2 research outputs found
Embedding 2-Dimensional Grids Into Optimal Hypercubes With Edge-Congestion 1 Or 2
This paper explores one-to-one embeddings of 2-dimensional grids into hypercubes. It is shown that each 2-dimensional grid can be embedded with edge-congestion 2 into its optimal hypercube (the smallest hypercube with at least as many nodes as the grid). Additionally, a technique is developed to embed many 2-dimensional grids into their optimal hypercubes with edge-congestion 1
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