highly fault-tolerant communication network models

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 1, JANUARY 1989 23

Highly Fault -Toler an t Communication Network Models

JAROSLAV OPATRNY, N. SRINIVASAN, AND VANGALUR S. ALAGAR

Abstract -Constructions of new classes of highly fault-tolerant communication network models based on balanced incomplete block designs are presented. For a block design with parameters (b , n, r, k , l ) the first two constructions give k-connected networks (ICl, b, n, r, k),(Kl,k, b, n , r, k ) with degree sequence ( r , k ) of diameter 3 or 4, 4 or 5 on b + n, n ( r + 1) nodes respectively. It is shown that removal of any (k - 1) nodes in these two classes of networks increases the diameter of the network at most by 1 and 3, respectively.

b, n, r, k)k-‘ with degree sequence ( r , k ) of diameter 4(k - I ) + 2 is obtained using the design (b, n , r, k , l ) repeatedly ( k - I ) times starting with the graph K, , k. The fault-tolerant properties of this network model are investigated and a comparison to other network models is done.

The network model

I. INTRODUCTION EMAND for high computing power has led to con- D structions of multiprocessor networks consisting of

autonomous processors, where each processor is connected by a communication link to a number of other processors. In such a network processors communicate through a communication path consisting of an alternating sequence of communication links and processors. For many applica- tions it is desirable that such a multiprocessor system have the following properties [4].

1) The communication network accommodates a large number of processors.

2) There is a limit on the number of communication lines connected to each processor.

3) The length of a communication path between any two processors should be small.

4) The network must be fault-tolerant, i.e. the perfor- mance of the network should not degrade rapidly if a few processors become faulty.

Modeling a network as a graph G = ( V ( G ) , E ( G ) ) in which nodes V ( G ) correspond to switching nodes or processors, and edges E ( G ) correspond to the communication links, we can study the above properties precisely in a graph-theoretic setting. We give below the relevant defini- tions. We follow graph terminology from [6].

8 I

Manuscript received October 10, 1985; revised August 7, 1987, and May 11, 1988. This paper was recommended by Associate Editor R. W. Liu.

J. Opatrny and V. S . Alagar are with the Department of Computer Science, Concordia University, Montreal, Canada H3G 1M8.

N. SriNvasan is with the Department of Computer Science, Concordia University, Montreal, P.Q., Canada H3G 1M8, on leave from the Depart- ment of Mathematics, A.M. Jain College, Madras 600 114, India.

IEEE Log Number 8824450.

The length of a shortest path between nodes x , y is denoted by d ( x , y ) . The diameter d ( G ) of G is defined to be m a { d ( x , y ) l x , y , E V(G)}. The degree sequence of a graph G is defined to be the list of distinct degrees of the nodes of G . A graph G is k-(node) connected if there exist k node-disjoint paths joining every pair of nodes of G. A graph G is said to be maximally connected if its connectivity is equal to the minimum of its degree sequence. The complete graph on k nodes and the complete bipartite graph on a partition with i and k nodes is denoted by K , and K j , ,, respectively.

Let X be a subset of the vertex set of graph G . The subgraph of G induced by X is the graph ( X , E ( G ) n

Let F C V ( G ) be a set of faulty nodes in G. We define the surviving graph G / F as a graph induced by V - F.

Depending on the desired behaviour of the communication network, several different fault-tolerant criteria can be defined. The simplest criterion is the number of disjoint communication paths between any two nodes of network, whch corresponds to the node connectivity of the network. The network should be maximally connected. Since in most networks the switching delay is critical, the length of the communication path should not increase substan- tially in the presence of faulty nodes. Thus d( G / F ) - d( G ) should be small. This property is sometimes referred to as “diameter stability” [ 91.

In some networks a pair of nodes is assigned a fixed path for communication between the nodes. This assign- ment of a path to every pair of nodes is called a routing. A node failure in a network with a fixed routing can inter- rupt routing between some nodes, say X and Y, which have not failed. Communication between X and Y may still be possible by X sending the message along all the routes from X which have not been interrupted, called surviving routes [2], and rebroadcasting the message at the end of each route. If there is a sequence of surviving routes, say from X to Z,, Z , to Z,,. . e , Z j to Y, message from X will reach Y after i rebroadcasts. Thus an impor- tant consideration for a design of a fault-tolerant network with fixed routing is the minimization of the maximum number of broadcasting of the message needed for communication between pair of nodes of the network in presence of faults. The above motivated the introduction of the concept of surviving route graph in [2] as defined below.

( X x X ) .

0098-4094/89/0100-0023$01.00 01989 IEEE

24 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 1, JANUARY 1989

A

R ( G , p ) / F where F = hB):

Fig. 1

For a communication network G ( V , E ) a routing p is a function which assigns to an element (x, y ) from V x I/ a fixed path between x and y . The routing will be called minimal if p ( x , y ) is a shortest path between x and y . The route graph R ( G , p ) of G has the same node set as G, and two nodes of the route graph are joined by a link if there is an assigned route by a routing p between the nodes. p ( x , y ) is said to avoid F, if no internal node of p ( x , y ) is a member of F.

The surviving route graph R ( G , p ) / F is the subgraph of R ( G , p ) induced by V ( G ) - F. The distance between two nodes in R ( G , p ) / F indicates, in a network with fixed routing, the number routes and the number of broadcasts necessary to send a message between the nodes if nodes in F are faulty. The diameter of the surviving route graph was studied for some networks by Broder et al. [2] and Dolev et al. [3]. Fig. l(a) shows network G with a minimal routing p and Fig. l(b) shows the surviving route graph R / F for a set of faulty nodes F.

In this paper we are presenting a new interconnecting scheme based on the theory of balanced incomplete block designs [5] for construction of fault-tolerant communication network models. Let b, n , r , k, A be positive integers. A balanced incomplete block design, (BIBD) is a set S of n elements and a family of subsets B,, B,; . - , B, of S called blocks such that:

(a) Each block contains exactly k elements of S; (b) Each element of S is contained in exactly r blocks;

(c) Every pair of distinct elements of S is contained in

This block design is denoted by ( b , n , r , k, A). A ( b , n , r , k, A)-design is called trivial if either n = 1 or n = k. In this paper we will consider only nontrivial BIBD’s. A BIBD with n = b is called symmetric. In such a design r = k and hence such designs are often called ( n , k, A) designs (see [5] ) . As an example, the blocks of a (7,3,1) symmetric BIBD are

(1,2,4), (2,3,5), (3,4,6) , (4,5,7), (5,6,1), (6,7,2), (7,1,3).

The following four properties of block designs will be

Property 1: Let B = { B,, B,, . . e , B , } be the blocks of

exactly A of these blocks.

used in the paper. (See [5] for proofs).

( b , n , r , k, 1)-design. IB, n BJI ~1 for every i and j , i # j .

Property 2: Let B = { B,, B,; . ., B , } be the blocks of

Then IB, n BJI =1 for every i and j , Property 3: An ( n (k), k, 1)-symmetric block design,

where n( k ) = k2 - k + 1, exists when (k - 1) is a power of a prime number.

an (n, k, 1) symmetric block design. i # j .

Property 4: r 2 k for any nontrivial BIBD. For any integer k, n ( k ) is defined to be k 2 - k + l . As seen from the above example of a balanced incom-

plete block design, the blocks of a BIBD form a “well- overlapping” system of sets. In Sections I1 and I11 we define three network models based on BIBD’s. In general, network model ( G , b , n , r , k) consists of b copies G,, G, , . * *, G, of graph G and nodes U , , U , , - . ., U,,. Node vJ is connected to a node of graph G, if block B, of ( b , n , r , k,l)-design contains vJ.

For ( b , n , r , k, 1)-design and graph K , we will construct a k-connected network ( K , , b, n , r , k) with degree sequence (k, r ) of diameter 3 or 4 on b + n nodes. For ( b , n , I , k, 1)-design and graph K,, we construct a k-connected network ( K , , b, n , r , k) with degree sequence ( r , k ) of diameter 4 or 5 on n(r + 1) nodes. It is shown that for any set F of faulty nodes, IF1 < k, the diameter of ( K , , b, n , r , k ) / F , ( K , , b, n , r , k)/F is at most 4,7, respectively. Also, when a minimum routing p is used, the surviving route graph of the above networks is of diameter < 2.

For symmetric block designs the above network models are regular graphs.

In Section I11 the network model (K,& b, n , r , k)k-‘ is considered. This network model is obtained by using BIBD-interconnection recursively k- , times to graph K, , k .

The ( K , , k , b , n , r , k),-‘ network model is maximally connected with degree sequence ( r , k) of diameter 4( k - i ) + 2. For any set F of faulty nodes of the network, IF1 < k, the surviving graph (K , , k , b, n , r , k),-’/F has a diameter that is at most 4(k - i ) + 5. Furthermore, for a minimal routing p in (K,,,, b, n , r , k),-l the diameter of the surviving route graph R(( K,, ,, b, n , r , k),-’, p ) / F is of diameter 2. Thus the networks introduce in this paper have very good fault-tolerant properties. A comparison of the size and

OPATRNY et al. : COMMUNICATION NETWORK MODELS 25

( 1 2 , 9 , 4 , 3 , 1 ) - d e s l g n - ( 1 . 2 . 1 1 , ( 3 , 5 , 7 ) , ( 1 . 4 . 7 ) , (4.5.6) , (1.6,S) , (7.8.9) ,

( 2 , 4 , 9 1 , ( 2 . 5 . 8 ) , ( 1 . 5 . 9 ) , (3.4.8) , (3.6.9) , ( 2 . 6 . 7 )

Fig. 2.

properties of the proposed networks to results in [l] and to modified shift register graphs [4] is done.

11. NETWORK MODELS (K,, b, n, r, k) AND

(Kk, b, n, r, k) Construction 1: Let B = { B,, B2,- . -, B b } be the blocks

of a (b, n, r , k, 1)-design on set S = { u l , u2 , - * e , U,}. Construct the graph ( K l , b, n , r , k ) having nodes { ul, u z , - . ., u b } U { U,, u 2 , - . 0 , U,} in which U, is adjacent to uj if ui E Bj. The resulting graph is bipartite with a degree sequence ( r , k).

Fig. 2 gives (12,9,4,3,1)-design and graph ( K , , 12,9,4,3).

Lemma 2.1: In the graph ( K , , b, n , r , k,l) there is exactly one path of length 2 between every pair of nodes at distance 2.

Proof: To prove the lemma it is sufficient to show that graph ( K , , b, n , r , k ) does not contain C,, the cycle of length 4. If the graph contains C, then it should be of the form U , , u j , U,, U, since G is bipartite. This implies that U,, U, E Bj n Bj, a contradiction to Property 1.

Lemma 2.2: The diameter of ( K , , b, n, r , k) is 3 if B, n Bj # rp for every i , j and is equal to 4 otherwise.

Proof: We have to consider the distances between the following pairs of nodes:

1) U , , j , i # j : U, and uj are connected by a path of length 2 consisting of nodes U,, U,, uj where t is the index of the block containing pair U,, uj;

2) U,, u j : if U, E Bj then d( U,, u j ) = 1, otherwise U,, uj are connected by a path ui ,u , ,u j ,u j of length 3 where U , , uj E B,;

3) ui, uj , i # j : if B, n Bj # rp then the nodes are connected by a path of length 2 consisting of nodes U,, us, uj where U, E B i n Bj; otherwise ui, uj are connected by a path of length 4 having nodes U,, U,, U,, U,, uj where U, E B,, U, E Bj and U,, U, E B,.

Thus d ( G ) = 3 or 4 and the proof is complete.

Lemma 2.3: There are k node-disjoint paths of length Q 4 between every pair of nodes in the graph

Proof: We have to consider the following three types of pairs of nodes of the graph:

1) U,, uj , i # j : Let B, = { U,,, ui2; *, U,,} and Bj = { uj,, uj2,. . -, uj,}. Let B,, be the block containing the pair U,,, ujt, 1 Q t Q k. Now the paths U,, U , , , U,,, uj,, uj; t = 1,2; e , k, are k node-disjoint paths of length < 4 between the two nodes.

2) U,, uj, i # j : Construct two sequences of pairwise distinct blocks B,,, Biz, . . . , B,,, and Bjl, Bj2, * . *, Bjk such that U, E B,,, uj E Bj, and B,, n Bj, # rp for 1 Q t Q k as follows :

(a) B,, = Bj, is the block containing the pair U,, uj. (b) For B,, take a block containing ui different from

B,,,. e , B,,-,. Now, B,, must contain an element us, such that U,, 4 Bj, U Bj2 U . . . U Bj,-,. For Bjt take the block containing the pair uj, U,,.

Now the paths U,, uil, uj, and U,, U,,, us,, ujt , uj; 2 Q t Q k are k node disjoint paths of length Q 4.

3) uj, U,: Let B, = { uil, ui2; . e , u i k } and let Bjp be the block containing the pair uip,uj; l ~ p ~ k. Now the paths, uj, ujp, U. U,, 1 Q p Q k, are k node-disjoint paths of length Q 3 between uj, U,. Thus the proof of the lemma is complete.

Corollary 2.1: The network ( K , , b, n, r , k) is maximally connected, and for any set F of faulty nodes in

( K l , b, n, r , k).

'P'

Lemma 2.3 implies the following corollary.

(K1, b, n, r , k), IF1 Q (k -1). d((K,,b,n,r,k)/F) Q 4 -

Theorem 2.1: Let G = ( K , , b, n , r , k) be a communication network with a minimal routing p . For any set F of faulty nodes, IF I Q k - 1,

d ( R ( G , P I P ) Q 2. Proof: We will abbreviate R ( G , p ) / F by R / F in this

proof. By Corollary 2.1, between any two nodes x , y of G / F there exists at least one path of length Q 4 between x and y in G / F .

1) Consider a pair of nodes x, y of G / F , dGIF(x , y ) = 2. Since G is bipartite and, by Lemma 2.1, any pair of nodes at distance 2 has a unique path between them, x and y are adjacent in R / F .

2) Let x, zl, z2 , z3 , y be a path of length 4 in G / F . By 1) node x is adjacent to z2 in R / F and node z2 is adjacent to y in R / F . Thus d R , F ( ~ , y ) Q 2.

3) Let x, zl, z2 , y be a path of length 3 in G / F . By 1) node x is adjacent to z 2 in R / F and, clearly, z2, y are adjacent in R / F . Thus d R I F ( x , y ) Q 2.

Thus d( R / F ) Q 2.

~


(K4,13,13,4,41

Fig.

When Construction I is applied to a symmetric block design, we obtain a regular communication network model whose properties can be summarized as follows.

Corollary 2.2: For every integer k such that (k - 1) is a power of a prime number, ( K , , n ( k ) , n ( k ) , k , k , l ) is a k-regular communication network model having the following properties:

G is k-connected; d(G) = 3;

IV(G)l= 2( k2 - k + 1).

Furthermore, for any set F of faults, F C V(G), IF1 Q k -1;

d ( G/F) G 4; d(R(G,p ) /F ) =2foranyminimalroutingp.

Proot It follows directly from Property 2, Property 3, Corollary 2.1, and Theorem 2.1.

The graph (K,, n ( k ) , n ( k ) , k, k) is isomorphic to the incidence graph of a projective plane of order k - 1 from [l], but the fault-tolerant properties of these graphs have not been investigated yet.

Construction 11: Let B,, B,,. . + , B, be the blocks of a (b, n , r, k, 1)-design on set S = { U , , U * , u 3 , . . a , U,, }. For each block B, = { uI1, uI2; . -, u I k } , 1 < i Q b construct a complete graph K , and label its nodes as uI l , , , u I2 , , ; . ., ulk , , . Corre- sponding to elements u l , u 2 , - + a , U,, of S take nodes u 1 , u 2 , * * * , u n and join uJ to all U,,,, l g i ~ b , l ~ j ~ n .

3.

Denote the resulting graph as ( K , , b, n , r, k). Ths graph has degree sequence (r , k) and n( r + 1) nodes. For every I ,

1 < i Q b, V(B,) will denote the set of nodes of K , of ( K , , b, n, r, k) corresponding to B,.

Fig. 3 shows (K,, 13,13,4,4). Theorem 2.2: There exists exactly one shortest path be-

tween every pair of nodes in the graph ( K k , b, n, r, k ) . Moreover,

d ( ( K k , b , n , r , k ) ) = 4 i f B,nB,#+foreveryi , j ;

= 5 otherwise. Proof: We have to consider the following pairs of nodes

1) U,,, U,,, t # p : By construction, U,, E V( B,) and umP

a) Let B,n BP=+ Since pair uI ,um belongs to exactly one block B, for some j , the shortest path between U,,, U,, is U,,, U , , uIJ , umJ, U,, ump of length 5 and this path is unique.

b) Let B, n B, # cp. Then I B, n BPI = 1 by Property 2. Let U , E B, n B,, and hence U,, E V( B,) and uSp E

bl) If both i # s and m # s then the path u , ~ , U,,, U,, U,,, umP is the only shortest path between u I f , umP and d( U,,, U,,) = 4.

b2) If either i = s or m =s, say i = s then U,,, U , , u l p , ump is the unique shortest path of length 3 between them.

of (K,, b, n, r, k , l ) :

E V( BPI.

WP).

OPATRNY et al. : COMMUNICATION NETWORK MODELS

b3) If both i = s and rn = s then U , , , U , , U , ~ is the unique distance path of length 2 between the nodes.

2) U,,, U,, i # J . a) If both U , , uJ E B, then d(u,, , U,) = 2 and U,,, U,,, U,

is the unique shortest path. b) If U, E B, and U, 4 B,, then let B, be the block

containing pair U,, U,. Now d(u,, , U,) = 4 and U,,, U,, U,,, uJs, uJ is the unique shortest path.

3) uJ, U , i # j : Let B, be the block containing pair U/, U,. Now U,, U/,, U,,, U , is the unique shortest path of length 3 between the nodes.

Thus d(( K,, b, n , r , k)) = 4 or 5 and either pair of nodes

A graph having this property is called geodetic [7]. Theorem 2.3: The graph ( K , , b, n , r , k ) is a k-connected

with every pair of its nodes having at least k node-disjoint paths of length Q 7.

Proof: We have to consider the following pairs of nodes

has unique path of minimum length.

of the graph:

1) uit, ump: By construction, U,[ E V( Bl), umP E V( Bp). Let V B , ) = { U l l l , ul2,; . -, u l k l } and U B , ) - - { umlP, urn2,,. . ., umkp} . Now the paths

U,,, ~ l , , , uiX, ulzq, u m z q , U,,, u m z p , u m p , 1 Q 2 Q k where { ulz, U , , } belongs to B, for some s are k node disjoint paths of length Q 7 between U,,, umP. (This includes all cases of form i , = m,.) By construction, U,, E V( B,). Let V( B,) = { U,,,, um2,; . ., umkP}. Consider the paths

belongs to B, for some s. These are all node disjoint and of length Q 5. Construct two sequences of pairwise disjoint blocks B,,, BI2; a , Blk, and B,,, BJ2; a , BJk such that U, E B,,, U, E B,,, and Blr n BJ, # +, 1 Q t Q k as follows: (a) B,, = B,, is the block containing the pair

(b) For B,, take a block containing U, different from Bll; - , B,,- , . Now, B , must contain an element U,, such that u s , 4 BJ2U BJ2U U BJr-l. For BJ, take the block containing the pair U / , U,,.

u m p , urnzp9 urnz, urnz,, 'is, U, where { urn, U, 1

U , , urn.

Now the path ui, u i , uj, il, uj, and paths ui, ui,;,, u, , ,~ , , U,,, U,,, ,, , uj , uj, 2 Q t Q k are k node-disjoint paths of length Q 6 between U , , uj.

Thus the graph is k-connected and each pair of nodes are joined by at least k node-disjoint paths of length Q 7.

Corollury 2.3: For any set F of faulty nodes, IF1 Q (k - l), in a network G = ( K , , b, n , r , k )

~ ( G / F ) Q 7 .

Theorem 2.4: Let G = ( K , , b, n , r , k) be a network with a minimal routing p. For any set F of faulty nodes of

21

G , IF1 Q (k - 11, d ( R ( G , p ) / F ) Q 2 .

Proof: We will abbreviate R ( G , p ) / F by R / F in t h s proof. By Theorem 2.2, G has the unique property that between every pair of nodes, there exists a unique shortest path. By Theorem 2.3, each pair of nodes has at least k node-disjoint paths of length Q 7. Hence IF1 Q (k - 1) implies that G / F is connected. Consider the following pairs of nodes in R ( G , p ) / F .

In G / F there is a path of the type U,,

U,,, up , U ] , u J ~ , U,,, U,. Now ~ ( u g , uj) is the path U , , U,,,, uJs ,u j , and p ( u j , u m ) is the path U / , U,,, U,,, U , and they avoid F. Hence in R / F , U, is adjacent to uJ, and uJ is adjacent to U,, and dR/F( U,, U,) Q 2. There exists a path of the type

and U , m G / F . In G , p(umP, U , ) is the

path U,,, u,,,~ U,,, U, and these paths avoid F. Thus in R / F , dRIF(ump, U,) = 2. There exists a path of the type uir, ' i z ~ 9 viz, uizs, um,s, urnz, ' m z p , ' m p between U,,, U,,. Clearly p(u , , , U,,,) and ~ ( u ~ , ~ , u ~ ~ ) are the segments of the above path and they avoid F. Thus pairs of nodes U,,, u,,,~, and u , ~ , , umP are adjacent in R / F and dR/F(U,,, U,,) Q 2.

u m p , u m f p , urnz, u m p , uis, U, between u m p

path u m p , u m z p , omz, and P ( u m z , U , > is the

Thus d( R / F ) Q 2 and the proof is complete. When Construction I1 is applied to a symmetric block

design, we obtain a regular communication network model whose properties can be summarized as follows:

Corollary 2.4: For every integer k such that (k -1) is a power of a prime number, the graph G = (Kk, n( k), n( k), k, k, 1) is a k regular communication network model having the following properties:

G is k-connected; d ( G ) = 4;

) V ( G ) 1 = ( k2 - k + 1)( k + 1).

Furthermore, for any set F of faults, F c V ( G ) , IF1 i k -1,

~ ( G / F ) Q 7; d ( R ( G , p ) / F ) Q 2 for the minimal routing p.

Proof: It follows directly from Property 3, Property 2, Corollary 2.3, and Theorem 2.4.

The classes of networks ( K , , b, n , r , k),( K,, b, n , r , k ) presented above are models of large networks of a small diameter having very good fault-tolerant properties. Clearly, the only network which can have a better surviving route graph is the one corresponding to a complete graph. They also support very large number of nodes as indicated in Table I.


TABLE I

k Number of nodes

3 14 4 26 5 42 6 62 8 114 9 146 10 182

28 65

126 217 513 730

1001

111. NETWORK MODEL ( K , ,, b, n, r, k) ~

To obtain network models of higher diameter, we will iterate the construction given in Section 11. Informally, the graph ( K j , k , b, n , r, k)' is equal to (K;,,, b, n, r, k ) and ( K , , k , b , n , r , k ) " = ( ( K i , k , b , n , r , k ) s - - ' , b , n , r , k ) . More specifically, this model is constructed as follows:

Construction 111: Let B,, B,; . ., B, be the blocks of a (b,n,r,k,l)-design on the set S = { u 1 , u 2 ; ~ - , u , , } . Let BJ = { u . , u . . e . , u j k } , 1 < j d b. The graph ( K i , k , b, n, r, k)' is constructed as follows: For each block BJ construct a complete bipartite graph Ki, and label the nodes of Ki, , of degree k as { q1, j , z2,', j , . . ., q1, j > and the nodes of

elements of U' , U , , . . ., U,, of S take nodes U',', u ~ , ~ , . * * , U ' , , ,

and join U',, to all u ~ , ' , ~ , l < t < n , l < x < b. Graph ( K i , j , b, n , r , k)' has degree sequence (r , k , i + l), and b(i + k ) + n nodes and it contains b copies of K,,,.

2 graph ( K,, k , b, n, r, k ) , is defined recursively as follows: Let w = b"'. For each block Bj construct graph (K,,, , b, n, r, k)'-' and relabel in the m'th copy of K;,,, 1 < m < w its nodes U,,, U,,; . ., uak as

J i 12'

degree i as { ujI,', j , uj,, ', j , . . . , ujk,', }. Corresponding to

For s

. . . ' a l , j , , m , j , 'az. j 2 , m, j ' , uakjk, m, j y its nodes ZP,, zs2> * * ' 3 ZS, as zs,, m , j , zB2, m, j , . . . , z ~ , , ~ , j . Furthermore relabel nodes uyl, U,,,,- . ., of this COPY of ( K ; , k , b, n, r, k ) as uyl, J , uy2, j , . . . . Corresponding to elements u l , U , ; . ., U,, of s take nodes U,,l,l, U , , ' , , , . . . , Us,', w , Us,,,', Us,,,,,' * . ) us,,, W 3 - . . j u,,,,,l? ~ , , , , , 2 ~ * . us,,, and join to U , , r , X , y for every t ,x, y , 1< t g n , l < x 6 w , l < y < b. It is easy to see that graph (K , , k , b, n , r, k ) s has degree sequence (r , k , i + s) and bs-'(b(i + k ) + sn) nodes.

Fig. 4 shows the construction of graph ( K,, k , b, n, r, k ) , Lemma 3.1: The diameter of graph (K;, , , b, n, r, k)" <

Proof: We will prove it by induction on s. Clearly, the graph ( K i , k , b, n, r, k)' can be obtained

from the graph (K,, b, n, r, k ) by replacing each copy of K , by a copy of K;,,. Thus any shortest path in ( Ki, k , 6, n , r, k)' can be obtained from a shortest path in (K, , b, n, r, k ) by replacing an edge in Kk by a path of length 2. Since at most two edges of any shortest path in (K, , b, n , r , k ) belong to some copies of K , and the path is of length G 4, d((K,, k , b, n, r, k ) ' ) < 6 .

Assume now that d( (K l , k , b, n, r , k)"') < 4(s - l ) + 2. Consider the following pairs of nodes of graph

4s + 2 .

( K l , k , b, n, r, k ) s :

(a) U,,, i,. ml. j , , ' a 2 , i 2 , m2, j 2 3 j ,# j,: Let B, be the block containing the pair i,, i,. Now, there exist a3, a4

b I

I I I I

Fig. 4.

for which paths U,,, i I , m l , j,' us, i , , m,, 'a,, i , , ml. t and u~,,;,, m,, 1,) us, i , , m,' U,.,, 1,. m,, r of length 2 join the nodes to a copy of (Ki+ b, n, r, k),-' and, therefore, the distance of the two nodes < 4 s + 2.

us, ;,, j , : If i, # i , then let B, be the block contalning the pair i l , i 2 . By the construction

Since U,,, ;,, j l , r and u, , ,~ , , belong to the same copy of ( Ki,,, b, n, r, k)"?, the distance of the two U nodes ~ 4 s . If i l = i 2 then the two nodes are adjacent to the same copy of ( K,, ,, b, n, r, k ) and the result easily follows.

(c) u,,,l,,ml,,,, us,;,, j , : If i, # i , then let B, be the block containing the pair il,i2. By the construc-

is at distance 2 from node ua2,il. m,, and u s , ; , , j , is adjacent to ua3, i , , m,, r for some (Y,, a3, m 2 , m 3 . Thus the two nodes are at distance < 4s + 1. If i,= i, then U , , ; , , j 2 is adjacent to a node

which is at distance < 4(s - 1) + 2 from U a q . i l , m3. ~1

'a,, i l , ml ,J l . (d) us,;,, j l , x, where x is either z , or U,,,,,, s1 < s:

Since x is adjacent to some us for some /3 and, by (c), d( us, U,, ;,, j l ) Q 4s + 3, the lemma follows.

Notice that x is adjacent to a node u , , , ~ , , ~ , , j ,

such that i , E Bjl and, therefore, x is at distance 3 from a node 'a,. i,, m3. j 1 which is at distance 6 4(s - 1) + 2 from ual, ;,, m,, j l by the induction hypothe- sis.

(b) U,, ;,,

us , i , , j,' (us,i2,J2) is adjacent to ua,,;,, j , , r ' (ua2 , i2 , j , , 1).

tion, U,,, i , , m,, j ,

x where x is either z , or U,,,,,, s1 < s: (e) ~ a l . l l . m l . j l ~

et al. : COMMUNICATION NETWORK MODELS 29 OPATRNY

(0 x, y where each of x, y is of type z, or u.~,.~, j l , s, < s: As shown in (e), x is at distance < 3 from a node U-. whch is at distance < 4( s - 1) + 2 from -. -.,‘.L, 11.2, ,I

y . This completes the proof.

Lemma 3.2: There are s + i node disjoint paths of length Q 4s + 6 between every pair of nodes of graph

Proof: The proof is very similar to proofs of Lemma 3.1 and Theorem 3.3 and is therefore omitted.

When Construction I11 is applied to a symmetric block design with s = k - i , we obtain a regular communication network model whose properties are summarized in the next theorem.

Theorem 3.1: For every integer k such that (k - 1) is a power of a prime number, G = ( K l , k , n ( k ) , n(k), k, k ) k - r is a k-regular communication network model having the following properties:

G is k-connected,

( K l , k , by n, r , k)”.

d ( G ) = 4 ( k - i ) + 2 ,

IV(G) I = 2 k ( k 2 - k + l ) k - i .

Furthermore, for any set F of faults, F E V(G) , IF1

d ( G / F ) < 4(k - i ) +6 .

k -1,

Proof: It follows directly from Property 3, Property 2, Lemma 3.1, and Lemma 3.2.

IV. CONCLUSIONS The three classes of communication network models

presented in this paper have very good fault-tolerant properties. They have very large number of nodes, since the size of the network models grows exponentially with the diameter. For example, some of the network models ( K , , n(k), n( k), k, k) are the largest known regular graphs of diameter 3 (see [l]), and ( K k , n(k), n ( k ) , k , k), k 2 3, and some of the models ( K j , k , n ( k ) , n ( k ) , k, k ) k - r in Table I1 are larger than DeBruijn graphs [4]. In general, our graph models are larger than DeBruijn graphs for low degrees or low diameters. Also, unlike DeBruijn graphs, the models presented in this paper are very naturally defined for odd degrees.

A very interesting property of our network models is their “herarchical structure.” A network of higher diameter consists of several networks of lower diameter, each of them interconnected in a similar manner. It should also be noted that graphs other than K,, K, , or can be used in building network models and, if necessary, not all

components need to be identical, which can provide a great variety of network models.

Thus, the new models presented in th s paper should be seriously considered for constructions of large fault-tolerant networks.

REFERENCES [l]

[2]

[3]

J.-C. Bermond, C. Delorme, and J.-J. Quisquater, “Tables of large graphs with given degree and diameter,” Inf. Proc. Lett., vol. 15, no. 1; pp. 10-13, 1982. A. Broder, M. Fischer, D. Dolev, and B. Simons, “Efficient fault tolerant routings in networks,” in Proc. 16th Annual ACM Svmp. on Theory of Computing, pp. 536-541, 1984. D. Dolev, J. Halpern, B. Simons, and R. Shong, “A new look at fault-tolerant network routine.” in Proc. 16th Ann. ACM Svmo. on

, 1

Theory in Computing, pp. 52g1535, 1984. A. Esfahanian and S. L. Hakimi, “Fault-tolerant routing in DeBruGn communication networks,” IEEE Trans. Computers, vol.

M. Hall, Jr., Combinatorial Theory. F. Harary, Graph Theory. 0. Ore, “Theory of graphs,” Amer. Math. Soc., Providence, RI, 1962 C. S. Raghavendra, M. Gerla, and A. Aviziemis, “Reliable loop topologies for large local computer networks,” IEEE Trans. Com- puters, vol. C-34, pp. 46-55, 1985. N. I. Rubin and J. Hartman, “On diameter stability of graphs communication networks,” Proc. Conj. Information Sciences and Systems, the Johns Hopkins Univ., Baltimore, MD, 1976. R. S. Wilkov, “Construction of maximally reliable communication networks with minimum transmission delay,” Proc. of 1970 IEEE Inr. Conf. Commun., vol. 6, pp. 4210-4215, June 1970.

34, pp. 777-788, 1985. New York: Wiley, 1967.

Reading MA: Addison-Wesley, 1969.

models, analysis of

Jaroslav Opatrny received the M.Comp.Sci. degree from Charles University, Prague, Czechoslo- vakia in 1968, and the Ph.D. degree in computer science from the University of Waterloo, Canada, in 1975.

From 1975 to 1977 he was an assistant professor at the University of Alberta, Edmonton, Canada. In 1977 he joined the Computer Science Department at the Concordia University, Mon- treal, Canada, where he is presently an associ- ated professor. His main interests are network

xithms, and graph theory.

N. Srinivasan received the B.S. degree from Vivekananda College, University of Madras, In- dia, in 1963, the M.Phi1. degree from Ramanujan Institute of Advanced Mathematics, Madras University in 1977, and the doctorate degree by the Indian Institute of Technology, Madras in 1983.

He was a Post-Doctoral fellow at the Concor- dia University, Montreal, Canada from 1984 to 1985. He is presently a professor of Mathematics in the A.M. Jain College, Madras University. His

main interests include geodetic graphs, Hamiltonian cycles, and fault- tolerant network models.


Vangalur S. Alagar received the undergraduate and postgraduate degrees in mathematics with highest honors and rank from Madras Univer- sity, India.

After serving as lecturer in the department of Mathematics, Vivekananda College, Madras University, he joined McGill University, Mon- treal, Canada, where he received the Ph.D. degree in Computer Science in 1974. Subsequently, he joined the department of Computer Science, Concordia University, Montreal, Canada, first as

Assistant Professor and currently as a Professor. His research interests include the general area of algorithmics, robotic algorithms and lan- guages, and formal aspects of computing.

Dr. Alagar is a member of the Association of Computing Machmery.

highly fault-tolerant communication network models

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