), so u and v are part of a small component
inΓn(S) − (F1∩ F2). Since F1andF2are distinct, we have|F1∩ F2| ≤ 3n − 8 when Γn(S) is a star graph and
|F1∩ F2| ≤ 3n − 9 otherwise. Thus in Theorem 2 cases (iv)–(vi) can’t occur, hence{u, v} forms a component K2 ofΓn(S) − (F1∩ F2), i.e. u is the unique neighbor of v inΓn(S) − (F1∩ F2). This is a contradiction since F1 is a conditional faulty set, but all the neighbors of v are faulty inΓn(S) − F1.
Lemma 6. LetG be a graph with δ(G) ≥ 2, and let F1 andF2 be any two distinct conditional faulty subsets of
V (G) with F2 ⊂ F1. Then(F1, F2) is a distinguishable conditional pair under the comparison diagnosis model.
Proof. Let u be any vertex of F1 − F2. Since F1 is a conditional faulty subset ofV (G), there is a vertex v of V (G) − F1such that(u, v) ∈ E(G) and there is a vertex
w of V (G)−F1such that(v, w) ∈ E(G). Since F2⊂ F1, neitherv nor w is in F2. By Theorem 1,(F1, F2) is a dis-tinguishable pair.
Now we can prove our main results:
Theorem 7. For n ≥ 4, let F1 and F2 be two dis-tinct conditional faulty subsets ofV (Γn(S)). Assume that
|F1| ≤ 3n − 7 and |F2| ≤ 3n − 7 when Γn(S) is a star graph, and|F1| ≤ 3n − 8 and |F2| ≤ 3n − 8 otherwise. Then(F1, F2) is a distinguishable conditional pair under the comparison diagnosis model.
Proof. By Lemma 6,(F1, F2) is a distinguishable pair if
F1⊂ F2orF2⊂ F1. Thus we assume that|F1− F2| ≥ 1 and|F2− F1| ≥ 1. Let A = F1∩ F2. Then we have
|A| ≤ 3n − 8 when Γn(S) is a star graph, and |A| ≤
3n − 9 otherwise. Let H be the maximum component of Γn(S) − A. By Lemma 5, every vertex in F1△F2 is in
H.
We claim thatH has a vertex v outside F1∪F2that has no neighbor inA. Since every vertex has degree n − 1,
vertices inA can have at most |A|(n − 1) neighbors in H.
There are at most2(3n − 7) − |A| vertices in F1∪ F2, and at most two vertices ofΓn(S)−A may not belong to H by Theorem 2. Since|A| ≤ 3n−8, we have n!−|A|(n−2)− 2(3n − 7) − 2 ≥ n! − (3n − 8)(n − 2) − 2(3n − 7) − 2 ≥ 4
whenn ≥ 4. Thus there must be vertices of H outside F1∪ F2having no neighbor inA; let v be such a vertex.
If v has no neighbor inF1∪ F2, then we can find a path of length at least 2 withinH to a vertex p in F1△F2. We may assume that p is the first vertex ofF1△F2 on this path, and let q and w be the two vertices on this path immediately before p (we may have v = q), so q and ware not inF1∪ F2. Then the edges(q, w) and (w, p) show that(F1, F2) is a distinguishable conditional pair. Now assume that v has a neighbor inF1△F2. Then since the degree of v is at least 3, and v has no neighbor inA,
there are three possibilities:
(1) v has two neighbors inF1− F2, (2) v has two neighbors inF2− F1, or
(3) v has at least one neighbor outsideF1∪ F2. In each case Theorem 1 implies that(F1, F2) is a distin-guishable conditional pair ofΓn(S) under the comparison diagnosis model, finishing the proof.
To summarize, with Proposition 4 and Theorem 7, we have the following result.
Theorem 8. Forn ≥ 4, tc(Γn(S)) = 3n−7 when Γn(S) is a star graph, andtc(Γn(S)) = 3n − 8 otherwise.
Remark: Theorem 3 can be proved similarly, indeed
much simpler, using that its connectivity isn − 1, proved
in [6].
5 Conclusions
In the real world, processors fail independently and with different probabilities. The probability that any faulty set contains all the neighbors of some processor is very small [31], so we are interested in the study of conditional diagnosability. A new diagnosis measure pro-posed by Lai et al. [20] requires that each processor of
a system is incident with at least one fault-free proces-sor. In this paper, we considered Cayley graphs generated by transposition trees, which are a generalization of the
n-dimensional star graph Sn, and showed that the condi-tional diagnosability ofΓn(S) is 3n − 8 under the com-parison model except when it is the star graph, for which the conditional diagnosability is3n − 7. This number is
about three times as large as the classical diagnosability. It would be interesting to find other conditional measures for network reliability under which diagnosability of such networks are even higher.
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