 
Summary: On the Complexity of Isometric Embedding in the Hypercube
David Avis
School of Computer Science
McGill University
3480 University
Montreal, Quebec
H3A 2A7
ABSTRACT
A finite metric is h  embeddable if it can be embedded isometrically in the
N cube (hypercube) for some N . It is known that the problem of testing whether
a metric is h  embeddable is NPComplete, even if the distances are restricted to
the set {2, 4, 6 }. Here we study the problem where the distances are restricted to
the set {1, 2, 3 } and give a polynomial time algorithm and forbidden submetric
characterisation. In fact, we show these metrics are h  embeddable if and only if
they are 11  gonal and the sum of the distances arround any triangle is even.
The socalled truncated metric case, where the distances are chosen from {1, 2} is
particularly simple, the only embeddable metrics arise from the graphs
K 1,n1 , K 2,2 , and 2K n ( K n with all distances 2).
1. Introduction
For an integer n and a finite set X = {x 1 , . . . , x n }, let (X, d) be a metric space. In other
