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,n-1 , 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

Source: Avis, David - School of Computer Science, McGill University

Collections: Computer Technologies and Information Sciences