Node-Independent Spanning Trees in Gaussian Networks

Transcription

1 4 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'16 Node-Independent Spanning Trees in Gaussian Networks Z. Hussain 1, B. AlBdaiwi 1, and A. Cerny 1 Computer Science Department, Kuwait University, Kuwait Department of Information Science, Kuwait University, Kuwait Abstract Gaussian network is known to be an alternative to toroidal network since it has the same number of nodes with less diameter, which makes it perform better than toroidal network. Spanning trees are said to be independent if all trees are rooted at the same node r and for any other node u, the nodes of the paths from r to u in all trees are distinct except the nodes r and u. In this paper, we investigate the problem of finding node independent spanning trees in Gaussian networks. Keywords: Circulant Graphs, Gaussian Networks, Spanning Trees, Independent Spanning Trees, Fault-Tolerant Routing. 1. Introduction The topology of an interconnection network plays a major role in achieving high performance computing in parallel systems. There are many varieties of these interconnection networks and some of them are popular such as hypercube, generalized hypercube, mesh, torus, De Bruijn, REFINE, and RMRN networks [][3][4][5][6][7][9]. An efficient interconnection topology called Gaussian network has been studied in [8][15][16]. The studies show that the Gaussian network can be a better alternative to the torus network since it has the same number of nodes but with less diameter. This network is briefly reviewed in Section. In parallel computing and distributed systems, a network can be represented as a graph G(V,E) wherev is the set of nodes and E is the set of edges. These nodes and edges represent processors and communication links between the processors in the network, respectively. A path from node s to node d in the graph is a sequence of edges, which connects a sequence of nodes from s to d. Two such paths are said to be independent if their nodes are different except the end nodes s and d, i.e. the intermediate nodes in the first path are distinct form the intermediate nodes in the second path. A spanning tree is a connected loop-free subgraph of graph G containing all the nodes of graph G. Spanning trees rooted at node r are said to be independent if the paths from r to any other node u in all trees are independent. Node independent spanning trees used to resolve important issues in network applications such as fault-tolerant broadcasting [11][13] and secure message distribution [17][18]. These applications are briefly described below: Consider that there exist t node independent spanning trees rooted at node r in a network N. Assume that the network N contains at most t 1 faulty nodes. Then, r can broadcast a message to every non-faulty node u in the network N with the existence of t 1 faulty nodes. Since the number of faulty nodes is less than t, thenat least one of those t node disjoint paths from r to u is fault free. Thus, every non-faulty node in the network N would receive the broadcasted message from r if all t node independent spanning trees are used to broadcast the message. Node independent spanning trees could be used to secure message distribution as follows. A message can be divided into t packets where each packet is sent by node r through a different spanning tree to its destination. Thus, each node in the network receives at most one of the t packets whereas the destination node receives all the t packets [14][18][19]. In [1], we have constructed two edge-disjoint nodeindependent spanning trees in dense Gaussian network, in which the network contains the maximum number of nodes for a given diameter k [16], where the depth of each tree is k, k 1. We also designed algorithms that can be used in fault-tolerant routing or secure message distribution where the source node in these algorithms is not restricted to a specific node; it could be any node in the network. In this paper, we investigate the problem of finding node independent spanning trees in Gaussian networks where the trees are not necessarily edge-disjoint. This paper is organized as follows. The Gaussian network is reviewed in Section. Section 3 presents the algorithms that illustrate the construction of node independent spanning trees in Gaussian networks. The communication overhead and the amount of work to construct the trees are discussed in Section 4. The paper is concluded in Section 5.. Background Gaussian networks are 4-regular symmetric networks where each node in the network has 4 neighbors. These networks are based on quotient rings of Gaussian integers Z[i] ={x + yi x, y Z} where i = 1 [10]. Z[i] is a Euclidean domain. The nodes of the network are elements of the residue class modulo some α Z[i]. The total number of nodes in the network is known as norm, N(α), ofthe Gaussian integer α = a + bi and is equal to a + b. Various representations of these residue classes are given in [1]. Each node in the network is labeled as x + yi. The nodes A and B are said to be adjacent, i.e. neighbors, if and only if

2 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'16 5 (A B) mod α is equal to ±1 or ±i. As described in [15], if N(α) is odd, for each node in the grid, the number D(s) of nodes at distance s such that 0 s k and t = a+b 1 is: D(s) = 1 4s 4(b s) if s =0 if 0 <s t if t < s < b Further, when N(α) is even, the number D(s) of nodes at distance s such that 0 s k and t = a+b is: 1 if s =0 4s if 0 <s<t D(s) = (b 1) if s = t () 4(b s) if t < s < b 1 if s = b when 0 <a= b, the distance distribution of the graph G b+bi is as follows: 1 if s =0 D(s) = 4s if 0 <s<b (3) b 1 if s = b Based on the above, the diameter of the network is k, which is equal to b when N(α) is even and to b 1 when N(α) is odd. Figure 1 shows an example of Gaussian network generated with α =3+4i. -1+i i 1+i -+i -1+i i 1+i +i i -i 1-i - Fig. 1: Gaussian network generated with α =3+4i The wrap-around links can be obtained by tiling the Gaussian network on an infinite grid. For example, Figure shows the Gaussian network generated with α =+ and the nodes wrap-around links are illustrated based on tiling the network. Consider the node 1+i where its 1 edge and i edge are connected to the nodes i and 1, respectively, which are within the basic grid. However, the +1 edge should be connected to node +i, which is not within the basic grid. Thus, this edge is considered as a wrap around link and it is (1) connected to node i, which is an equivalent to node +i modulo +. Similarly, +i edge of node 1+i is connected to node 1+i, whose corresponding node in the basic grid is 1 i. Fig. : Tiling Gaussian network generated with α =+ In this paper, we deal with dense diameter-optimal graph, which is isomorphic to the Gaussian network G(α k ),where α k = k +(k +1)i, sincegcd(k, k +1) = 1. We denote G k =(V k,e k )=G(α k ), k Node Independent Spanning Tree Construction In this section, we describe how to construct four node independent spanning trees in Gaussian network G k. We give algorithms that construct four different spanning trees and then, from the figures, we show that these trees are independent and the exact proof of the independence of these trees will be in an extended version of this paper. Algorithm 1 constructs four paths of length 1 where each path connects the node 0 to one of its neighbors (children of root), which is considered as the first step for constructing the four node independent spanning trees. The variables used in the following algorithms are described as follows. treeno = 1,, 3, and 4 determines the first, second, third, and fourth node independent spanning tree, respectively. root is the root node and current is the current node where both being of the form x + yi; root.x or root.y (current.x or current.y) describe the coordinates of the node in the network. The root.child1, root.child, root.child3, and root.fourth are the links, in respective order, to the first, second, third, and fourth children of the node root. Thecurrent.Child1, current.child, and current.child3 are the links to the first, second, and third children of node current, respectively, and current.parent is the link to the parent node of the node current. The network generator is α = k +(k +1)i; k is the network diameter in

3 6 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'16 dense Gaussian networks. In all algorithms, all operations of + and are done mod α. Algorithm 1 alltrees: Construct four node independent spanning trees based on a network generator α = a + bi 1: root.x 0 : root.y 0 3: root.child1 root.x +1 4: root.child root.y + i 5: root.child3 root.x 1 6: root.child4 root.y i 7: send through +1 packet (root, root.child1, 1) 8: send through +i packet (root, root.child, ) 9: send through 1 packet (root, root.child3, 3) 10: send through i packet (root, root.child4, 4) Algorithm does the configuration for a node when it receives the packet (parent, current, treeno). For example, it initializes the current node and it sets its parent node. After that, based on the value of treeno, it calls the corresponding function to construct the targeted tree. Algorithm Node initialization based on the received packet (parent, current, treeno) 1: k b 1 : current.parent parent 3: current.child1 Nil 4: current.child Nil 5: current.child3 Nil 6: if treeno =1then 7: call Tree1(parent, current) 8: else if treeno =then 9: call Tree(parent, current) 10: else if treeno =3then 11: call Tree3(parent, current) 1: else 13: call Tree4(parent, current) 14: end if Algorithm 3 sets the child nodes and forwards the packet to them based on the directions corresponding to the first spanning tree. The first tree uses the following edges. k edges are in +1 direction where k of them on the path from node 0 to node k and the other k are from node k to node ki and from node ( 1+ji) to node (ji) for j =1,,...,k 1. Further, k edges are used in i direction from node (m ni) mod α to node (m (n+1)i) mod α for m =1,,...,k and n =0, 1,...,k 1. Also, there are k edges are used in +i direction from node (m + ni) mod α to node (m +(n + 1)i) mod α for m =1,,...,kand n =0, 1,...,k 1. Thus, a total of k +k edges are used in the first spanning tree, which are sufficient to form a spanning tree of all k +k+1 nodes of the network. From figures 3, 4, 5, and 6, it is clear that the four spanning trees are symmetric. Also, it is obvious that the second tree in Figure 4 is a one rotation to the counter clockwise of the first tree in Figure 3. Similarly, the third and fourth trees as seen in figures 5 and 6, respectively, are two and three rotations to the counter clockwise of the first tree, respectively. Thus, it follows that the algorithms 4, 5, and 6 are similar to the Algorithm 3 except that the direction of sending the packet is set according to the rotation based on its corresponding tree. Note that, the intersected edges between the first and third (also, second and fourth) spanning trees are used in opposite directions. That is, the path between any nodes u and v in the first spanning tree is independent from the third spanning tree. The similar argument applies to the second and fourth spanning trees. Thus, the first and third (second and fourth) spanning trees are independent. Moreover, the first spanning tree uses mostly ±i (vertical) edges, which are not used in the second tree, while the second spanning tree uses mostly ±1 (horizontal) edges, which are not used in the first tree. Also, note that the intersected edges between the first and second spanning trees are used in opposite directions. That is, the path between any nodes u and v in the first spanning tree is independent from the second spanning tree. The similar argument applies to the third and fourth spanning trees. Thus, the first and second (third and fourth) spanning trees are independent. A similar to the above argument can be applied to show that the first and the fourth (the second and the third) spanning trees are independent. Thus, we get four independent spanning trees. -1+i i 1+i -+i -1+i i 1+i +i i -i 1-i - Fig. 3: First spanning tree After constructing the trees, we can execute tree broadcasting to send a message from the root node to every other node in the network in k steps.

4 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'16 7 Algorithm 3 Tree1(parent, current): Invoked by a function call in Algorithm 1: if current.y = 0 and current.x > 0 and current.x k then : current.child1 current.y i 3: send through i packet (current, current.child1, 1) 4: current.child current.x+1 5: send through +1 packet (current, current.child, 1) 6: current.child3 current.y+i 7: send through +i packet (current, current.child3, 1) 10: current.child1 current.y i 11: if current.y i and current.y (k 1)i and current.x = 1 then 1: current.child current.x+1 14: if current.child1.y 0 then 15: send through i packet (current, current.child1, 1) 19: current.child3 current.y+i 0: if current.child3.y i then 1: send through +i packet (current, current.child3, 1) : end if 3: end if Algorithm 4 Tree(parent, current): Invoked by a function call in Algorithm 1: if current.x = 0 and current.y > 0 and current.y ki then : current.child1 current.x+1 3: send through +1 packet (current, current.child1, ) 4: current.child current.y+i 5: send through +i packet (current, current.child, ) 6: current.child3 current.x 1 7: send through 1 packet (current, current.child3, ) 10: current.child1 current.x+1 11: if current.x k +1 and current.x 1 and current.y = i then 1: current.child current.y+i 14: if current.child1.x 0 then 15: send through +1 packet (current, current.child1, ) 19: current.child3 current.x 1 0: if current.child3.x 1 then 1: send through 1 packet (current, current.child3, ) : end if 3: end if 4. Construction Complexity Each tree needs k parallel construction steps. In the following two subsections we will derive the communication overhead and the amount of work to construct a single tree. In the third subsection, we will derive the total complexity to construct all four trees. 4.1 Communication Complexity We will enumerate the construction steps from 0 to k 1. The number of messages generated in the i th step of communication is: Comm(i) = { i +1, 0 i k (k i)+1, k +1 i k 1 Thus, the total messages generated in each tree construc- (4) tion is: = = = Comm(i) (5) k (i +1)+ k i + k 1+ ((k i)+1)) (6) 4k i + 1 (7) k(k +1) = ( )+(k +1)+4k(k 1) k ( i i)+(k 1) (8) i=1 i=1 = 5k ((k 1)(k)) k(k +1) k ( ) (9) = 5k k (3k 3k) (10) = k +k (11) Note that, this one less than the number of nodes in G k since each node receives a message except the root node. 4. Local Computation Each node needs 6 local operations for assignments, comparisons, and sending the packets. 9 of these operations

5 8 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'16-1+i i 1+i -+i -1+i i 1+i +i i -i 1-i - Fig. 4: Second spanning tree -1+i i 1+i -+i -1+i i 1+i +i Algorithm 5 Tree3(parent, current): Invoked by a function call in Algorithm 1: if current.y = 0 and current.x k and current.x 1 then : current.child1 current.y+i 3: send through +i packet (current, current.child1, 3) 4: current.child current.x 1 5: send through 1 packet (current, current.child, 3) 6: current.child3 current.y i 7: send through i packet (current, current.child3, 3) 10: current.child1 current.y+i 11: if current.y ( k +1)i and current.y i and current.x = 1 then 1: current.child current.x 1 14: if current.child1.y 0 then 15: send through +i packet (current, current.child1, 3) 19: current.child3 current.y i 0: if current.child3.y i then 1: send through i packet (current, current.child3, 3) : end if 3: end if -1-i -i 1-i - Fig. 5: Third spanning tree are performed in Algorithm and 17 of them are performed when calling one of the following algorithms 3, 4, 5, and 6. Hence, the number of local computations in step i is 6Comm(i). Indetails,itis: { 6(i +1), 0 i k (1) 6((k i)+1), k +1 i k 1 Thus, the total amount of work over all phases is 5k +5k. 4.3 Total Complexity All four trees can be constructed in k parallel steps. The total communication overhead to construct all trees is: = 4 Communication Complexity for One T ree (13) = 8k +8k (14) The total amount of computation work needed to construct all trees is: = 4 Local Computations for One T ree (15) = 08k + 08k (16) 5. Conclusions In this paper, we briefly described the definition of graphs, independent paths, node independent spanning trees, and Gaussian network. Then, we gave algorithms that construct four node independent spanning trees in Gaussian networks. The depth of each tree is k. Further, we have computed the construction complexity for the given algorithms. In our future work, we would like to extend this work and implement algorithms for parallel construction of node independent spanning trees and fault-tolerant routing from a given source node to a given destination node. References [1] B. AlBdaiwi, Z. Hussain, A. Cerny, and R. Aldred, Edge-disjoint node-independent spanning trees in dense gaussian networks, The Journal of Supercomputing, pp. 1 19, 016. [Online]. Available: [] H. R. Arabnia and J. W. Smith, A reconfigurable interconnection network for imaging operations and its implementation using a multistage switching box, in Proceedings of the 7th annual international high performance computing conference. The, 1993, pp

6 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'16 9 Algorithm 6 Tree4(parent, current): Invoked by a function call in Algorithm 1: if current.x = 0 and current.y >= ki and current.y < 0 then : current.child1 current.x 1 3: send through 1 packet (current, current.child1, 4) 4: current.child current.y i 5: send through i packet (current, current.child, 4) 6: current.child3 current.x+1 7: send through +1 packet (current, current.child3, 4) 10: current.child1 current.x 1 11: if current.x 1 and current.x k 1 and current.y = i then 1: current.child current.y i 14: if current.child1.x 0 then 15: send through 1 packet (current, current.child1, 4) 19: current.child3 current.x+1 0: if current.child3.x 1 then 1: send through +1 packet (current, current.child3, 4) : end if 3: end if [3] H. Arabnia and S. Bhandarkar, Parallel stereocorrelation on a reconfigurable multi-ring network, The Journal of Supercomputing, vol. 10, no. 3, pp , [4] S. Bhandarkar and H. Arabnia, The hough transform on a reconfigurable multi-ring network, Journal of Parallel and Distributed Computing, vol. 4, no. 1, pp , [5] S. M. Bhandarkar and H. R. Arabnia, The REFINE multiprocessor theoretical properties and algorithms, Parallel Computing, vol. 1, no. 11, pp , [6] S. M. Bhandarkar, H. R. Arabnia, and J. W. Smith, A reconfigurable architecture for image processing and computer vision, International Journal of Pattern Recognition and Artificial Intelligence, vol. 09, no. 0, pp. 01 9, [7] J. Duato, S. Yalamanchili, and L. Ni, Interconnection Networks: An Engineering Approach, 1st ed. Los Alamitos, CA, USA: IEEE Computer Society Press, [8] M. Flahive and B. Bose, The topology of Gaussian and Eisenstein- Jacobi interconnection networks, IEEE Transactions on Parallel and Distributed Systems, vol. 1, no. 8, pp , August 010. [9] A. Grama, A. Gupta, G. Karypis, and V. Kumar, Introduction to Parallel Computing, nd ed. Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc., 003. [10] K. Huber, Codes over Gaussian integers, IEEE Transactions on Information Theory, vol. 40, no. 1, pp , Jan [11] A. Itai and M. Rodeh, The multi-tree approach to reliability in distributed networks, Inf. Comput., vol. 79, no. 1, pp , Oct [Online]. Available: [1] C. J. P. J. H. Jordan, Complete residue systems in the gaussian integers, Mathematics Magazine, vol. 38, no. 1, pp. 1 1, [Online]. Available: [13] M. S. Krishnamoorthy and b. Krishnamurthy, Fault diameter of interconnection networks, Comput. Math. Appl., vol. 13, -1+i i 1+i -+i -1+i i 1+i +i i -i 1-i Fig. 6: Fourth spanning tree no. 5-6, pp , Apr [Online]. Available: [14] J.-C. Lin, J.-S. Yang, C.-C. Hsu, and J.-M. Chang, Independent spanning trees vs. edge-disjoint spanning trees in locally twisted cubes, Information Processing Letters, vol. 110, no. 10, pp , 010. [15] C. Martinez, R. Beivide, E. Stafford, M. Moreto, and E. Gabidulin, Modeling toroidal networks with the Gaussian integers, IEEE Transactions on Computers, vol. 57, no. 8, pp , Aug [16] C. Martinez, E. Vallejo, R. Beivide, C. Izu, and M. Moreto, Dense Gaussian networks: Suitable topologies for on-chip multiprocessors, International Journal of Parallel Programming, vol. 34, pp , 006. [17] A. Rescigno, Vertex-disjoint spanning trees of the star network with applications to fault-tolerance and security, Information Sciences, vol. 137, no. 1-4, pp , 001. [18] J.-S. Yang, H.-C. Chan, and J.-M. Chang, Broadcasting secure messages via optimal independent spanning trees in folded hypercubes, Discrete Applied Mathematics, vol. 159, no. 1, pp , 011. [Online]. Available: [19] J.-S. Yang, J.-M. Chang, and H.-C. Chan, Independent spanning trees on folded hypercubes, in Proceedings of the th International Symposium on Pervasive Systems, Algorithms, and Networks, ser. ISPAN 09. Washington, DC, USA: IEEE Computer Society, 009, pp