The Journal of Supercomputing, 37, 23–45, 2006 C 2006 Springer Science + Business Media, LLC. Manufactured in The Netherlands.
The Grid-Pyramid: A Generalized Pyramid Network M.R. HOSEINYFARAHABADY IPM School of Computer Science, Tehran, Iran
[email protected]
H. SARBAZI-AZAD∗ azad@{sharif.edu, ipm.ir} IPM School of Computer Science, Tehran, Iran; Sharif University of Technology, Tehran, Iran
Abstract. The Pyramid network is a desirable network topology used as both software data-structure and hardware architecture. In this paper, we propose a general definition for a class of pyramid networks that are based on grid connections between the nodes in each level. Contrary to the conventional pyramid network in which the nodes in each level form a mesh, the connections between these nodes may also be according to other grid-based topologies such as the torus, hypermesh or WK-recursive. Such pyramid networks form a wide class of interconnection networks that possess rich topological properties. We study a number of important properties of these topologies for general-purpose parallel processing applications. In particular, we prove that such pyramids are Hamiltonian-connected, i.e. for any arbitrary pair of nodes in the network there exists at least one Hamiltonian path between the two given nodes, and pancyclic, i.e. any cycle of length 3, 4 . . . and N , can be embedded in a given N -node pyramid network. It is also proven that two link-disjoint Hamiltonian cycles exist in the torus-pyramid and hypermesh-pyramid networks. Keywords: Interconnection networks, Pyramid, Mesh, Torus, WK-recursive mesh, Hypermesh, Hamiltonian-connectivity, Hamiltonian path, Hamiltonian cycle, Pancyclicity, Link-disjoint Hamiltonian cycles
1.
Introduction
An interconnection network can be modeled by an undirected graph in which a processor is represented by a node, and a communication channel between two nodes is represented by an edge between the corresponding nodes. The mesh, torus, hypercube, WK-recursive mesh, and hypermesh are examples of grid-based interconnection network topologies. Many efficient algorithms for these networks have been developed and their important topological properties have been reported in the literature [1–10, 18–25]. The conventional pyramid network (a mesh-pyramid) is one of the important network topologies as it has been used in both hardware architectures and software structures for parallel computing, graph theory, digital geometry, machine vision, and image processing [1–7]. Moreover, fault-tolerant properties of this network [2] make it a promising network for reliable computing. Pancyclicity, Hamiltonian-connectedness and the possession of link-disjoint Hamiltonian cycles, are three important properties of network topologies that have been studied vastly in the past [1, 11–25]. The pancyclicity of a network represents its capacity for cycles of different lengths to be embedded in it. A network is pancyclic if ∗ Corresponding
author.
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any cycle of length 3, 4 . . . and N can be embedded within it, where N is the size of network. In a Hamiltonian-connected network, there exists at least one Hamiltonian path between any arbitrary pair of nodes in the network. In a network that has link-disjoint Hamiltonian cycles, Hamiltonian cycles can be identified in the network that do not have any common links. In this paper, we generalize the definition of the conventional pyramid network. While the definition of the standard pyramid network is based on the 2-D mesh topology for interconnecting the nodes in each level of its hierarchical structure, other topologies can be employed for this purpose. In this study, we are interested in determining the characteristics of the pyramid network when other grid-based topologies such as the torus, WK-recursive mesh, and hypermesh are employed. In Section 2, we formally define the grid-based pyramid and some useful notations and definitions to be used in subsequent sections. In Section 3, some of the basic properties of the grid-pyramids are studied and a routing algorithm for grid-pyramids is proposed in Section 4. The fact that these new topologies are Hamiltonian-connected and pancyclic is proven in Sections 5 and 6. Moreover, we prove, in Section 7, that two link-disjoint Hamiltonian cycles exist in the torus-pyramid and hypermesh-pyramid networks. Finally, Section 8 concludes this study. 2.
The grid-based pyramid
In this section, we formally define the grid-based pyramid networks. Definition 1 An a × b mesh, Ma,b , is a set of nodes V (Ma,b ) = {(x, y) | 1 ≤ x ≤ a, 1 ≤ y ≤ b} where nodes (x1 , y1 ) and (x2 , y2 ) are connected by an edge iff | x1 − x2 | + |y1 − y2 | = 1[1]. Definition 2 The a × b 2-D torus network, denoted as Ta×b , consists of a set of nodes V (Ta×b ) = {(x, y) | 1 ≤ x ≤ a, 1 ≤ y ≤ b}, where each node (x1 , y1 ) is connected to its four neighbouring nodes (x1 ± 1 mod a, y1 ) and (x1 , y1 ± 1 mod b). Definition 3 An a × b hypermesh, HMa,b , is a set of nodes V (H Ma,b ) = {(x, y) | 1 ≤ x ≤ a, 1 ≤ y ≤ b} where nodes (x1 , y1 ) and (x2 , y2 ) are connected by an edge iff x1 = x2 or y1 = y2 . Definition 4 An L-level WK-recursive network based on 4-tuple nodes, denoted by W K (4,L) , consists of a set of nodes V (W K (4,L) ) = {a L a L−1 . . . a1 |0 ≤ ai < 4}. The node with address schema A = (a L a L−1 . . . a1 ) is connected to 1) all the nodes with addresses (a L a L−1 . . . a2 k) that 0 ≤ k < 4, k = a1 , as sisters nodes and 2) node (a L a L−1 . . . a j+1 a j−1 (a j ) j ) if for one j, 1 ≤ j < 4; a j−1 = a j−2 = a j−3 = . . . = a1 and a j = a j−1 , as cousin node. Notation (a j ) j denotes j consecutive a j ’s. Figure 1 shows M4×4 , T4×4, H M4×4 , WK(4,2) . It is based upon the abovementioned definitions that the grid-pyramid is defined. Definition 5 A grid-based pyramid of n levels, denoted by PG,n , consists of a set of nodes V (PG,n ) = {(k, x, y) | 0 ≤ k ≤ n, 1 ≤ x, y ≤ 2k }. A node (k, x, y) ∈ V (PG,n )
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
Figure 1.
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The topologies of a). M4×4 , b) T4×4 , c) W K 4,2 , and d) H M4× .
is said to be a node at level k. All the nodes in level k form a 2k × 2k grid network G, which can be one of the grid-based networks: mesh, torus, hypermesh, or WKrecursive mesh (G ∈ {M, T, H M, W K }). The resulted pyramids can then be denoted as PM,n , PT,n , PH M,n and PW K ,n , respectively. n k n+1 There are a total of N = −1)/3 nodes in a PG,n . The nodes k=0 4 = (4 k at level k ≤ n are connected as an G(2 , 2k ). We refer to the adjacent nodes at the same level as sister nodes. It is also connected to node (k + 1, 2x−1, 2y), node (k + 1, 2x, 2y − 1), node (k + 1, 2x−1, 2y−1), and node (k + 1, 2x, 2y), for 0 ≤ k< n, in level k + 1, as child nodes. Also, it is connected to node (k − 1, x2 , 2y ), in level k − 1 as its father node [10]. The apex node in PG,n is the node with address (0, 1, 1), which can alternatively be denoted as PG,n . It is adjacent to nodes (1, 0, 0), (1, 0, 1), (1, 1, 0) and (1, 1, 1), which are its children. So the degree of the apex is 4. The corner nodes of PG,n are nodes (n, 1, 1), (n, 2n , 1), (n, 2n , 2n ) and (n, 1, 2n ) are denoted by PG,n , PG,n , PG,n and PG,n , respectively. A WKpyramid network, PW K ,n , can be defined in a similar manner. It consists of a set
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Figure 2.
HOSEINYFARAHABADY AND SARBAZI-AZAD
A Torus pyramid of level 2 and its apex, corner nodes and edges.
of nodes V (PW K ,n ) = {(k, ak ak−1 . . . a1 ) | 1 ≤ k ≤ n, 0 ≤ ai < 4} {(0, 0)}. A node (k, ak ak−1 . . . a1 ) is said to be a node at level k and is connected to nodes (k + 1, ak ak−1 . . . a1 0), (k + 1, ak ak−1 . . . a1 1), (k + 1, ak ak−1 . . . a1 2), (k + 1, ak ak−1 . . . a1 3). All the nodes in level k form a 2k × 2k WK-recursive network. The apex node is (0, 0) which is denoted by PW K ,n . The edges connecting the levels of pyramid are in the same topology as said in advance. The addresses of corner nodes are (n, 0n ), (n, 1n ), (n, 2n ) and (n, 3n ) where a n denotes n consecutive a’s. Figure 1 shows examples of different 4 × 4 grid-based networks. (k,x,y)
Definition 6 A sub-pyramid of PG,n is denoted as PG,n,m where G is either mesh, (k,ak ak−1 ...a1 ) torus or hypermesh or PW K ,n,m when G is WK-recursive network. It indicates a sub-pyramid in the main pyramid of level n, with apex (k, x, y) or with apex (k, ak ak−1 . . . a1 ) and m levels, where m ≤ n − k. For example, we can write either (0,1,1) (1,1,1) (1,1,2) V (PG,n ) = V (PG,n,n−1 ) ∪ V (G 2 n × 2 n ), or V (PG,n ) = V (PG,n,n−1 ) ∪ V (PG,n,n−1 )∪ (1,2,1) (1,2,2) V (PG,n,n−1 ) ∪ V (PG,n,n−1 ) ∪ {PG,n }. Definition 7 Corner edges are edges connected to corner nodes. A level-k corner edge is (0,1,1) a corner edge in sub-pyramid PG,n,k , 1 ≤ k ≤ n. Figure 2 shows a torus pyramid of level 2 and its apex, corner nodes and edges. Recursive definition of pyramids is useful when studying their properties. Two recursive definitions exist for the pyramid network [24]. Method 1 To construct a pyramid with L + 1 levels, PG,L+1 , we can connect four level-L pyramids and a node (as the apex, PG,L+1 ) as follows. Consider four pyramids 1 2 3 4 PG,L , PG,L PG,L and PG,L relatively placed at north-east, north-west, south-west, and
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
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Figure 3. Two recursive construction methods to built a mesh pyramid network, (a) method 1 and (b) method 2. i south-east positions. Connect the apex node to the apexes of all PG,L , for 1 ≤ i ≤ 4. Then connect all nodes located at the borders of the network G at level k, 1 ≤ k ≤ L, in 1 2 3 4 the four pyramids PG,L , PG,L PG,L and PG,L to form a 2k+1 × 2k+1 desirable G network. The resultant network is a PG,L+1 network composed of four PG,L networks and an apex node. Figure 3-a displays this recursive construction method.
Method 2 To construct a pyramid PG,L+1 , we can connect a PG,L and a G network, G2 L+1 × 2 L+1 , as follows. When G is a mesh, torus or hypermesh network, connect node (L , x, y), 1 ≤ x, y ≤ 2 L , to the nodes (2x − 1, 2y), (2x, 2y − 1), (2x − 1, 2y − 1), (2x, 2y). If G is a WK-recursive network, then connect node (L , a L a L−1 . . . a1 )), 0 ≤ ai < 4, to nodes (L +1, a L a L−1 . . . a1 0)), (L +1, a L a L−1 . . . a1 1)), (L +1, a L a L−1 . . . a1 2)) and (L + 1, a L a L−1 . . . a1 3)). Figure 3-b shows the construction of an (L + 1)-level mesh-pyramid, PM,L+1 , from an L-level mesh-pyramid, PM,L , and a 2 L+1 × 2 L+1 mesh, M2 L+1 ×2 L+1 , according to this method. 3.
Basic properties
The node degree (N D ) is defined as the number of physical channels emanating from a node. This attribute is a measure of a node’s I/O complexity [20]. For the Mesh-based Pyramid, PG,n , node degree depends on node level. Each node in the n-th layer, has exactly N D,G + 1 neighbours, which the N D,G notation indicates the node degree of the mention node in G network. Each node in the middle layers has just N D,G + 5 channels and the degree of apex node is 4. The diameter is commonly used to measure to compare the static network performance of a system. The diameter of a network is the maximum inter-node distance, i.e. the maximum number of links that must be traversed to send a message to all nodes along a shortest path. The smaller the diameter of a network is, the less time it takes to send a message from one node to the node farthest away. The diameter of PG,n with N = (4n+1 − 1)/(3) nodes is D = 2n − 1 ≈ log N , if G is either Mesh, Torus or WK-Recursive and the diameter of PH M,n is D = n − 1 ≈ 0.5 log N . So the diameter of the PG,n is O(log N). A comparison of network cost (which
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Figure 4. Two recursive construction methods to built a mesh pyramid network, (a) method 1 and (b) method 2.
is defined as the multiplication of diameter and node degree) for several different network topologies is shown in [26] where PG,n is of much less network cost than conventional mesh, hypercube, k-ary n-cube and WK-recursive networks. 4.
Routing algorithm in the grid-based pyramids
We propose a routing algorithm for PG,n which is planned with consideration of the standard routing algorithm for G networks, RG . We present a simple routing algorithm for PG,n named it Up-Down routing as shown in Figure 4. In this algorithm, it is examined whether the level of the source and the destination nodes are identical or not. In the first case, if levels are identical, the algorithm recursively calls itself and finds the minimum path between two routes, first the path which starts from the source node, goes to the father node and goes along a path between the source’s father and the destination’s father node which is returned by recursively calling the UDRPG,n (father of S, father of D) algorithm, goes to the destination’s father and finally ends at the destination node. The second path is the one that is positioned in layer k of the pyramid and is returned by calling the RG (S, D) algorithm on mesh-based network of layer k. In the second case, when the levels of the nodes are not identical, the algorithm first checks whether the source node is an ancestor of the destination or not. If this is true, the source node sends the message directly to its child and this is repeated
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
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until it reaches to the destination node. In another situation, when the source node is not an ancestor of the destination node, two paths must be compared. The first path is between the source node and the father of the destination node, added up to the link between the father of destination and the destination node. The second path is the one that goes from the source node to its father node, then goes along a path between the source’s father and the destination node which is returned by calling RG (S , D) function. At the end, the path with minimum length (between these two paths) can be chosen. 5.
Hamiltonian path and cycles in PG,N
In this section, the property of Hamiltonian-connectedness of grid-pyramids is studied. We prove that any grid-pyramid network is Hamiltonian-connected. Definition 8 A given network G = (V, E), is Hamiltonian-connected if it contains a Hamiltonian path starting at any node x ∈ V and ending at any node y ∈ V − {x} [20]. Theorem 1 [1]. Any mesh pyramid of level n, PM,n contains a Hamiltonian path starting at any node x ∈ W = {PM,n , PM,n , PM,n , PM,n , PM,n } and ending at any node y ∈ W − {x}. Because any form of mesh-based pyramid of level n excluding PW K ,n can embed a mesh pyramid of level n, PM,n , the theorem 1 is true for any mesh-based pyramid of level n. Theorem 2 [19]. The WK-recursive mesh is Hamiltonian-connected. Theorem 3 Every level-L WK-pyramid network, PW K ,n , contains a Hamiltonian path starting at a node x ∈ W = {(0, 1), (L , (k L )) | 1 ≤ j, k ≤ 4} and ending at any node y ∈ W − {x}. Proof: With induction on L. the following paths determine the Hamiltonian path between two nodes (0, 1) and (1, j) or between (1, k) and (1, k + j), for some 1 ≤ k ≤ 4 and −k + 1 ≤ j ≤ 4 − k, which prove the theorem when L = 1 (HPW K P,L (x,y) denotes a Hamiltonian path starting at node x and end in y in an L-level WK-pyramid network). HPWKP,1(0,1),(1, j) = < (0, 1) || (1, j + 1 mod 4) || (1, j + 2 mod 4) || . . . || (1, j − 1 mod 4) || (1, j) > HPWKP,1(1,k),(1, j) = < (1, k) || (0, 1) || (1, k + 1 mod 4) || (1, k + 2 mod 4) || . . . || (1, k + j − 1 mod 4) || (1, k + j + 1 mod 4) || (1, k + j + 2 mod 4) || . . . || (1, k − 1 mod 4) || (1, k + j mod 4) > When L is greater than 1, two different cases must be considered:
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HOSEINYFARAHABADY AND SARBAZI-AZAD
1. When the source node address is (0, 1) and the destination node address is (L , (k L )). 2. When neither the source node nor the destination node have addresses of the form (0, 1), i.e. both of them have addresses of the form (L + 1, ( j L+1 )). Case 1: We construct the Hamiltonian path as follows: H PWKP,L ((0, 1), (L , k L )) =< (0, 1) || H PWKP,L−1 ((0, 1), (L − 1, k + 1 L−1 )) || (L−1, (k + 1 mod 4) L−1 ) || (L , (k + 1 mod 4) L ) || H PW K ,L ((L , k + 1 L ), (L , k L )) || (L , k L ) >. The H PWK,L ((L , k + 1 L ), (L , k L )) notation represents the Hamiltonian path in a WK-recursive network, the layer L of WK-Pyramid network, between nodes (L , (k + 1 mod 4) L ) and (L , k L ). As a result of theorem 2, such a Hamiltonian path exists. Case 2: Without loss of generality, let us assume that the address of node x, the source node, is (L,( j L )) and the address of y, the destination node, is (L,( j + k) L ) for some 1 ≤ j ≤ 4 and − j + 1 ≤ k ≤ 4 − j. We only present proof of the case in which k is 1 or 2; the case in which k is equal to 3 can be demonstrated in a similar manner. In this case, the following path determines a Hamiltonian path between such nodes. H PWKP,L ((L , j L ), (L , ( j + 1) L )) = < (L , j L ) || H PWKP,L−1 ((L , j L ), (1, j)) || (1, j) || (0, 1) || (1, j + 3 mod 4) || H PWKP,L−1 ((1, j + 3 mod 4), (L , j + 3( j + 2) L−1 )) || (L , j + 3( j + 2) L−1 ) || (L , j + 2( j + 3) L−1 ) || H PWKP,L−1 ((L , j + 2( j + 3) L−1 ), (1, j + 2 mod 4)) || (1, j + 2) || (1, j + 1) || H PWKP,L−1 ((1, j + 1), (L , j + 1 L )) || (L , ( j + 1) L ) = y > . H PWKP,L ((L , j L ), (L , ( j + 2) L )) = < (L , j L ) || H PK P,L−1 ((L , j L ), (1, j)) || (1, j) || (0, 1) || (1, j + 3 mod 4) || H PWKP,L−1 ((1, j + 3 mod 4), (L , j + 3( j + 1) L−1 )) || (L , j + 3( j + 1) L−1 ) || (L , j + 1( j + 3) L−1 ) || H PWKP,L−1 ((L , j + 1( j + 3) L−1 ), (1, j + 1)) || (1, j + 1) || (1, j + 2) || H PWKP,L−1 ((1, j + 2), (L , ( j + 2) L )) || (L , ( j + 1) L ) = y >. Now, the proof of theorem 3 is completed. Lemma 1 Any mesh pyramid PM,n contains a Hamiltonian path starting from any node x ∈ V(PM,n ) (except from the apex node) and ending at the apex node, PM,n . Proof: With induction on the number of levels, L. For L = 1 the lemma holds by Theorem 1. Let’s consider a pyramid of level L + 1 constructed using Method 1. There are two cases which must be considered: (1) when the starting node, x, is the apex of any i 1 2 3 4 of pyramids PM,L , i = 1, 2, 3, 4, i.e. x ∈ ζ = {PM,L , PM,L , PM,L , PM,L },
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
31
i 1 and (2) when the starting node is not an apex of any of pyramids PM,L i.e. x ∈ V (PM,L )∪ 2 3 4 V (PM,L ) ∪ V (PM,L ) ∪ V (PM,L ) − ζ. 1 Case 1: Without lose of generality, lets assume x = PM,L . In this case, the Hamiltonian path, denoted by H PM,L+1 (x, y), from x to y in a pyramid network of L + 1 levels, can be constructed as follows:
H PM,L+1 (x, y) = < x || H PM,L (x, (L + 1, 2 L , 2 L + 1)) || (L + 1, 2 L , 2 L + 1) || (L + 1, 2 L , 2 L ) || H PM,L ((L + 1, 2 L , 2 L ), (1, 1, 1)) || (1, 1, 1) || (1, 2, 1) ||H PM,L ((1, 2, 1), (L + 1, 2 L+1 , 2 L )) || (L + 1, 2 L+1 , 2 L ) || (L + 1, 2 L+1 , 2 L + 1) || H PM,L ((L + 1, 2 L+1 , 2 L + 1), (1, 2, 2)) || (1, 2, 2) || (0, 1, 1) = y >. 1 2 3 4 This path starts from x, traverses consecutively PM,L , PM,L , PM,L and PM,L subpyramids and finally ends in y, as the last node of constructed Hamiltonian path. The symbol “ ||” is utilized to display the order of nodes in the Hamiltonian path. i Case 2: In this case, the starting node is not an apex of any of sub pyramids PM,L . 1 1 Without lose of generality, we assume that x∈ V (PM,L ) − {PM,L }. A Hamiltonian path from x to the apex node of PM,L+1 can be made as follows:
H PM,L+1 (x, y) = < x || H PM,L (x, (1, 1, 2)) || (1, 1, 2) || (1, 1, 1) || H PM,L ((1, 1, 1), L + 1, 2 L , 2 L + 1) || (L + 1, 2 L , 1) || (L + 1, 2 L + 1, 1) || H PM,L (L + 1, 2 L + 1, 1), (L + 1, 2 L+1 , 2 L )) || (L + 1, 2 L+1 , 2 L ) || (L + 1, 2 L+1 , 2 L + 1) || H PM,L (L + 1, 2 L+1 , 2 L + 1), (1, 2, 2)) || (1, 2, 2) || (0, 1, 1) = y >. This H PM,L+1 path starts from x and consecutively traverses sub-pyramids 1 2 3 4 PM,L , PM,L , PM,L and PM,L . Figure 5 graphically shows the construction methods which have been used to prove Lemma 1. (a) case 1. and (b) case 2. Lemma 2 The mesh pyramid PM,n contains a Hamiltonian path which starting at an j i arbitrary node x ∈ V (PM,L ), for i = 1, 2, 3, 4, and ending at any node y ∈ V (PM,L ), j = 1, 2, 3, and 4, and j = i ( while considering the construction method 1). Proof: With induction on the number of levels, L. Without loss of generality, we 1 2 assume that x ∈ V (PM,L ) and y ∈ V (PM,L ). The remaining cases can be treated similarly. We must consider three different cases, as follows:
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Figure 5.
Construction methods uses in the proof of Lemma 1, (a) Case 1. and (b) Case 2.
1 2 Case 1: x = PM,L and y = PM,L . The Hamiltonian path can be constructed in this manner:
H PM,L+1 (x, y) = < x || H PM,L (x, (1, 1, 2)) || (1, 1, 2) || (0, 1, 1) || (1, 2, 2) || H PM,L ((1, 2, 2), (L + 1, 2 L+1 , 2 L + 1)) || (L + 1, 2 L+1 , 2 L + 1) || (L + 1, 2 L+1 , 2 L ) || H PM,L ((L + 1, 2 L+1 , 2 L ), (1, 2, 1)) || (1, 2, 1) || (1, 1, 1) || H PM,L ((1, 1, 1), y) || y >. 1 4 3 The mentioned path successively goes across the sub-pyramids PM,L , PM,L , PM,L and
2 . PM,L
1 2 Case 2: x = PM,L and y = PM,L . The Hamiltonian path in the (L + 1)-level 1 4 3 2 , PM,L , PM,L and PM,L mesh-pyramid network can be constructed by traversing PM,L sub-pyramids in sequence:
H PM,L+1 (x, y) =< x || H PM,L (x, (L + 1, 2 L , 2 L+1 )) || (L + 1, 2 L , 2 L+1 ) || (L + 1, 2 L + 1, 2 L+1 ) || H PM,L ((L + 1, 2 L + 1, 2 L+1 ), (1, 2, 2)) || (1, 2, 2) || (0, 1, 1) || (1, 2, 1) || H PM,L ((1, 2, 1), (L + 1, 2 L + 1, 1)) || (L + 1, 2 L + 1, 1) || (L + 1, 2 L , 1) || H PM,L ((L + 1, 2 L , 1), y) || (1, 1, 1) = y >.
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
Figure 6.
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Construction methods used in the proof of Lemma 2, (a) Case 1, (b) Case 2, and (c) Case 3.
1 2 1 Case 3: x = PM,L and y = PM,L . The following path, which traverses the PM,L , 4 3 2 PM,L , PM,L and PM,L sub pyramids in order, proves this case:
H PM,L+1 (x, y) = < x || H PM,L (x, (L + 1, 2 L , 2 L+1 )) || (L + 1, 2 L , 2 L+1 ) || (L + 1, 2 L + 1, 2 L+1 ) || H PM,L ((L + 1, 2 L + 1, 2 L+1 ), (L + 1, 2 L+1 , 2 L + 1)) || (L + 1, 2 L+1 , 2 L + 1) || (L + 1, 2 L+1 , 2 L ) || H PM,L ((L + 1, 2 L+1 , 2 L ), (1, 2, 1)) || (1, 2, 1) || (0, 1, 1) || (1, 1, 1) || H PM,L ((1, 1, 1), y) || y >. Figure 6 graphically shows the construction methods used to prove the Lemma 2.
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Figure 7.
A Hamiltonion path in sub-pyramid PL1 and links E1 and E2.
Lemma 3 There is a Hamiltonian path which starts at any arbitrary node x ∈ i i V (PM,L ), i = 1, 2, 3, 4, and end at any node y ∈ V (PM,L ), x = y (i.e. the x node y nodes belong to the same sub pyramid). Proof: We prove the lemma with induction on L (the number of levels in the pyramid). Without loss of generality, assume that i = 1. Other cases (i = 2, 3, and 4) can be similarly treated. There are two cases which must be considered. 1 1 Case 1: x = PM,L and y = PM,L . According to Lemma 1 and induction hypothesis, 1 we can make a Hamiltonian path on PM,L which starts from x and end at y nodes that 1 1 we denote it by H PM,L (x, y). Obviously the apex node PM,L belongs to this path via two links, namely E1 and E2, as shown in Figure 7. We assume that the E1 link is 1 the same link which connects the apex node PM,L (or node with address (1, 2, 1) in 1 PM,L+1 ) to the node z = (1, 2, 1) of PM,L (or equally z = (2, 2, 3) in PM,L+1 ). We can 1 thus conclude that H PM,L (x, y) is the union of three parts: 1 H PM,L (x, y) = < P A1M,L (x, (2, 2, 3)) ∪ E1 ∪ P A1M,L ((1, 2, 1), y) >.
The notation P A1M, L (α, β) represents a path which starts at node α and ends at node β 1 in pyramid PM,L . 1 Considering the above Hamiltonian path in PM,L , we can build the following Hamiltonian path in PM,L+1 to obtain proof of this case: H PM,L+1 (x, y) =< x P A1M, L (x, (2, 2, 3)) (2, 2, 3) (2, 3, 3) H PM,L ((2, 3, 3), (1, 2, 2)) (1, 2, 2) (0, 1, 1) (1, 2, 1) H PM,L (1, 2, 1), (L + 1, 2 L + 1, 2 L ))
(L + 1, 2 L + 1, 2 L ) (L + 1, 2 L , 2 L ) H PM,L ((L + 1, 2 L , 2 L ), (1, 1, 1))
(1, 1, 1) (1, 1, 2) P A M, 1 L ((1, 2, 1), y) y >.
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
Figure 8.
35
Methods used in the proof of Lemma 3, (a) Case 1 and (b) Case 2.
1 Case 2: x = P 1 M,L . We can make a Hamiltonian path on PM,L which starts at x and 1 1 end at y nodes, which we denote as H PM, L (x, y). Since node x is the apex node PM,L , this node belongs to this path via one links, namely E1. Once more, we assume that the 1 E1 link is the same link which connects the apex node PM,L (or node with address (1, 1 2, 1) in PM,L+1 ) to the node z = (1, 2, 1) of PM,L (or equally z = (2, 2, 3) in PM,L+1 ). 1 We can thus write the H PM, L (x, y) as the union of two parts, as follows: 1 H PM,L (x, y) = < P A1M,L (x, (2, 2, 3)) ∪ E1 >.
Now, the following path in PM,L+1 completes the proof of this case. H PM,L+1 (x, y) = < y P A1M,L (y, (2, 2, 3)) (2, 2, 3) (2, 3, 3)
H PM,L ((2, 3, 3), (1, 2, 2) (1, 2, 2) (0, 1, 1) (1, 2, 1) H PM,L ((1, 2, 1), (L + 1, 2 L + 1, 2 L )) (L + 1, 2 L + 1, 2 L ) (L + 1, 2 L , 2 L )
H PM,L ((L + 1, 2 L , 2 L ), (1, 1, 1)) (1, 1, 1) (1, 1, 2) = x >. Figure 8 graphically shows the construction methods used in the proof of Lemma 3. Theorem 4 Any level-n mesh pyramid,PM,n contains a Hamiltonian path starting with x ∈ V (PM,n ) and ending at y ∈ V (PM,n ) − {x}. Proof: All possible different cases were considered in Lemmas 1, 2, and 3. Regarding these three lemmas, the proof of Theorem 4 for any grid-pyramid (except for WK-pyramid) is completed. Because these mesh-based pyramid of level n, i.e. PT,L+1
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HOSEINYFARAHABADY AND SARBAZI-AZAD
and PHM,L+1 can embed a mesh-pyramid of level n with the same number of nodes, Theorem 4 can be extended to these mesh-based pyramids. In the next theorem, we prove that the WK-pyramid has the same property, such that any PG,n contains a Hamiltonian path starting with x ∈ V (PG,n ) and ending at y ∈ V (PG,n ) − {x}, i.e. any PG,n is Hamiltonian-connected. Theorem 5 Any level-n WK-pyramid,PW K ,n contains a Hamiltonian path starting at an arbitrary node x ∈ V (PWK,n ) and ending at another node, y ∈ V (PWK,n ) − {x}. Proof: With a poof similar to that of Theorem 4, we should consider 3 different cases which stated in following lemmas (Lemmas 4, 5, and 6). Lemma 4 Any WK-pyramid PWK,n contains a Hamiltonian path starting at any node x ∈ V (PWK,n ) (except from the apex node) and ending at the apex node, PWK,n . Proof: With induction on the number of levels, n, there are two cases that must be i considered: (1) when the starting node, x, is the apex of any of pyramids PWK,L ,i = 1 2 3 4 1, 2, 3, 4, i.e. x ∈ ζ = {PWK,L , PWK,L , PWK,L , PWK,L }, and (2) when the starting i 1 2 node is not an apex of any of the pyramids PWK,L i.e. x ∈ V (PWK,L ) ∪ V (PWK,L )∪ 3 4 V (PWK,L ) ∪ V (PWK,L ) − ζ . 1 Case 1: Without lose of generality, let us assume x = PWK . The Hamiltonian path can be constructed as follows:
H PWK,L+1 (x, y) = < x H PWK,L (x, (L + 1, 12 L )) (L + 1, 12 L )
(L + 1, 2 L 1) H PWK,L ((L + 1, 2 L 1), (1, 2)) (1, 2) (1, 3) H PWK,L ((1, 3), (L + 1, 34 L )) (L + 1, 34 L ) (L + 1, 4 L 3) H PWK,L ((L + 1, 4 L 3), (1, 4))
(1, 4) (0, 1) = y >. 1 1 Case 2: Let us assume that x ∈ V (PWK,L ) − {PWK,L }. A Hamiltonian path from x to the apex node of PWK,L+1 can be built as follows:
H PM,L+1 (x, y) = < x H PWK,L (x, (1, 1)) (1, 1) (1, 2) H PWK,L ((1, 2), (L + 1, 23 L )) (L + 1, 23 L ) (L + 1, 3 L 2) H PWK,L (L + 1, 3 L 2), (L + 1, 34 L )) (L + 1, 34 L ) (L + 1, 4 L 3) H PWK,L (L + 1, 4 L 3), (1, 4))
(1, 4) (0, 1) = y >. Lemma 5 The WK-pyramid PWK,n contains a Hamiltonian path starting at an arbitrary j i node x ∈ V (PWK,L ), for i = 1, 2, 3, 4, and ending at any node y ∈ V (PWK,L ), j = 1, 2, 3, and 4, and j = i.
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
37
Proof: With induction on the number of levels, L, without loss of generality, we assume 1 2 that x ∈ V (PM,L ) and y ∈ V (PM,L ). The remaining cases can be treated similarly. We must consider three different cases as stated below: 1 2 Case 1: x = PWK,L and y = PWK,L . The Hamiltonian path can be constructed in this manner:
H PWK,L+1 (x, y) = < x H PWK,L (x, (1, 1)) (1, 1) (0, 1) (1, 3)
H PWK,L ((1, 3), (L + 1, 34 L )) (L + 1, 34 L ) (L + 1, 4 L 3)
H PW K ,L ((L + 1, 4 L3 ), (1, 4)) (1, 4) (1, 2) H PWK,L ((1, 2), y) y >. l 2 Case 2: x = PM,L and y = PM,L . The desirable Hamiltonian path can be constructed as:
H PWK,L+1 (x, y) = < x H PWK,L (x, (L + 1, 14 L )) (L + 1, 14 L )
(L + 1, 4 L 1) H PWK,L ((L + 1, 4 L 1), (1, 4)) (1, 4) (0, 1) (1, 3)
H PWK,L ((1, 3), (L + 1, 3 L 2)) (L + 1, 3 L 2) (L + 1, 2 L 3)
H PWK,L ((L + 1, 2 L 3), (1, 2)) (1, 2) = y >. 1 2 Case 3: x = PWK,L and y = PWK,L . The following path, which traverses the 1 4 3 2 PWK,L , PWK,L , PWK,L and PWK,L sub pyramids in order, prove this case :
H PWK,L+1 (x, y) = < x H PWK,L (x, (L + 1, 14 L )) (L + 1, 14 L )
(L + 1, 4 L 1) H PWK,L ((L + 1, 4 L 1), (L + 1, 43 L )) (L + 1, 43 L )
(L + 1, 3 L 4) H PWK,L ((L + 1, 3 L 4), (1, 3)) (1, 3) (0, 1) (1, 2)
H PWK,L ((1, 2), y) y > . i Lemma 6 There is a Hamiltonian path starting at node x ∈ V (PWK,L ), and ending at i any node y ∈ V (PW K ,L ), x = y (i.e. the x and y nodes belong to the same sub pyramid).
Proof: This is proved with induction on L. Assume that i = 1. There are two cases that must be considered. 1 1 Case 1: x = PWK,L and y = PWK,L . Considering Lemma 3 and induction hypothesis, 1 1 we can make a Hamiltonian path on PM,L between x and y, H PWK,L (x, y). Thus, the 1 apex node PWK,L belongs to this path via two links, namely E1 and E2. We assume that 1 the address of E1 link is ((1, 1), (2, 13)) in PWK,L+1 . So, we can write the H PWK,L (x, y) as the union of three parts, like this: 1 1 1 H PWK, L (x, y) = < P AWK, L (x, (2, 13)) ∪ E1 ∪ P AWK, L ((1, 1), y) >.
The notation P A1WK, L (α, β) represents a path which starts from node α and end in node 1 . The address of node α and β represent as they appear in PWK,L+1 . β in pyramid PWK,L
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HOSEINYFARAHABADY AND SARBAZI-AZAD
1 With considering the above Hamiltonian path in PWK,L , the following Hamiltonian path in PWK,L+1 can be built.
H PWK,L+1 (x, y) = < x P A1WK, L (x,(2, 13)) (2, 13) (2, 31)
H PWK,L ((2, 31),(1, 3)) (1, 3) (0, 1) (1, 2) H PWK,L (1, 2), (L + 1, 2 L 4))
(L + 1, 2 L 4) (L + 1, 42 L ) H PWK,L ((L + 1, 42 L ), (1, 4)) (1, 4) (1, 1)
P A1WK, L ((1, 1), y) y >. 1 1 Case 2: We assume that x = PM,L . The Hamiltonian path on PWK,L pyramid which 1 starts from x and end at y nodes, denoted by H PWK, L (x, y) can be built. The node x = (1, 1) belongs to this path via one link, namely E1. Once more, we assume that the 1 address of E1 link is ((1, 1), (2, 13)) in PWK,L+1 . So the H PWK, L (x, y) can be written as the union of two parts, like this:
1 1 H PWK, L (x, y) = < P AWK, L (x, (2, 13)) ∪ E1 >.
The following path in PWK,L+1 is Hamiltonian and starts at x and ends at y. H PWK,L+1 (x, y) = < y P A1WK, L (y, (2, 13)) (2, 13) (2, 31)
H PWK,L ((2, 31), (1, 3)) (1, 3) (0, 1) (1, 2) H PWK,L ((1, 2), (L + 1, 2 L 4)) (L + 1, 2 L 4) (L + 1, 42 L ) H PWK,L ((L + 1, 42 L ), (1, 4))
(1, 4) (1, 1) = x > . Considering these three lemmas, proof of theorem 5 is completed. Considering Theorems 4 and 5, the Hamiltonian-connectivity of grid-pyramid networks can be concluded.
6.
Pancyclicity of pyramids
The study of Pancyclicity of networks has gained much attention in the literature [11–17]. Bondy [11] has proved that if the minimum degree of a network of size N is N /2, then it is a pancyclic graph. In [14], it was proven that if a network of size N has a Hamiltonian cycle < x1 , x2 , x3 . . . x N > where degr ee(x1 ) + degr ee(x N ) ≥ N , then it is pancyclic. It is also shown in [11] that if a graph, G, is a non-bipartite Hamiltonian network of size N ≥ 102 with a minimum degree of 2N /5, then G is pancyclic. If network G of size N has minimum degree N /4 + 250 and contains a triangle and a Hamiltonian cycle, then G is pancyclic [13]. In [11], it is shown that every Hamiltonian network G of size N with a minimum number of edges N 2 /4 is pancyclic. Zhang [12] has shown that if G is a Hamiltonian network with N vertices of maximum and minimum degree (G) and δ(G), then it is pancyclic if (G) + δ(G) ≥ N . None of the above general case studies is applicable to the grid-pyramid network. In [25], it is proven that a pyramid is pancyclic. In this section, we deal with these specific
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
39
networks directly and use an algorithmic approach that proves any grid-pyramid to be pancyclic. Definition 9 Let G = (V, E) and n = |V (G)|. If G contains cycles of 3, 4 . . . n nodes, then G is called to be pancyclic [16]. Theorem 6 The grid-pyramid (except for the WK-pyramid) is pancyclic. Proof: We proceed with induction on L (the number of levels). When L = 1, we have n = 5. Construction of cycles with 3, 4, and 5 nodes in PM,1 , is as follows: (we denote a cycle with length in an n-level G-based pyramid network by C,G,n .). C3,M,1 : C4,M,1 : C5,M,1 : Now, we assume that the claim holds for L , meaning that we can have cycles of length 3, 4, 5 . . . and (4 L+1 −1)/3. We then show that the claim also holds for (L + 1)-level pyramids. In other words, we can find cycles of 3, 4, 5. . . and (4 L+2 − 1)/3 nodes in PM,L+1 . As {3, 4, 5, . . . , (4 L+2 − 1)/3} = {3, 4, 5, . . . , (4 L+1 − 1)/3}∪{(4 L+1 − 1)/3+ 1, . . . , (4 L+2 − 1)/3}. We only have to show the possibility of constructing cycles with length l ∈ γ = {(4 L+1 −1)/3 + 1, . . . , (4 L+2 −1)/3}, as the first set of cycle lengths is covered by induction hypothesis. The following lemma is the key point of building these cycles. Lemma 7 The mesh M2L+1 × 2L+1 contains all paths of length grater than 2 which traverse the edge ε = < (1, 1), (1, 2) > and end either in node (1,3) or in node (1,4). Proof: As illustrated in Figure 9, a Hamiltonian path, P A4L+2 , can be constructed with length 4 L+2 in a 2 L+1 × 2 L+1 mesh network in the layer L + 1 of PM,L+1 . This path starts from node (1, 2) and ends at node (1, 3). In addition, the edge ε in the mesh network belongs to this path. The path with length 4 L+2 − 1 is shown in this figure, too. The latter path, P A4L+2 − 1, starts from node (1, 2) and ends at mesh node (1, 3). The algorithm, shown in Figure 10, builds all paths with length 2, 3 . . . and 2 L+1 × 2 L+1 by pruning 2 edges in each step from paths P A4L+2 and P A4L+2 − 1. Some paths returned by this algorithm have been shown in Figure 11. Obviously, the paths which are returned by the mentioned algorithm satisfy all conditions of lemma 7 and the proof is completed. Both the cycles of length (4 L+1 − 1)/3 and (4 L+1 − 4)/3, C(4L+1 −1)/3 and C(4L+1 −4)/3 respectively, in PM,L include the corner edge < (L, 1, 1), (L, 1 ,2) >. To construct all
40
Figure 9.
Figure 10.
HOSEINYFARAHABADY AND SARBAZI-AZAD
The path in mesh M2L−1 ×
L−1 2
The path in mesh M2L−1 ×
with length 4 L+2 or 4 L+2 − 1 nodes.
L−1 2
with length 4 L+2 or 4 L+2 − 1 nodes.
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
Figure 11. Construction of a path of length 5 ≤ l ≤ 2 L+1 × 2 L+1 in M2L+l × either (1, 4) or (1, 3).
41
L+1 , between nodes (1, 2) and 2
cycles of length l ∈ γ = {(4 L+1 − 1)/3 + 3, . . . , (4 L+2 − 1)/3}, the paths built by lemma 7 can be added to the C(4L+1 −1)/3 and C(4L+1 −4)/3 with the following method. If the sought path is of length ζ then let ζ = ζ − (4 L+1 − 1)/3. Say P = Mesh Path(h, ζ ). Now remove the edge < (L, 1, 1), (L, 1, 2) > from C(4L+1 −1)/3 if ζ is even and remove this edge from C(4L+1 −4)/3 if ζ is odd. So the old cycle splits to 2 new paths. Say this new path C(4 L+1 −1)/3 and C (4 L+1 −1)/3 or C (4 L+1 −4)/3 and C (4 L+1 −4)/3 respectively. The path sought can be built by : < C(4L+1 −1)/3 (L, 1, 1) (L+1, 1, 2) P (L + 1, 1, 3)
C(4 L+1 −1)/3 > if ζ is even or can be written as < C (4 L+1 −4)/3 (L , 1, 1) (L + 1, 1, 2)
P (L + 1, 1, 4) C(4L+1 −4)/3 > if ζ is odd. As yet, we have constructed all paths with length l ∈ γ (excluding two lengths (4 L+1 − 1)/3 + 1 and (4 L+1 − 1)/3 + 2 in a PM,L+1 . To construct these two paths, consider a cycle of length (4 L+1 − 1)/3 − 2 in PM,L , namely C(4L+1 −7)/3 . Again, this cycle can be constructed such that it holds the corner edge ε = < (L, 1, 1), (L, 1, 2) >. Therefore, this cycle can be written as joining three separated part as C(4L+1 −7)/3 = < C(4 L+1 −7)/3 ε C (4 L+1 −7)/3 >.
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HOSEINYFARAHABADY AND SARBAZI-AZAD
The following cycles possess lengths of (4 L+1 − 1)/3 + 1 and (4 L+1 − 1)/3 + 2 in the PM,L+1 . C(4L+1 +2)/3 = < C(4 L+1 −7)/3 (L, 1, 1) (L + 1, 1, 1) (L + 1, 1, 2)
(L + 1, 1, 3) (L, 1, 2) C(4L+1 −7)/3 >. C(4L+1 +5)/3 = < C(4 L+1 −7)/3 (L, 1, 1) (L + 1, 1, 1) (L + 1, 1, 2)
(L + 1, 1, 3) (L + 1, 1, 4) (L, 1, 2) C(4L+1 −7)/3 >.
In this manner cycles of length l ∈ ξ = {3, 4, . . . (4 L+1 − 1)/3 + 1, . . . , (4 L+2 − 1)/3} have been constructed in the PM,L+1 network and thus the theorem is proved. As the torus pyramids and hyper-mesh pyramids can embed the mesh pyramid within themselves, it can be concluded that these networks are also Hamiltonian-connected networks. 7.
The link disjoint hamiltonian cycles
In this section, we prove that any torus-pyramid and hypermesh-pyramid holds two link-disjoint Hamiltonian cycles. The reasons for the interest in the link-disjoint Hamiltonian cycles are various. One of the most important usages of these cycles arises in transmitting messages in the network. By routing messages through these disjoint Hamiltonian cycles, communication problems such as all-to-all broadcasting and all-to-all scattering can be realized optimally [20]. Definition 10 [20] Two cycles C1 and C2 in network G are said to be link-disjoint Hamiltonian if both of them are Hamiltonian and there is no link e ∈ G such that e ∈ C1 and e ∈ C2 . Theorem 7 [20] Any torus network, Tn,m , has two link disjoint Hamiltonian cycles. Lemma 8 Any torus network, T2n , 2 n , has two link-disjoint paths C1 and C2 such that edges e1 = < (1, 1), (1, 2) > and e2 = < (1, 2), (1, 3) > belong to C1 and edges e3 = < (1, 1), (2, 1) > and e4 = < (2, 1), (3, 1) > belong to C2 . Proof: Consider the following path T2n , 2 n in C1 : < ei ei+1 > for 1 ≤ i ≤ 4n −1. The index i indicates the rank number of edges in the cycle. Edge ei can be built in this way:
ei = < (c + 1, 2n + 1 − c), (c + 2, 2n + 1 − c) > if i mod 2n = 0 and c = i/2n . ei = < (c + 1, r − c), (c + 1, r + 1 − c) > if i mod 2n = r = 0 and c = [i/2n ]. In a similar manner, the second path consists of the remaining edges in T2n , 2 n which do not appear in C1 . Thus: C2 = E(T2n , 2 n ) − C1 .
THE GRID-PYRAMID: A GENERALIZED PYRAMID NETWORK
43
Figure 12. Two disjoint link Hamiltonian cycles in T4,4 network . The edges < (1, 1), (1, 2) >, < (1, 2), (1, 3) >, < (1, 1), (2, 1) > and < (2.1), (3.1) > are highlighted.
Figure 13. Two disjoint link Hamiltonian cycles in P1,2 network to prove the induction base of proof of theorem 8.
Figure 12 shows two Hamiltonian cycles which are link-disjoint in a T4, 4 network. It is clear that the edges < (1, 1), (1, 2) > and < (1, 2), (1, 3) > belong to C1 and edges and belong to C2 . Theorem 8 Any torus pyramid contains two link-disjoint Hamiltonian cycles. Proof: This can be proved by induction on L. For L = 1, the two link-disjoint Hamiltonian cycles are shown in Figure 13. A torus pyramid with L + 1 levels can be considered as the union of two networks. The first is a L-level torus pyramid which includes two link disjoint Hamiltonian cycles, namely, C1,L and C2,L , and the second is a torus network,
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HOSEINYFARAHABADY AND SARBAZI-AZAD
T2L+1 , 2L+1 which according to lemma 8 contains two link disjoint Hamiltonian cycles, namely C1,T and C2,T . Assume that the link ε1 = < (L, 1, 1), (L, 1, 2) > in PT,L+1 belongs to cycle C1,L and the link ε2 = < (L, 1, 1), (L, 2, 1) > belongs to C2,L . We can write: C1,L = < C1,L
ε1 C1,L > and C2,L =< C2,L
ε2 C2,L >. Where Ci,L or Ci,L indicates part of cycle Ci,L in the pyramid. In addition, as C1,T and C2,T respectively contain links e = < (1, 2), (1, 3) > and e = < (2, 1), (3, 1) >, they can be written as C1,T = < e C1,T > and C2,T = < e C2,T >.
The two link disjoint Hamiltonian cycles within PT,L+1 can thus be built as: C1,L+1 = < C1,L
< (L , 1, 1), (L + 1, 1, 2) > C1,T
< (L + 1, 1, 3), (L , 1, 2) > C1,L >, C2,L+1 = < C2,L < (L , 1, 1), (L + 1, 2, 1) > C2,T
< (L + 1, 3, 1), (L , 2, 1) > C2,L >.
It is apparent that the C1,L+1 cycle contains ε1 = < (L + 1, 1, 1), (L + 1, 1, 2) > in PT,L+1 and the C2,L+1 cycle contains ε2 = < (L + 1, 1, 1), (L + 1, 2, 1) >, such that the abovementioned assumption remains true for the next level. This theorem is also true for the hypermesh-pyramid because of the possibility of embedding a torus-pyramid in the hypermesh-pyramid network directly.
8.
Conclusions and future work
Many graph topologies have been proposed for interconnection networks in the past. In this paper, we introduced a new topology for pyramid networks based on grid connections between the nodes in each level of pyramids. While in the conventional mesh-pyramid network, the nodes in each level form a mesh topology, we suggest that the connections between these nodes can also be a torus, hypermesh, or WK-recursive network, defining a family of pyramid networks, called grid-pyramids. A number of important properties of the grid-pyramids such as Hamiltonian-connectness, Pancyclicity and possessing two link-disjoint Hamiltonian cycles were investigated here. A comparison between the conventional mesh pyramid and these new classes of pyramid shows that the properties of the new ones are superior to conventional topology. The future work, in this line, includes a thorough performance evaluation of the GridPyramids using extensive simulation experiments for different working conditions. In addition, an implementation of some well-known algorithms for these topologies is under study.
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References 1. H. Sarbazi-Azad, M. Ould-khaoua, L. M. Mackenzie. Algorithmic construction of Hamiltonians in Pyramids, Information Processing Letters, 80:75–79, (2001). 2. F. Cao, D. F. Hsu. Fault-tolerance properties of pyramid networks, IEEE Transactions on Computers, 48:88–93, (1999). 3. A. Dingle, H. Sudborough. Simulation of binary trees and x-trees on pyramid networks, In Proc. IEEE Symp. Parallel & Distributed Processing, 220–229, (1992). 4. T. H. Lai, W. White. Mapping pyramid algorithms into hypercube, Journal of Parallel and Distributed Computing, 9:42–54, (1990). 5. R. Miller, Q. Stout. Data movement techniques for the pyramid computer, SIAM Journal of Computing, 16:38–60, (1987). 6. S. G. Ziavras, M. A. Siddiqui. Pyramid mapping onto hypercube for computer vision, Concurrency: Practice & Experience, 471–489, (1993). 7. Jing-Fu Jeng, et al. Image Shrinking and Expanding on a Pyramid, IEEE Transaction on parallel and distributed systems, 4:1291– 1296, (1993). 8. C. K. Y Ng, Embedding pyramids into 3D meshes, Journal of Parallel & Distributed Computing, 36:173– 184, (1996). 9. A. Dingle, H. Barada, Optimum embedding of end-around meshes into pyramid networks, In Proc. Int’l Symp. Parallel Processing, 112–118, (1991). 10. F. T. Leighton, Introduction to parallel algorithms and architectures: arrays, trees, hypercubes, Morgan Kaufmann Publishers, New York, (1992). 11. J. A Bondy, Pancyclic graphs I, J. Combin. Theory Ser. B, 11:80–84, (1971). 12. S. M. Zhang, Pancyclism and bipancyclism of Hamiltonian graphs, J. Combin. Theory, 60:159–168, (1994). 13. S. S. Brandt, R. J. Faudree and W. Goddard, Weakly pancyclic graphs, J. Graph Theory, 27: 141–176, (1998). 14. E. F. Schmeichal and S. L. Hakimi, A cycle structure theorem or Hamiltonian graphs. Combin. Theory Ser. B, 45:99–107, (1988). 15. Mekkia Kouider and Antoni Marczyk, On pancyclism in hamiltonian graphs, Discrete Mathematics, 251:119–127, (2002). 16. J. A. Bondy, Pancyclic Graphs: Recent Results, Infinite and Finite sets (Colloq. Keszthely, 1973; Dedicated to P. Erdos on his 60th birthday), Colloq. Math. Soc., 181–187, (1975). 17. R. Haggkvist, R. J. Faudree, and R. H. Schelp, Pancyclic graphs-connected Ramsey number, Ars Combin. 11:37–49, (1991). 18. Della Vecchia, G. C. Sanges, Recursively Scalable Networks for Message Passing Architectures, In Proc. Conf. Parallel Processing and Applications, 33–40, (1987). 19. Sheng Fu, Hamiltonian-connectedness of the WK-Recursive network, 7th international Symposium on Parallel Architecture, Algorithms and Networks, 569–574, (2004). 20. J. Duato, S. Yalamanchili, L. M. Ni, Interconnection networks: An engineering approach, Morgan Kaufmann Publishers, New York, (2003). 21. B. Bollob´as, A. Thomason, Weakly Pancyclic Graphs, Journal of Combinatorial Theory, Ser. B, 7:121– 137, (1999). 22. H. J. Broersma, A note on the minimum size of a vertex pancyclic graph, Discrete Mathematics 164:29-32, (1997). 23. A. Dingle, H. Barada, Optimum embedding of end-around meshes into pyramid networks, In Proc. Int’l Symp. Parallel Processing, 112–118, (1991). 24. M. R. Hoseiny, H. Sarbazi-Azad, On the Hamiltonian Connectedness of Pyramids, 4th IASTED International Conference on Communication Systems and Networks, (2005). 25. Ruei-Yu Wu, Dyi-Rong Duh, Pancyclicity of Pyramid Networks. In International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA), 1523–1526, (2004). 26. M.R. Hoseiny and H. Sarbazi-Azad, WK-Recursive Pyramid: A High Performance Network Topology, In Proceedings of 8th IEEE International Symposium on Parallel Architectures, Algorithms and Networks (IEEE-ISPAN’05), 312–317, (2005).
