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Algorithms 2018, 11(7), 93; https://doi.org/10.3390/a11070093

Layered Graphs: Applications and Algorithms

1
Department of Computer Science and Engineering, Amrita Vishwa Vidyapeetham, Amritapuri 690525, India
2
Department of Computer Science, University of Texas at Dallas, Richardson, TX 75083, USA
*
Author to whom correspondence should be addressed.
Received: 4 June 2018 / Revised: 17 June 2018 / Accepted: 21 June 2018 / Published: 28 June 2018
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Abstract

The computation of distances between strings has applications in molecular biology, music theory and pattern recognition. One such measure, called short reversal distance, has applications in evolutionary distance computation. It has been shown that this problem can be reduced to the computation of a maximum independent set on the corresponding graph that is constructed from the given input strings. The constructed graphs primarily fall into a class that we call layered graphs. In a layered graph, each layer refers to a subgraph containing, at most, some k vertices. The inter-layer edges are restricted to the vertices in adjacent layers. We study the MIS, MVC, MDS, MCV and MCD problems on layered graphs where MIS computes the maximum independent set; MVC computes the minimum vertex cover; MDS computes the minimum dominating set; MCV computes the minimum connected vertex cover; and MCD computes the minimum connected dominating set. MIS, MVC and MDS run in polynomial time if k=Θ(log|V|). MCV and MCD run in polynomial time ifk=O((log|V|)α), where α<1. If k=Θ((log|V|)1+ϵ), for ϵ>0, then MIS, MVC and MDS run in quasi-polynomial time. If k=Θ(log|V|), then MCV and MCD run in quasi-polynomial time. View Full-Text
Keywords: NP-complete; layered graph; quasi-polynomial time; dynamic programming; independent set; vertex cover; dominating set; string transformations; social networks NP-complete; layered graph; quasi-polynomial time; dynamic programming; independent set; vertex cover; dominating set; string transformations; social networks
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Chitturi, B.; Balachander, S.; Satheesh, S.; Puthiyoppil, K. Layered Graphs: Applications and Algorithms. Algorithms 2018, 11, 93.

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