Let G and H be graphs, with β£V(H)β£β₯β£V(G)β£, and f:V(G)βV(H) a one to one map of their vertices. Let dilation(f)=max{distHβ(f(x),f(y)):xyβE(G)}, where distHβ(v,w) is the distance
between vertices v and w of H. Now let B(G,H) = minfβ{dilation(f)}, over all such maps f.
The parameter B(G,H) is a generalization of the classic and well studied
"bandwidth" of G, defined as B(G,P(n)), where P(n) is the path on n
points and n=β£V(G)β£. Let [a1βΓa2βΓβ―Γakβ]
be the k-dimensional grid graph with integer values 1 through aiβ in
the i'th coordinate. In this paper, we study B(G,H) in the case when G=[a1βΓa2βΓβ―Γakβ] and H is the hypercube
Qnβ of dimension n=βlog2β(β£V(G)β£)β, the hypercube of
smallest dimension having at least as many points as G. Our main result is
that B([a1βΓa2βΓβ―Γakβ],Qnβ)β€3k,
provided aiββ₯222 for each 1β€iβ€k. For such G, the bound
3k improves on the previous best upper bound 4k+O(1). 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
We show that any real valued matrix A can be rounded to an integer one B such that the error in all 2 Γ 2 (geometric) submatrices is less than 1.5, that is, we have |aij - bij| < 1 and for all i,j