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DOI: 10.7155/jgaa.00076
LowDistortion Embeddings of Trees
Vol. 7, no. 4, pp. 399409, 2003. Regular paper.
Abstract We prove that every tree T=(V,E) on n vertices with edges of unit length
can be embedded in the plane with distortion O(√n);
that is, we construct a mapping f V→R^{2} such that
ρ(u,v) ≤ f(u)−f(v) ≤ O(√n)·ρ(u,v)
for every u,v ∈ V,
where ρ(u,v) denotes the length of the path from u to v
in T. The embedding
is described by a simple and easily computable
formula. This is asymptotically
optimal in the worst case. We also construct interesting optimal embeddings
for a special class of trees (fans consisting of paths of the same length
glued together at a common vertex).
