Building PQR trees in almost-linear time

Abstract

AbstractIn 1976, Booth and Leuker invented the PQ trees as a compact wayof storing and manipulating all the permutations on n elements thatkeep consecutive the elements in certain given sets C 1 ,C 2 ,...,C m .Such permutations are called valid. This problem finds applicationsin DNA physical mapping, interval graph recognition, logic circuitoptimization and data retrieval, among others. PQ trees constructiontime is linear on the size of the input sets. In 1995, Meidanis andMunuera created the PQR trees, a natural generalization of PQ trees.The difference between them is that PQR trees exist for every setcollection, even when there are no valid permutations. The R nodesencapsulate subsets where the consecutive ones property fails. In thisnote we present an almost-linear time algorithm to build a PQR treefor an arbitrary set collection. Keywords: Trees, analysis of algorithms 1 Introduction Given a collection of m subsets C 1 ,C 2 ,...,C m of a set U of n elements,the consecutive ones problem consists in answering whether there is a validpermutation of the elements in U, that is, a permutation that keeps theelements of each C