378 research outputs found
Embedding multidimensional grids into optimal hypercubes
Let and be graphs, with , and a one to one map of their vertices. Let , where is the distance
between vertices and of . Now let = , over all such maps .
The parameter is a generalization of the classic and well studied
"bandwidth" of , defined as , where is the path on
points and . Let
be the -dimensional grid graph with integer values through in
the 'th coordinate. In this paper, we study in the case when and is the hypercube
of dimension , the hypercube of
smallest dimension having at least as many points as . Our main result is
that
provided for each . For such , the bound
improves on the previous best upper bound . Our methods include
an application of Knuth's result on two-way rounding and of the existence of
spanning regular cyclic caterpillars in the hypercube.Comment: 47 pages, 8 figure
Mapping unstructured grid problems to the connection machine
We present a highly parallel graph mapping technique that enables one to solve unstructured grid problems on massively parallel computers. Many implicit and explicit methods for solving discretizated partial differential equations require each point in the discretization to exchange data with its neighboring points every time step or iteration. The time spent communicating can limit the high performance promised by massively parallel computing. To eliminate this bottleneck, we map the graph of the irregular problem to the graph representing the interconnection topology of the computer such that the sum of the distances that the messages travel is minimized. We show that, in comparison to a naive assignment of processors, our heuristic mapping algorithm significantly reduces the communication time on the Connection Machine, CM-2
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