Browsing by Subject "Graph coloring"

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    Coloring for distributed-memory-parallel Gauss-Seidel algorithm
    (2019-09) Koçak, Onur
    Gauss-Seidel is a well-known iterative method for solving linear system of equations. The computations performed on Gauss-Seidel sweeps are sequential in nature since each component of new iterations depends on previously computed results. Graph coloring is widely used for extracting parallelism in Gauss-Seidel by eliminating data dependencies caused by precedence in the calculations. In this thesis, we present a method to provide a better coloring for distributed-memoryparallel Gauss-Seidel algorithm. Our method utilizes combinatorial approaches including graph partitioning and balanced graph coloring in order to decrease the number of colors while maintaining a computational load balance among the color classes. Experiments performed on irregular sparse problems arising from various scientific applications show that our model effectively reduces the required number of colors thus the number of parallel sweeps in the Gauss-Seidel algorithm.
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    Linear colorings of simplicial complexes and collapsing
    (Academic Press, 2007) Civan, Y.; Yalçın, E.
    A vertex coloring of a simplicial complex Δ is called a linear coloring if it satisfies the property that for every pair of facets (F1, F2) of Δ, there exists no pair of vertices (v1, v2) with the same color such that v1 ∈ F1 {set minus} F2 and v2 ∈ F2 {set minus} F1. The linear chromatic numberlchr (Δ) of Δ is defined as the minimum integer k such that Δ has a linear coloring with k colors. We show that if Δ is a simplicial complex with lchr (Δ) = k, then it has a subcomplex Δ′ with k vertices such that Δ is simple homotopy equivalent to Δ′. As a corollary, we obtain that lchr (Δ) ≥ Homdim (Δ) + 2. We also show in the case of linearly colored simplicial complexes, the usual assignment of a simplicial complex to a multicomplex has an inverse. Finally, we show that the chromatic number of a simple graph is bounded from above by the linear chromatic number of its neighborhood complex. © 2007 Elsevier Inc. All rights reserved.