838 research outputs found
From Hall's Matching Theorem to Optimal Routing on Hypercubes
AbstractWe introduce a concept of so-called disjoint ordering for any collection of finite sets. It can be viewed as a generalization of a system of distinctive representatives for the sets. It is shown that disjoint ordering is useful for network routing. More precisely, we show that Hall's âmarriageâ condition for a collection of finite sets guarantees the existence of a disjoint ordering for the sets. We next use this result to solve a problem in optimal routing on hypercubes. We give a necessary and sufficient condition under which there are internally node-disjoint paths each shortest from a source node to any others(sâ©œn) target nodes on ann-dimensional hypercube. When this condition is not necessarily met, we show that there are always internally node-disjoint paths each being either shortest or near shortest, and the total length is minimum. An efficient algorithm is also given for constructing disjoint orderings and thus disjoint short paths. As a consequence, Rabin's information disposal algorithm may be improved
Properties and Algorithms of the (n,k)-Arrangement Graphs and Augmented Cubes
The (n, k)-arrangement graph was first introduced in 1992 as a generalization of the star graph topology. Choosing an arrangement topology is more efficient in comparison with a star graph as we can have a closer number of nodes to what is needed. Also it has other advantages such as a lower degree and a smaller diameter, depending on k. In this thesis we investigate the problem of finding k(n â k) disjoint paths from a source node to k(nâk) target nodes in an (n, k)-arrangement interconnection network such that no path has length more than diameter+(nâk)+2, where diameter is the maximum length of shortest path between any two nodes in the graph. These disjoint paths are built by routing to all neighbors of the source node and fixing specific elements in each of the k positions of the node representation in an (n, k)-arrangement graph. Moreover, a simple routing is presented for finding n disjoint paths between two nodes which are located in different sub-graphs. The lengths are no more than d(t, s) + 4, for d(t, s) being the shortest path length between two nodes s and t. This routing algorithm needs O(n^2) time to find all n these paths.
In addition to arrangement graphs, we also study augmented cubes, first introduced in 2002, a desirable variation of the hypercube. An augmented cube of dimension n has a higher degree and a lower diameter in comparison with the hypercube. We introduce an O(n^3) algorithm for finding disjoint shortest paths from a single source node to 2n â 1 different target nodes
Adaptive fault-tolerant routing in hypercube multicomputers
A connected hypercube with faulty links and/or nodes is called an injured hypercube. To enable any non-faulty node to communicate with any other non-faulty node in an injured hypercube, the information on component failures has to be made available to non-faulty nodes so as to route messages around the faulty components. A distributed adaptive fault tolerant routing scheme is proposed for an injured hypercube in which each node is required to know only the condition of its own links. Despite its simplicity, this scheme is shown to be capable of routing messages successfully in an injured hypercube as long as the number of faulty components is less than n. Moreover, it is proved that this scheme routes messages via shortest paths with a rather high probabiltiy and the expected length of a resulting path is very close to that of a shortest path. Since the assumption that the number of faulty components is less than n in an n-dimensional hypercube might limit the usefulness of the above scheme, a routing scheme is introduced based on depth-first search which works in the presence of an arbitrary number of faulty components. Due to the insufficient information on faulty components, the paths chosen by the above scheme may not always be the shortest. To guarantee that all messages be routed via shortest paths, it is proposed that every mode be equipped with more information than that on its own links. The effects of this additional information on routing efficiency are analyzed, and the additional information to be kept at each node for the shortest path routing is determined. Several examples and remarks are also given to illustrate the results
A multipath analysis of biswapped networks.
Biswapped networks of the form have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of vertex-disjoint paths joining any two distinct vertices in , where is the connectivity of . In doing so, we obtain an upper bound of on the -diameter of , where is the diameter of and the -diameter. Suppose that we have a deterministic multipath source routing algorithm in an interconnection network that finds mutually vertex-disjoint paths in joining any distinct vertices and does this in time polynomial in , and (and independently of the number of vertices of ). Our constructions yield an analogous deterministic multipath source routing algorithm in the interconnection network that finds mutually vertex-disjoint paths joining any distinct vertices in so that these paths all have length bounded as above. Moreover, our algorithm has time complexity polynomial in , and . We also show that if is Hamiltonian then is Hamiltonian, and that if is a Cayley graph then is a Cayley graph
Shortest path routing algorithm for hierarchical interconnection network-on-chip
Interconnection networks play a significant role in efficient on-chip communication for multicore systems. This paper introduces a new interconnection topology called the Hierarchical Cross Connected Recursive network (HCCR) and a shortest path routing algorithm for the HCCR. Proposed topology offers a high degree of regularity, scalability, and symmetry with a reduced number of links and node degree. A unique address encoding scheme is proposed for hierarchical graphical representation of HCCR networks, and based on this scheme a shortest path routing algorithm is devised. The algorithm requires 5(k-1) time where k=logn4-2 and k>0, in worst case to determine the next node along the shortest path
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Shortest paths in orthogonal graphs
Orthogonal graphs were introduced as a simple but powerful tool for the description and analysis of a class of interconnection networks. Routing, and hence finding shortest paths between any two nodes of an orthogonal graph, becomes an important problem. It is shown in this paper that routing in this class of graphs reduces to a node covering problem in the bipartite coverage graph of the orthogonal graph. A minimum cover clearly leads to a shortest path. In general, the problem of finding the mĂnimum node cover in a bipartite graph is NP-complete. However, the bipartite coverage graphs corresponding to orthogonal graphs have a regular pattern of edges. This allows the development of a routing algorithm which results in a minimum cover. The procedure executes in polynomial time in the number of bit-nodes of the bipartite graph. It therefore results in a shortest path algorithm whose time complexity is quadratic in the logarithm of the number of nodes in the original orthogonal graph
Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes
The -dimensional hypercube network is one of the most popular
interconnection networks since it has simple structure and is easy to
implement. The -dimensional locally twisted cube, denoted by , an
important variation of the hypercube, has the same number of nodes and the same
number of connections per node as . One advantage of is that the
diameter is only about half of the diameter of . Recently, some
interesting properties of were investigated. In this paper, we
construct two edge-disjoint Hamiltonian cycles in the locally twisted cube
, for any integer . The presence of two edge-disjoint
Hamiltonian cycles provides an advantage when implementing algorithms that
require a ring structure by allowing message traffic to be spread evenly across
the locally twisted cube.Comment: 7 pages, 4 figure
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