Constructing Node-Disjoint Paths in Enhanced Pyramid Networks

Constructing Node-Disjoint Paths in Enhanced Pyramid Networks Hsien-Jone Hsieh 1 and Dyi-Rong Duh 2, * 1,2 Department of Computer Science and Information Engineering National Chi Nan University, Puli, Nantou Hsien, Taiwan 54561 1 s1321902@ncnu.edu.tw 2 drduh@ncnu.edu.tw Abstract. Chen et al. in 2004 proposed a new hierarchy structure, called the enhanced pyramid network (EPM, for short), by replacing each mesh in a pyramid network (PM, for short) with a torus. Recently, some topological properties and communication on the EPMs have been investigated or derived. Their results have revealed that an EPM is an attractive alternative to a PM. This study investigates the node-disjoint paths between any two distinct nodes and the upper bound of the ω-wide-diameter of an EPM. This result shows that the EPMs have smaller ω-wide-diameters than the PMs. Keywords: Enhanced pyramid networks, pyramid networks, fault-tolerance, wide diameter, node-disjoint paths, container, interconnection networks. 1 Introduction Pyramid networks (PMs, for short) have conventionally been adopted for image processing [6, 10], computer vision [6], parallel computing [5] and network computing [1]. A PM is a hierarchy structure based on meshes. Recently, there are many researches on the PMs, such as Hamiltonicity [2, 12, 16], pancyclicity [2, 16], fault tolerance [1], routing [7, 15], and broadcasting [8]. Note that the node degree of a PM is from 3 to 9, and both its node connectivity and edge connectivity are 3 [1, 16]. For establishing a PM in expandable VLSI chips, each of its nodes should be configured as a 9-port component or too many different components should be designed and fabricated. In other words, those nodes of degree less than 9 have unused ports. These ports can be used for further expansion or I/O communication. To modify a wellknown network a little bit such that the resulting network has better topological properties. Chen et al. [4] in 2004 proposed a variant network of the PM, named the enhanced pyramid network, by reconnecting some of the unused ports. An enhanced pyramid network (EPM, for short), suggested by Chen et al. [4], is a supergraph of a pyramid network with the same node set. In other words, a PM is a spanning subgraph of an EPM. The EPM can be constructed by replacing each mesh of the PM with a torus. Therefore, the hardware cost of the EPM would be slightly more expensive than the PM because some extra edges have to be added in the VLSI * Corresponding author. C. Jesshope and C. Egan (Eds.): ACSAC 2006, LNCS 4186, pp. 380 386, 2006. Springer-Verlag Berlin Heidelberg 2006

Constructing Node-Disjoint Paths in Enhanced Pyramid Networks 381 chips. Some topological properties and communication on the EPMs, including the number of nodes/edges, node connectivity, edge connectivity, diameter, routing algorithm, and a simple broadcasting algorithm, have been determined or derived [4]. Their results show that the EPM has better topological properties than the PM such as larger node/edge connectivity and better fault-tolerance ability. The topological structure of an interconnection network (network, for short) can be modeled by a graph [3, 10, 12, 13, 17]. The vertices and edges of a graph respectively correspond to nodes and edges of an interconnection network. The length of a path is the number of edges, which the path passes through. Given two nodes s and t in a network G, the distance between them, denoted by d G (s, t), is the length of their shortest path. The diameter of a network G, denoted by d(g), is defined as the maximum of d G (s, t) among all pairs of distinct nodes in G. A ω-wide container, denoted by C ω (s, t), is a set of node-disjoint paths of width ω. The length of C ω (s, t), denoted by l(c ω (s, t)), is the length of the longest path in C ω (s, t). The total length of C ω (s, t), denoted by l T (C ω (s, t)), is the sum of the lengths of the ω paths in the C ω (s, t). The ω- wide distance between s and t in G, denoted by d ω (s, t), is minimum among all l(c ω (s, t)). The ω-wide diameter of G, written as d ω (G), is the maximum of ω-wide distance among all pairs of distinct nodes in G. Obviously, d G (s, t) d ω (s, t), d(g) d ω (G), and d ω (s, t) d ω (G). Parallel transmission is a one-to-one communication in G such that the message can be transmitted from the source node to the destination node via a container between them to enhance the transmission performance and/or improve fault tolerance ability of the communication. Notably, the parallel transmission delay is bounded above by d ω (G). This work first constructs a ω-wide container between any two distinct nodes in an EPM and then based on the constructed containers the upper bound of the ω-wide diameter of the EPM can be determined. The rest of this paper is organized as follows. Section 2 describes the structure and terms of an EPM, and some notations and definitions in graphs. Section 3 first constructs a ω-wide container between any two distinct nodes in an EPM and then determines the upper bound of the ω-wide diameter of the EPM. Finally, the conclusion is made in Section 4. 2 Preliminaries This section first formally describes the structures of the PM and EPM and then defines some notations and definitions in graphs that are used in the rest of this paper. The node set of a mesh M(m, n) is V(M(m, n)) = {(x, y) 0 x<m, 0 y<n}. Two nodes (x 1, y 1 ) and (x 2, y 2 ) are joined by an edge iff x 1 x 2 + y 1 y 2 = 1, where (x 1, y 1 ) and (x 2, y 2 ) belong to V(M(m, n)). The n-layer pyramid network, denoted by PM[n], is a hierarchy structure based on meshes. The node set of PM[n] is V(PM[n]) = {(k; x, y) 0 k n, 0 x, y<2 k }, where n 0. Note that the node (0; 0, 0) is the PM[0]. A node (k; x, y) V(PM[n]) is said to be a node at layer k with the coordinate (x, y). The nodes at layer k are connected as a M(2 k, 2 k ). The EPM is a conjunction of a quad tree and tori. The node set of a torus T(m, n) is V(T(m, n)) = {(x, y) 0 x<m, 0 y<n}. Let {u, v} denote an edge connecting nodes u and v of a network. The edge set E(T(m, n)) = E(M(m, n)) {{(x, 0), (x, n 1)}, {(0, y), (m 1, y)} 0 x<m, 0 y<n}. In other words, an EPM can be constructed by replacing

382 H.-J. Hsieh and D.-R. Duh each mesh of a PM with a torus. The nodes at layer k are connected as a T(2 k, 2 k ). Notice that a M(2, 2) is also a T(2, 2) in some sense. An EPM of n layers is denoted by EPM[n]. The node set of EPM[n] is V(EPM[n]) = {(k; x, y) 0 k n, 0 x, y<2 k }, where n 2. In general, the node (0; 0, 0) is called the apex of EPM[n] (apex, for short). For ease of discussion, we define some symbols in the following. For a node v = (k; x, y) at layer 1 k n in EPM[n], the coordinate of its parent, denoted by P(v), is given by (k 1; x/2, y/2 ). Conversely, v is a child of P(v). Moreover, each node in EPM[n] has a parent (four children) except the apex (the nodes at layer n). More generally, we recursively define the h th ancestor of v, denoted by P h (v), as follows: (1) h=1, P 1 (v) = P(v) is simply the parent of v. (2) h>1, P h (v) = (k h; x/2 h, y/2 h ) is the parent of P h 1 (v). For a node v = (k; x, y) at layer 0 k<n in EPM[n], the coordinates of its four children are given by (k+1; 2x, 2y), (k+1; 2x+1, 2y), (k+1; 2x, 2y+1), and (k+1; 2x+1, 2y+1). Conversely, v is the parent of its children. For simplicity, let (a) b denote a modulo b. For a node v = (k; x, y) at layer 2 k n in EPM[n], (k; (x+1) 2 k, y), (k; x, (y+1) 2 k), (k; (x 1) 2 k, y), and (k; x, (y 1) 2 k) are the coordinates of its four siblings, and they are also denoted by S 0 (v), S 1 (v), S 2 (v), and S 3 (v), respectively. 3 Node-Disjoint Paths Before describing how to construct ω node-disjoint paths between any pair of nodes in EPM[n], some lemmas are first presented. These lemmas related to how to routing paths in EPM[n] are stated in Subsection 3.1. By the aid of these lemmas, a shortest path between any pair of nodes is first constructed and then the other ω 1 paths can be built based on the shortest path. Subsections 3.2 describes how to construct these ω paths, where 2 ω 4. 3.1 Routing in EPM[n] Given two paths P 1 and P 2, let P 1 P 2 denote joining P 2 to the tail of P 1. Let P G (u, v) denote a path between two nodes u and v in a network G. Let l(p) denote the length of a path P. Some results derived by Chen et al. [4] are described first. Lemma 1 [4]. The node connectivity of EPM[n] is 4. By Menger s theorem [17] and Lemma 1, there is a 4-wide container between any pair of nodes in EPM[n]. Lemma 2 [4]. If d T(2 k, 2 k )(u, v) < 2+d T(2 k 1, 2 k 1 )(P(u), P(v)), d T(2 k, 2 k )(u, v) = d EPM[n] (u, v). Lemma 3 [4]. Given two nodes u=(k u k; x u, y u ) and v=(k v k; x v, y v ), 0 k v k u <n, there is a shortest P EPM[n] (u, v) having the form as P EPM[n] (u, P k u k v +i (u)) P T(2 k, 2 k )(P k u k v +i (u), P i (v)) P EPM[n] (P i (v), v), where 0 i<k v.

Constructing Node-Disjoint Paths in Enhanced Pyramid Networks 383 By Lemma 3, a shortest path P EPM[n] (u, v) can be constructed and it is the first constructed path when a container is built. Since a shortest path between any two nodes has been established, the diameter of EPM[n] can be easily obtained and stated in the following lemma. Lemma 4 [4]. d(epm[n]) = 2n, where n 2. After constructing a ω-wide container between any pair of nodes in a network, the upper bound of the ω-wide diameter of the network can be determined as the maximum length among the constructed ω-wide containers. The first path P 1 (s, t), constructed by Lemma 3, is shortest and its length is at most d(epm[n]) = 2n. Then the other ω 1 paths P 2,..., P ω, can be constructed based on P 1, they disjoint to each other and also disjoint to P 1. Notably, the other ω 1 paths are never shorter than P 1, and then a method is proposed to share P 1 with the others, in order to shorten the length of the ω-wide container. Given two nodes u and v at layer k u and k v of EPM[n], respectively. They have at most four siblings S i (u) and S j (v), respectively, where 0 i, j 3. P(u) (P(v)) also has at most four siblings S i (P(u)) (S j (P(v))), 0 i (j) 3. Let Q ij (u, v) denote a path from S i (u) to S j (v) along siblings of P 1 (u, v). We can recursively construct Q ij (u, v)as follows: (1) v is P(u): Q ii (u, P(u)) is a shortest path from S i (u) to S i (P(u)) excluding P 1 (u, P(u)). (2) v is P h (u), where 2 h k u 2: Q ii (u, P h (u)) = Q ii (u, P h 1 (u)) Q ii (P h 1 (u)), P h (u)). Note that Q ii (u, P h (u)) might not be a shortest path from S i (u) to S i (P h (u)) in EPM[n] excluding P 1 (u, P h (u)). (3) Let w=(k w ; x w, y w )=P k u k w (u), z=(k z ; x z, y z )=P k v k z (v), if 2 k w =k z min{k u, k v }: Q ij (u, v) = Q ii (u, w) P T(2 kw, 2 kw) (S i (w), S j (z)) Q jj (v, z). Obviously, there are 4 Q ii (u, P(u))s can be constructed. For each Q ii (u, P(u)), it is represented by π 1 (π 2 ) if l(q ii (u, P(u))) = 1 (2). l(q ii (u, P(u)))+l(Q mm (u, P(u))) = 3, where m is (i+2) 4. Also, if Q ii (u, P h (u)) has h 1 π 1 s and h 2 = h h 1 π 2 s, then Q mm (u, P h (u)) has h 2 π 1 s and h 1 π 2 s, where m is (i+2) 4. Therefore, l(q ii (u, P h (u)))=h+h 2, l(q mm (u, P h (u)))=h+h 1, and l(q ii (u, P h (u)))+l(q mm (u, P h (u))) = 3h. The shortest one of Q ii (u, P h (u)) and Q mm (u, P h (u)) is denoted by Q S (u, P h (u)). Otherwise, the longest of them is denoted by Q L (u, P h (u)). Therefore, l(q S (u, P h (u))) 3h/2 = h+ h/2, and l(q L (u, P h (u))) 2h. That is, there are at most h/2 (h) π 2 s in a Q S (u, P h (u)) (Q L (u, P h (u))). Q S is also denoted as Q SE (Q SO ) if i is even (odd). 3.2 ω-wide Container This subsection constructs a ω-wide container between any pair of nodes in EPM[n]. For ease of discussion, let s=(k s ; x s, y s ), t=(k t ; x t, y t ), w=(k w ; x w, y w )=P k s k w (s), z=(k z ; x z, y z )=P k t k z (t) be four nodes in EPM[n] for n 2, where 2 k w =k z min{k s, k t }. Lemma 5 [9]. l(q S (s, w)+d T(2 2, 2 2 )(S i (w), S j (z))+l(q S (z, t) 2n+2 n/2 4, l(q S (s, w)+d T(2 2, 2 2 )(S i (w), S j (z))+l(q L (z, t) 3n+ n/2 5, and l(q L (s, w)+d T(2 2, 2 2 )(S i (w), S j (z))+l(q S (z, t) 3n+ n/2 5.

384 H.-J. Hsieh and D.-R. Duh Lemma 6. l(c 2 (s, t)) and l(c 3 (s, t)) 2n+2 n/2 2, l(c 4 (s, t)) 3n+ n/2 3, l T (C 2 (s, t)) 4n+2 n/2 2, l T (C 3 (s, t)) 6n+4 n/2 4, and l T (C 4 (s, t)) 10n+2 n/2 6. Proof. For ω-wide container, 2 ω 4, by Lemma 3 and Lemma 4, a shortest path P 1 (s, t) can be first constructed and l(p 1 (s, t)) 2n. Second, a Q ij (s, t) is constructed as the second path P 2 (s, t) = {s, S i (s)} Q SE (s, w) P T(2 2, 2 2 )(S i (w), S j (z)) Q SE (z, t) {S j (t), t}. By Lemma 5, l(p 2 (s, t)) 2n+2 n/2 2. Thus, l(c 2 (s, t)) 2n+2 n/2 2 and l T (C 2 (s, t)) 4n+2 n/2 2. A C 3 (s, t) can be constructed by adding the third path P 3 (s, t)={s, S i (s)} Q SO (s, w) P T(2 2, 2 2 )(S i (w), S j (z)) Q SO (z, t) {S j (t), t} into the original C 2 (s, t). l(p 3 (s, t)) 2n+2 n/2 2. Hence, l(c 3 (s, t)) 2n+2 n/2 2 and l T (C 3 (s, t)) 6n+4 n/2 4. For constructing a C 4 (s, t), P 3 (s, t)={s, S i (s)} Q SO (s, P n 2 (s)) P T(2 2, 2 2 )(S(P n 2 (s)), S(P n 2 (t))) Q L (P n 2 (t), t) {S j (t), t} and P 4 (s, t)={s, S(s)} Q L (s, P n 2 (s)) P T(2 2, 2 2 )(S(P n 2 (s)), S(P n 2 (t))) Q SO (P n 2 (t), t) {S j (t), t}. By Lemma 5, both l(p 3 (s, t)) and l(p 4 (s, t)) 3n+ n/2 3, and l T (C 4 (s, t)) 10n+2 n/2 6 is maximum. Note that all P T(2 2, 2 2 )s disjoint to each other. The C ω (s, t) constructed in the proof of Lemma 6 is too long and l(p ω (s, t))s l(p 1 (s, t)) are too large, for 2 ω 4. A C ω (s, t) can be shortened by rerouting its paths such that each path shares some part of the P 1 (s, t). The X(P EPM[n] (u, P 2 (u)), Q ii (u, P 2 (u))) is an operation to reroute the two node-disjoint paths P EPM[n] (u, P 2 (u)) and Q ii (u, P 2 (u)) in EPM[n] by using some nodes near to them, where u V(P 1 (s, P n 2 (s)). After rerouting, one path from u to S i (P 2 (u)) is denoted by R 1, the other path from S i (u) to P 2 (u) is denoted by R 2, and they are still disjoint to each other. The rerouting would little increase the sum of the lengths of the paths. l(r 1 ) l(p EPM[n] (u, P 2 (u))) and l(r 2 ) l(q ii (u, P 2 (u))) are represented by c 1 and c 2, respectively. The main idea of rerouting is to share the shortest path of the original container with the other paths such that the longest path in C ω (u, v) is only one longer than the shortest path and then l(c ω (u, v)) can be minimized. Lemma 7. X(P EPM[n] (u, P 2 (u)) and Q ii (u, P 2 (u))) can be completed within 2 layers, and c 1 =2, c 2 = 0 or 1. Proof. Given a node u=(k; x, y), there are 4 Q ii (u, P 2 (u)))s. Without lost of generality, only X(P EPM[n] (u, P 2 (u)), Q 00 (S i (u), S i (P 2 (u)))) is discussed. Let x=(x k 1 x k 2 x 2 0x 0 ) 2 in binary, and the bit 0 of y is y 0, Q 00 (P(u), P 2 (u)) is a π 2. Hence, P(u)=(k 1; (x k 1 x k 2 x 2 0) 2, y/2 ), P 2 (u)=(k 2; (x k 1 x k 2 x 2 ) 2, y/4 ), H 0 (u)=(k; (x k 1 x k 2 x 2 10) 2, y), H 0 (P(u))=(k 1; (x k 1 x k 2 x 2 1) 2 +1, y/2 ), S 0 (P 2 (u))=(k 2; (x k 1 x k 2 x 2 ) 2 +1, y/4 ), a=(k; 2 x/2 +2, 2 y/2 +1), and b=(k; 2 x/2 +2, 2 y/2 +2). Before rerouting, l(p 1 (u, P 2 (u))) = 2 and l(q 00 (u, P 2 (u))) = 4 x 0. In order to keep disjoint to the other Q ii (u, P 2 (u)))s, after rerouting, R 2 must be P T(2 k, 2 k )(S 0 (u), a) {a, b} P(b, P 2 (u)). l(r 1 ) = 4 and l(r 2 ) = 5 x 0 y 0. Hence, c 1 = 4 2 = 2, c 2 = (5 x 0 y 0 ) (4 x 0 )=1 y 0 = 0 or 1. Theorem 8. d 2 (EPM[n]) 2n+ n/2 1, for n 2. d 3 (EPM[n]) 2n+ (2n+2)/3 (2n+ 2n/3 ) if n is even (odd), for n 4. d 4 (EPM[n]) 2n+ 3n/4 (2n+ (3n 1)/4 ) if n is even (odd), for n 4. Proof. This theorem can be proved by rerouting the C ω (s, t) constructed in the proof of Lemma 6.

Constructing Node-Disjoint Paths in Enhanced Pyramid Networks 385 Case 1. (for C 2 (s, t)): Only the longest C 2 (s, t) should be considered. After reconstruction, the new paths are P' 1 (s, t)=p EPM[n] (s, w) P T(2 2, 2 2 )(w, S j (z)) Q jj (z, t) {S j (t), t} and P' 2 (s, t)={s, S i (s)} Q ii (s, w) P T(2 2, 2 2 )(S i (w), z) P EPM[n] (z, t). The lengths of P' 1 (s, t) and P' 2 (s, t) are at most 2n+ n/2 1. Case 2. (for C 3 (s, t)): Assume there are h s and h t π 2 s in Q SE (s, w) and Q SO (t, z), respectively, the lengths of h s and h t are at most (n 2)/2. Let h sx = 2 h s /3 1, h tx = 2 h t /3 1. After applying X(P EPM[n] (P hsx 2 (s), P hsx (s)), Q SE (P hsx 2 (s), P hsx (s))) and X(P EPM[n] (P htx 2 (t), P htx (t)), Q SO (P htx 2 (t), P htx (t))), the new paths are P' 1 (s, t), P' 2 (s, t), and P' 3 (s, t) with lengths l' 1, l' 2, and l' 3, respectively. By Lemma 6 and Lemma 7, l' 1, l' 3 2n+ (2n+2)/3 (2n+ 2n/3 ) and l' 2 2n+ (2n+2)/3 (2n+ 2n/3 ) if n is even (odd), for n 4. Case 3. (for C 4 (s, t)): After applying 3 times of rerouting, the new paths are P' 1 (s, t), P' 2 (s, t), P' 3 (s, t), and P' 4 (s, t) with lengths l' 1, l' 2, l' 3, and l' 4, respectively. Similarly, l' 1, l' 4 2n+ 3n/4 (2n+ (3n 1)/4 ), l' 2, l' 3 2n+ 3n/4 (2n+ (3n 1)/4 ) if n is even (odd), for n 4. It is easy to check that d 2 (EPM[1]) = d 3 (EPM[1]) = 2, d 3 (EPM[2]) = d 4 (EPM[2]) = 4, d 3 (EPM[3]) = 6, and d 4 (EPM[3]) = 7. 4 Concluding Remarks This work has revealed that d 2 (EPM[n]) 2n+ n/2 1, for n 2, d 3 (EPM[n]) 2n+ (2n+2)/3 (2n+ 2n/3 ), n is even (odd), for n 4, and d 4 (EPM[n]) 2n+ 3n/4 (2n+ (3n 1)/4 ), n is even (odd), for n 4. In [1], d 2 (PM[n]) =3n 1, and d 3 (PM[n]) 10n/3+6. Therefore, an EPM has smaller ω-wide-diameter than the pertinent PM. The ω-path transmission delay of a network is bounded below by its ω-wide diameter. The lower bound of d ω (EPM[n]) is still unknown. We are going to determine the lower bound of d ω (EPM[n]) and we claim that it is equal to its upper bound. If this is true, d ω (EPM[n]) can be eventually obtained. Acknowledgments. The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC-93-2213-E-260-005-. References 1. Cao, F., Du, D.-Z., Hsu, D. F., Teng, S.-H.: Fault tolerance properties of pyramid networks. IEEE Transactions on Computers 48 (1999) 88 93 2. Chen, Y.-C.: Pancycles and Hamiltonian connectedness of the pyramid network with one node or one edge fault. Master Report, Department of Computer Science and Information Engineering, National Chi Nan University (2003) 3. Chen, W.-M. Chen, G.-H. Hsu, D.-F.: Generalized diameters of the mesh of trees. Theory of Computing Systems 37 (2004) 547-556

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