An Almost Linear-Time Algorithm for Graph Realization

Bixby, Robert E.; Wagner, Donald K.

doi:10.1287/moor.13.1.99

Cited by 63 publications

(30 citation statements)

References 21 publications

(24 reference statements)

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“…With this restatement we see that all the constraints on a legal labeling function can be expressed as linear equations over GF [2]. The number of variables is at most m 2 and the number of equations is at most…”

Section: If (C C ) Is Ofmentioning

confidence: 91%

“…In order to complete the analysis of the algorithm, we show that the algorithm assigns a legal assignment to B. , then one of two cases can be true: 2). In this case, we must have that c and c has the same color, and therefore cannot share an edge.…”

Section: Lemma 42 If the Algorithm Reports Infeasibility Then Therementioning

confidence: 99%

“…Gusfield [10] introduces a polynomial time algorithm for the PPH problem. His algorithm uses as a black box an algorithm to recognize graphic matroids [21,2]. This algorithm is very complicated and impractical.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Reconstruction of Haplotype Structure via Perfect Phylogeny

Eskin

Halperin

Karp

2003

J. Bioinform. Comput. Biol.

102

106

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Each person's genome contains two copies of each chromosome, one inherited from the father and the other from the mother. A person's genotype specifies the pair of bases at each site, but does not specify which base occurs on which chromosome. The sequence of each chromosome separately is called a haplotype. The determination of the haplotypes within a population is essential for understanding genetic variation and the inheritance of complex diseases. The haplotype mapping project, a successor to the human genome project, seeks to determine the common haplotypes in the human population.Since experimental determination of a person's genotype is less expensive than determining its component haplotypes, algorithms are required for computing haplotypes from genotypes. Two observations aid in this process: first, the human genome contains short blocks within which only a few different haplotypes occur; second, as suggested by Gusfield, it is reasonable to assume that the haplotypes observed within a block have evolved according to a perfect phylogeny, in which at most one mutation event has occurred at any site.We present a simple and efficient polynomial-time algorithm for inferring haplotypes from the genotypes of a set of individuals assuming a perfect phylogeny. Using a reduction to 2-SAT we extend this algorithm to handle constraints that apply when we have genotypes from both parents and child. We also present a hardness result for the problem of removing the minimum number of individuals from a population to ensure that the genotypes of the remaining individuals are consistent with a perfect phylogeny.Our algorithms have been tested on real data and give biologically meaningful results.

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Section: If (C C ) Is Ofmentioning

confidence: 91%

Section: Lemma 42 If the Algorithm Reports Infeasibility Then Therementioning

confidence: 99%

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Efficient Reconstruction of Haplotype Structure via Perfect Phylogeny

Eskin

Halperin

Karp

2003

J. Bioinform. Comput. Biol.

102

106

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“…The time for the reduction is O(nm), and the graph realization problem can be solved by several published methods. The method in [3] is based on a general algorithm due to Lofgren, and runs in O(nmα(nm)) time. That algorithm is the basis for the worst-case time bound established in [18], but we found it to be too complex to implement.…”

Section: Algorithm and Program Gpphmentioning

confidence: 99%

An Overview of Combinatorial Methods for Haplotype Inference

Gusfield

2004

Computational Methods for SNPs and Haplotype Inference

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A current high-priority phase of human genomics involves the development of a full Haplotype Map of the human genome [23]. It will be used in large-scale screens of populations to associate specific haplotypes with specific complex genetic-influenced diseases. A key, perhaps bottleneck, problem is to computationally infer haplotype pairs from genotype data. This paper follows the talk given at the DIMACS Conference on SNPs and Haplotypes held in November of 2002. It reviews several combinatorial approaches to the haplotype inference problem that we have investigated over the last several years. In addition, it updates some of the work presented earlier, and discusses the current state of our work.

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“…Glover and Greenberg [1987] propose using netforms to enhance the operation of expert systems. A growing body of research is dedicated to detecting network structures that may be hidden inside more complex models [Bixby 1984;Bixby and Wagner 1985;Brown, McBride and Wood 1985;Truemper 1983]. This has led more recently to research into creating these and other netform structures in problem settings where they may not otherwise be found [Glover and Klingman 1988a].…”

Section: Future Directionsmentioning

confidence: 99%

Netform Modeling and Applications

Glover¹,

Klingman²,

Phillips³

1990

Interfaces

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Many real-world applications have profited from netform innovations in both modeling and solution strategies. Practical experience shoves that advances in netform modeling and solution strategies overcome many of the difficulties in conceptual design and problem solving of previous approaches to system optimization. Moreover, they provide the type of technologies required of truly useful decision-planning tools, technologies that facilitate modeling, solution, and implementation. The ultimate test and worth of computer-based planning models, however, depends on their use by practitioners. In this tutorial, we show how certain algebraic models can be viewed graphically using netform modeling and describe several large practical problems we have solved. Some of our insights can make it easier for practitioners to take advantage of these technologies.

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An Almost Linear-Time Algorithm for Graph Realization

Abstract: Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information.

Cited by 63 publications

References 21 publications

Efficient Reconstruction of Haplotype Structure via Perfect Phylogeny

Efficient Reconstruction of Haplotype Structure via Perfect Phylogeny

An Overview of Combinatorial Methods for Haplotype Inference

Netform Modeling and Applications

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