## PC trees and circular-ones arrangements

Venue: | Theoretical Computer Science |

Citations: | 34 - 4 self |

### BibTeX

@ARTICLE{Hsu_pctrees,

author = {Wen-lian Hsu and Ross M. Mcconnell},

title = {PC trees and circular-ones arrangements},

journal = {Theoretical Computer Science},

year = {},

volume = {296},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

A 0-1 matrix has the consecutive-ones property if its columns can be ordered so that the ones in every row are consecutive. It has the circular-ones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutive-ones orderings of the columns of a matrix that has the consecutive-ones property. We give an analogous structure, called a PC tree, for representing all circular-ones orderings of the columns of a matrix that has the circular-ones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1

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Citation Context ...ied PQ tree and a simpler incremental update of the tree. Klein and Reif [12] constructed efficient parallel algorithms for manipulating PQ trees. Hsu gave a simple test that is not based on PC trees =-=[11]-=-. In this paper, we give a mathematical characterization of a tree, called a PC tree, for representing the circular-ones arrangements of a matrix. No representation of the set of circular-ones arrangm... |

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Citation Context ...tly by a program. The literature on problems related to PQ trees is quite extensive. Korte and Moehring [13] considered a modified PQ tree and a simpler incremental update of the tree. Klein and Reif =-=[12]-=- constructed efficient parallel algorithms for manipulating PQ trees. Hsu gave a simple test that is not based on PC trees [11]. In this paper, we give a mathematical characterization of a tree, calle... |

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Citation Context ...dular decomposition or substitution decomposition of the graph [9, 18]. Other examples of rooted tree families include module-like structures in hypergraphs, matrices, and sets of intervals on a line =-=[17, 6, 7, 15, 16]-=-. Definition 2.3 Let an unrooted set family be a set family F on domain V with the following properties: 5 b c a g z d f es{1,2,3,4,5,6} {1,2,3,4,5} {4,5,6} {1,2,3,6} {1,2,3} {1} {2} {3} {4,5} {6} {4}... |

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(Show Context)
Citation Context ... of a matrix has previously been given. PC trees were originally used as a data structure to find planar embeddings of planar graphs and as a data structure for computing the PQ tree much more simply =-=[19, 10]-=-. It was presented as a rooted tree, but we show here that there are mathematical reasons to treat it as an unrooted tree. One of these is that doing so allows the PC tree to represent all circular-on... |

14 |
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Citation Context ...operties: 1. V and its singleton subsets are members of F; 2. Whenever X, Y ∈ F and X − Y , Y − X, and Y ∩ X are nonempty, then X ∩ Y , X ∪ Y , and X∆Y = (X − Y ) ∪ (Y − X) are also in F. Theorem 2.2 =-=[17, 6, 7]-=- For a rooted set family F, there is a representation with a rooted tree whose nodes are labeled degenerate and prime and whose leaves are the singleton subsets of V . X ∈ F iff it is the union of the... |

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Citation Context ...operties: 1. V and its singleton subsets are members of F; 2. Whenever X, Y ∈ F and X − Y , Y − X, and Y ∩ X are nonempty, then X ∩ Y , X ∪ Y , and X∆Y = (X − Y ) ∪ (Y − X) are also in F. Theorem 2.2 =-=[17, 6, 7]-=- For a rooted set family F, there is a representation with a rooted tree whose nodes are labeled degenerate and prime and whose leaves are the singleton subsets of V . X ∈ F iff it is the union of the... |

8 | PC-trees vs. PQ-trees
- Hsu
- 2001
(Show Context)
Citation Context ... of a matrix has previously been given. PC trees were originally used as a data structure to find planar embeddings of planar graphs and as a data structure for computing the PQ tree much more simply =-=[19, 10]-=-. It was presented as a rooted tree, but we show here that there are mathematical reasons to treat it as an unrooted tree. One of these is that doing so allows the PC tree to represent all circular-on... |

1 |
The Theory of Two-Structures. World Scientific
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Citation Context ...family iff all vertices not in X are adjacent to the same subset of X. Another example are module-like structures in matrices that are invariant under certain types of transformations on the matrices =-=[5]-=-. There the decomposition tree is called a plane tree. 3 The PC Tree and its relationship to the PQ tree We view a circular-ones arrangement as a circular permutation of the columns, where the last co... |