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Cycle Breaking in Wormhole Routed Computer Communication Networks
"... Because of its simplicity, low channel setup times, and high performance in delivering messages, wormhole routing has been adopted in second generation multicomputing environments [13]. Furthermore, irregular topologies formed by adhoc interconnection of low cost workstations provide cost effectiv ..."
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Because of its simplicity, low channel setup times, and high performance in delivering messages, wormhole routing has been adopted in second generation multicomputing environments [13]. Furthermore, irregular topologies formed by adhoc interconnection of low cost workstations provide cost effective alternative to massively parallel computing platforms. Switches used in these networks of workstations, NOWs, implement wormhole routing [1, 12]. However, due to a number of channels being held up while requesting others, wormhole routing is susceptible to deadlocks. In this paper we investigated deadlockfree routing algorithms in wormhole routed irregular network topologies. We used a modified version of the Turn Prohibition algorithm [4] to break cycles and prevent channel deadlocks. We then used shortest path algorithm to determine the routing tables for each computational node in the topology, avoiding prohibited turns along the paths from source to destination. We used EMA to import topologies into Opnet and simulated for message delivery using both our approaches and for the competing Up/Down [1] approach. We then repeated this sequence for hundreds of different topologies and determine the average latencies for all algorithms. Our results show that the modified turn prohibition based routing has outperformed both, the original Turn Prohibition and the Up/Down algorithms.
On identification in Z 2 using translates of given patterns
 J.UCS
, 2003
"... Abstract: Given a finite set of patterns, i.e., subsets of Z 2. What is the best way to place translates of them in such a way that every point belongs to at least one translate and no two points belong to the same set of translates? We give some general results, and investigate the particular case ..."
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Abstract: Given a finite set of patterns, i.e., subsets of Z 2. What is the best way to place translates of them in such a way that every point belongs to at least one translate and no two points belong to the same set of translates? We give some general results, and investigate the particular case when there is only a single pattern and that pattern is a square or has size at most four.
A new algorithm for finding minimal cyclebreaking sets of turns in a graph
 Journal of Graph Algorithms and Applications
, 2006
"... We consider the problem of constructing a minimal cyclebreaking set of turns for a given undirected graph. This problem is important for deadlockfree wormhole routing in computer and communication networks, such as Networks of Workstations. The proposed Cycle Breaking algorithm, or CB algorithm, g ..."
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We consider the problem of constructing a minimal cyclebreaking set of turns for a given undirected graph. This problem is important for deadlockfree wormhole routing in computer and communication networks, such as Networks of Workstations. The proposed Cycle Breaking algorithm, or CB algorithm, guarantees that the constructed set of prohibited turns is minimal and that the fraction of the prohibited turns does not exceed 1/3 for any graph. The computational complexity of the proposed algorithm is O(N 2 ∆), where N is the number of vertices, and ∆ is the maximum node degree. The memory complexity of the algorithm is O(N∆). We provide lower bounds on the minimum size of cyclebreaking sets for connected graphs. Further, we construct minimal cyclebreaking sets and establish bounds on the minimum fraction of prohibited turns for two important classes of graphs, namely, tpartite graphs and graphs with small degrees. The upper bounds are tight and demonstrate the optimality of the CB algorithm for certain classes of graphs. Results of computer simulations illustrate the superiority of the proposed CB algorithm as compared to the wellknown and the widely used Up/Down technique.
Cycles Identifying Vertices and Edges in Binary Hypercubes and 2dimensional Tori
"... A set of subgraphs C1 , C2 , . . . , Ck in a graph G is said to identify the vertices (resp. the edges) if the sets fj : v 2 C j g (resp. fj : e 2 C j g) are nonempty for all the vertices v (edges e) and no two are the same set. We consider the problem of minimizing k when the subgraphs C i are ..."
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A set of subgraphs C1 , C2 , . . . , Ck in a graph G is said to identify the vertices (resp. the edges) if the sets fj : v 2 C j g (resp. fj : e 2 C j g) are nonempty for all the vertices v (edges e) and no two are the same set. We consider the problem of minimizing k when the subgraphs C i are required to be cycles or closed walks. The motivation comes from maintaining multiprocessor systems, and we study the cases when G is the binary hypercube, or the twodimensional pary space endowed with the Lee metric.
On the Identification of Vertices Using Cycles
, 2003
"... A set of cycles C 1 ; : : : ; C k in a graph G is said to identify the vertices v if the sets fj : v 2 C j g are all nonempty and dierent. In this paper, bounds for the minimum possible k are given when G is the graph Z p endowed with the Lee or Hamming metric or G is a complete bipartite graph. ..."
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A set of cycles C 1 ; : : : ; C k in a graph G is said to identify the vertices v if the sets fj : v 2 C j g are all nonempty and dierent. In this paper, bounds for the minimum possible k are given when G is the graph Z p endowed with the Lee or Hamming metric or G is a complete bipartite graph.
UNICAST MESSAGE ROUTING IN СOMMUNICATION NETWORKS WITH IRREGULAR TOPOLOGY
"... Abstract. In this paper we consider the problem of deadlockfree unicast wormhole routing in computer and communication networks with irregular topologies. An example of such networks are Network of Workstations (NOWs). In general, the topology of these networks can be quite random. Several methods ..."
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Abstract. In this paper we consider the problem of deadlockfree unicast wormhole routing in computer and communication networks with irregular topologies. An example of such networks are Network of Workstations (NOWs). In general, the topology of these networks can be quite random. Several methods exist in the literature for wormhole routing in networks/multiprocessors with a regular topology, such as a ndimensional mesh, but very few papers have been published on wormhole routing for irregular networks. Some of these existing techniques require complex signaling hardware at the routers or result in a large amount of congestion at some specific links. The problem of deadlockfree routing consists of two parts. First, all deadlocks must be eliminated. An usual way of doing this, both for regular and irregular topologies, is to forbid some turns. The second part, which is the focus of this paper, is the problem of selecting an optimal (usually the shortest) path after the restrictions on routing have been formulated. We propose three efficient approaches for solving this problem. These approaches (local, global and mixed) differ in a way distances in the network graph are estimated using local information stored in the routers. Our approach for nonadaptive unicast deadlockfree wormhole routing provides for message paths very close to the shortest ones and more uniform distribution of the traffic between communication links in the system. Initial simulation results presented in the paper indicate that the proposed approaches are promising in terms of both throughput and scalability. Key Words. Wormhole routing, deadlock elimination, multiprocessor systems. 1.
ABSTRACT The Generalized Turn Prohibition Model for Multicast Routing in Irregular Networks
"... In this paper a universal model for breaking cycles is described. A method applicable to nondirected graphs is first developed and subsequently applied to communication networks. Loops or cycles are particularly important in communication networks. Specifically, communication networks using wormhole ..."
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In this paper a universal model for breaking cycles is described. A method applicable to nondirected graphs is first developed and subsequently applied to communication networks. Loops or cycles are particularly important in communication networks. Specifically, communication networks using wormhole routing techniques [1,2,3] and local area networks with backpressure flow control mechanisms [4, 5, 6] are particularly vulnerable to network states known as deadlock and livelock. An efficient approach to handle deadlocks and livelocks in such networks is based on preventing them by selecting routing policies that are deadlock and livelock free. Cycles in channel dependency graph [7] in such networks have been used as an indicator for vulnerability for deadlocks. Construction of the channel dependency graph is at best difficult for all but very small networks. Our model provides a simpler alternative. It operates in the network graph domain instead of the channel dependency graph. We present a simple routing strategy for deadlock and livelock free unicasting and multicasting by using minimal number of turn prohibitions to break all cycles in the network graph. Performance of the model is evaluated by simulation experiments for irregular communication networks using unicast and multicast wormhole routing.