383 research outputs found
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The cyclic cutwidth of mesh cubes
This project\u27s purpose was to understand the workings of a new theorem introduced in a professional paper on the cutwidth of meshes and then use this knowledge to apply it to the search for the cyclic cutwidth of the n-cube
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Generalization of edge general position problem
The edge geodesic cover problem of a graph is to find a smallest number
of geodesics that cover the edge set of . The edge -general position
problem is introduced as the problem to find a largest set of edges of
such that no edges of lie on a common geodesic. We study this dual
min-max problems and connect them to an edge geodesic partition problem. Using
these connections, exact values of the edge -general position number is
determined for different values of and for different networks including
torus networks, hypercubes, and Benes networks.Comment: This research is supported by Kuwait University, Kuwai
Optimal communication algorithms for hypercubes
Cover title.Includes bibliographical references.Supported by NSF with matching funds from Bellcore, Inc. ECS-8519058 ECS-8552419 Supported by the ARO. DAAL03-86-K-0171 Supported by the AFOSR. AFOSR-88-0032by D.P. Bertsekas ... [et al.]
Shortest Path Routing on the Hypercube with Faulty Nodes
Interconnection networks are widely used in parallel computers. There are many topologies for interconnection networks and the hypercube is one of the most popular networks. There are a variety of different routing paradigms that need to be investigated on the hypercube. In this thesis we investigate the shortest path routing between two nodes on the hypercube when some nodes are faulty and cannot be used. In this thesis the shortest path between two nodes is considered as the Hamming distance of them.
Regarding the shortest path problem in a faulty hypercube, some efficient algorithms have been proposed when each processor (node) has limited information regarding the status of other processors (whether they are faulty or not). There are also some proposed algorithms for the case where there is no limitation on the data of each processor but they are not efficient and are exponential in terms of number of faulty nodes and dimension of the hypercube.
To check whether there is a shortest path between two given nodes in a faulty hypercube, we propose a polynomial algorithm with time complexity of O(n^2 * m^2) where n is the dimension of the hypercube and m is the number of faulty nodes. Our algorithm only requires the source node to know the state of all other nodes. The proposed algorithm first checks whether there is a shortest path from the source node to the target node and then it can construct it efficiently.
Our idea is based on a so-called ordering and permutation model of paths in the hypercube. We use a constructive approach to find the path which is a permutation as well. We then use inclusion-exclusion and dynamic programming techniques to make our method efficient.
We also propose an algorithm for counting all possible shortest paths in the hypercube
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