Generalized Polychotomic Encoding: A Very Short Bit-Vector Encoding of Tree Hierarchies

Abstract

A well-known method to represent a partially ordered set P consists in associating to each element of P a subset of a fixed set S = {1,...,k} such that the order relation coincides with subset inclusion. Such an embedding is called a bit-vector encoding of P. Such encodings are economical with space and comparisons between elements can be performed efficiently via subset inclusion tests. As a consequence, they have found applications in databases, knowledge representation, distributed computing or object-oriented programming. The main issue consists in minimizing the size of the encoding, i.e. the cardinal of S, in order to get the best storage space and comparison speed. This smallest size is called the 2-dimension of P. Its computation is known to be NP-hard in the general case [1] and the complexity is open for trees which are an important class of orders encountered in practice. Finding heuristics which provide encodings of small size is challenging and it has yielded many works in the general case and in the particular case of trees. Our paper presents a new algorithm for trees which improves all previously known heuristics for trees. © Springer-Verlag Berlin Heidelberg 2008.