$(p,\,q)$-nuclear and $(p,\,q)$-integral operators

Factoring N = pq

We discuss the problem of factoring N = pq and survey some approaches. We then present a specialized factoring algorithm that runs in time Õ(q0.31), which is comparable to the runtime Õ(p) of the factoring algorithm for integers of the form N = pq presented in [1]. We then survey the factoring algorithm of [1] and discuss the number of advice bits needed for it to run in polynomial time. Furthe...

The Dwarf Nova Pq Andromedae

Breakpoint Distance and PQ-Trees

The PQ-tree is a fundamental data structure that can encode large sets of permutations. It has recently been used in comparative genomics to model ancestral genomes with some uncertainty: given a phylogeny for some species, extant genomes are represented by permutations on the leaves of the tree, and each internal node in the phylogenetic tree represents an extinct ancestral genome, represented...

Minimax Programs, Bitonic Columns and Pq Trees

1 Overview Among hard combinatorial optimization problems there is an increasing body of special cases known in which optimal solutions can be obtained eeciently. These special cases may be expressed as constraints on graph structure, such as chordal or interval graphs 4, 6] which allow linear-time solution of problems like vertex coloring, or Monge matrix edge weights which allow eecient solut...

Planarity Algorithms via PQ-Trees

We give a linear-time planarity test that unifies and simplifies the algorithms of Shih and Hsu and Boyer and Myrvold; in our view, these algorithms are really one algorithm with different implementations. This leads to a short and direct proof of correctness without the use of Kuratowski’s theorem. Our planarity test extends to give a uniform random embedding, to count embeddings, to represent...