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Computing Maximum Hamiltonian Paths in Complete Graphs with Tree Metric

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Fun with Algorithms (FUN 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7288))

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Abstract

We design a linear time algorithm computing the maximum weight Hamiltonian path in a weighted complete graph K T , where T is a given undirected tree. The vertices of K T are nodes of T and weight(i, j) is the distance between i, j in T. The input is the tree T and two nodes u, v ∈ T, the output is the maximum weight Hamiltonian path between these nodes. The size n of the input is the size of T (however the total size of the complete graph K T is quadratic with respect to n). Our algorithm runs in \(\mathcal{O}(n)\) time. Correctness is based on combinatorics of alternating sequences. The problem has been inspired by a similar (but much simpler) problem in a famous book of Hugo Steinhaus.

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References

  1. Steinhaus, H.: One Hundred Problems in Elementary Mathematics. Dover Publications (September 1, 1979)

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  2. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley (1974)

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  3. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)

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  4. Gibbons, A., Rytter, W.: Efficient parallel algorithms. Cambridge University Press (1988)

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw, Poland

    Wojciech Rytter & Bartosz Szreder

  2. Dept. of Math. and Informatics, Copernicus University, Toruń, Poland

    Wojciech Rytter

Authors
  1. Wojciech Rytter
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  2. Bartosz Szreder
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Editor information

Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Col. By Dr., K1S 5B6, Ottawa, ON, Canada

    Evangelos Kranakis

  2. Department of Mathematics and Computer Science, Wesleyan University, 06459, Middleton, CT, USA

    Danny Krizanc

  3. Dipartimento di Scienze Ambientali, Informatica e Statistica, Università Ca’ Foscari, Via Torino 155, 30172, Mestre (Ve), Italy

    Flaminia Luccio

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© 2012 Springer-Verlag Berlin Heidelberg

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Rytter, W., Szreder, B. (2012). Computing Maximum Hamiltonian Paths in Complete Graphs with Tree Metric. In: Kranakis, E., Krizanc, D., Luccio, F. (eds) Fun with Algorithms. FUN 2012. Lecture Notes in Computer Science, vol 7288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30347-0_34

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  • DOI: https://doi.org/10.1007/978-3-642-30347-0_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30346-3

  • Online ISBN: 978-3-642-30347-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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