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Directional Routing via Generals stNumberings
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
"... We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on ..."
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We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in di#erent directions from the same source are nodedisjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3connected graphs have 3directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to kconnected graphs. We also showthat a directed acyclic graph (dag) that is kconnected to a set of sinks has a kdirectional embedding in (k  1)space with the sink set as the extreme nodes.
Optimal Independent Spanning Trees on Hypercubes
, 2004
"... Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertexdisjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is ..."
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Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertexdisjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is optimal if the average path length of the trees is the minimum. Any kdimensional hypercube has k independent spanning trees rooted at an arbitrary vertex. In this paper, an O(kn) time algorithm is proposed to construct k optimal independent spanning trees on a kdimensional hypercube, where n = 2 k is the number of vertices in a hypercube.
Parallel Construction of Independent Spanning Trees on Multidimensional Tori
 THE 24TH WORKSHOP ON COMBINATORIAL MATHEMATICS AND COMPUTATION THEORY
"... A set of spanning trees rooted at vertex r in G is called independent spanning trees (IST) if for each vertex v in G, v = r, the paths from v to r in any two trees are different and vertexdisjoint. If the connectivity of G is k, the IST problem is to construct k IST rooted at each vertex. The IST ..."
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A set of spanning trees rooted at vertex r in G is called independent spanning trees (IST) if for each vertex v in G, v = r, the paths from v to r in any two trees are different and vertexdisjoint. If the connectivity of G is k, the IST problem is to construct k IST rooted at each vertex. The IST problem has found applications in faulttolerant broadcasting, but it is still open for general graph with connectivity greater than four. Obokata et al. [IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences E79A (1996) 1894–1903] have proved that the IST problem can be solved on multidimensional tori. However, their construction algorithm forbids the possibility of parallel processing. In this paper, we shall propose a parallel algorithm that is based on the Latin square scheme to solve the IST problem on multidimensional tori.