Summary: Metric Extensions and the L 1 Hierarchy
David Avis
McGill University
Montreal, Quebec
Hiroshi Maehara
Ryukyu University
Nishihara, Okinawa
ABSTRACT
A finite semimetric is L 1 - embeddable if it can be expressed as a non­
negative combination of Hamming semimetrics. A finite semimetric is called
hypermetric if it satisfies the (2k + 1) - gonal inequalities which naturally gener­
alize the triangle inequality. It is known that all L 1 - embeddable semimetrics
are hypermetric and the metric induced by K 7 - P 3 is hypermetric but not
L 1 - embeddable. In the first part of the paper we show that there are infinite
metrics that are hypermetric but are not L 1 - embeddable, answering a question
of M. Deza. We introduce the r - extension of a semimetric: this is the addition
of new points from which all of the distances are r. We show that all 2­exten­
sions of K 7 - P 3 are hypermetric. Unfortunately, 2 - extensions of hypermetrics
in which all of the distances are either one or two may not be hypermetric in gen­
eral. We illustrate this by showing that the metric induced by the star K 1,5 (which

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

Collections: Computer Technologies and Information Sciences