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Worst-case bounds for subadditive geometric graphs

Authors:

Marshall Bern,

David EppsteinAuthors Info & Claims

SCG '93: Proceedings of the ninth annual symposium on Computational geometry

Pages 183 - 188

https://doi.org/10.1145/160985.161018

Published: 01 July 1993 Publication History

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Abstract

We consider graphs such as the minimum spanning tree, minimum Steiner tree, minimum matching, and traveling salesman tour for n points in the d-dimensional unit cube. For each of these graphs, we show that the worst-case sum of the dth powers of edge lengths is O(log n). This is a consequence of a general “gap theorem”: for any subadditive geometric graph, either the worst-case sum of edge lengths is O(n^d–1)/d) and the sum of dth powers is O(log n), or the sum of edge lengths is Ω(n). We look more closely at some specific graphs: the worst-case sum of d powers is O(1) for minimum matching, but Ω(log n) for traveling salesman tour, which answers a question of Snyder and Steele.

References

[1]

M. Ajtai, J. Koml6s, and G. Tusn~dy. On optimal matchings. Combinatorica 4 (1984), 259-264.

Google Scholar

[2]

E. N. Gilbert and it. O. Pollak. Steiner minimal trees. SIAM J. Applied Math. 16 (1968), 1-29.

Digital Library

Google Scholar

[3]

Howard J. Karloff. How long can a Euclidean traveling salesman tour be? SIAM J. Disc. Math. 2 (1989), 91-99.

Digital Library

Google Scholar

[4]

Peter W. Shor. Random planar matching and bin packing. Ph.D. Thesis, Mass. Inst. of Technology, 1985.

Google Scholar

[5]

Timothy Law Snyder. Lower bounds for rectilinear Steiner trees in bounded space. Inf. Proc. Letlets 37 (1991), 71-74.

Digital Library

Google Scholar

[6]

Timothy Law Snyder. Worst-case minimal rectilinear Steiner trees in all dimensions, to appear in Disc. and Comp. Geometry.

Google Scholar

[7]

Timothy Law Snyder and J. Michael Steele. A Priori inequalities for the Euclidean traveling salesman problem. 8th A CM Syrup. on Computational Geometry, 1992, 344-349.

Digital Library

Google Scholar

[8]

J. Michael Steele. Subadditive Euclidean functionals and nonlinear growth in geometric probability. Annals of Probability 9 (1981), 365-376.

Crossref

Google Scholar

[9]

J. Michael Steele. Probabilistic and worst case analyses of classical problems of combinatorial optimization in Euclidean space. Math. of Operations Research 15 (1990), 749-770.

Digital Library

Google Scholar

[10]

J. Michael Steele and Timothy Law Snyder. Worst-case greedy matchings in the unit d-cube. Networks 20 (1990), 779-800.

Crossref

Google Scholar

[11]

J. Michael Steele and Timothy Law Snyder. Worst-case growth rates of some classical problems of combinatorial optimization. SIAM J. Compuling 18 (1989), 278-287.

Digital Library

Google Scholar

[12]

Jeffrey D. Ullman. Computational Aspects of VSLi. Computer Science Press, 1984.

Digital Library

Google Scholar

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Edelsbrunner HIvanov AKarasev R(2015)Current Open Problems in Discrete and Computational GeometryModeling and Analysis of Information Systems10.18255/1818-1015-2012-5-5-1719:5(5-17)Online publication date: 4-Mar-2015
https://doi.org/10.18255/1818-1015-2012-5-5-17
Curticapean RKünnemann M(2015)A Quantization Framework for Smoothed Analysis of Euclidean Optimization ProblemsAlgorithmica10.1007/s00453-015-0043-573:3(483-510)Online publication date: 1-Nov-2015
https://dl.acm.org/doi/10.1007/s00453-015-0043-5
Huang LLi J(2015)Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic PointsAutomata, Languages, and Programming10.1007/978-3-662-47672-7_74(910-921)Online publication date: 20-Jun-2015
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Index Terms

Worst-case bounds for subadditive geometric graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Trees
  2. Mathematical analysis
    1. Mathematical optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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SCG '93: Proceedings of the ninth annual symposium on Computational geometry

July 1993

406 pages

ISBN:0897915828

DOI:10.1145/160985

Chairman:
Chee Yap

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 July 1993

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May 18 - 21, 1993

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Edelsbrunner HIvanov AKarasev R(2015)Current Open Problems in Discrete and Computational GeometryModeling and Analysis of Information Systems10.18255/1818-1015-2012-5-5-1719:5(5-17)Online publication date: 4-Mar-2015
https://doi.org/10.18255/1818-1015-2012-5-5-17
Curticapean RKünnemann M(2015)A Quantization Framework for Smoothed Analysis of Euclidean Optimization ProblemsAlgorithmica10.1007/s00453-015-0043-573:3(483-510)Online publication date: 1-Nov-2015
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Phillips JKhanna S(2013)ε-samples for kernelsProceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms10.5555/2627817.2627933(1622-1632)Online publication date: 6-Jan-2013
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Kamousi PChan TSuri SHurtado Fvan Kreveld M(2011)Stochastic minimum spanning trees in euclidean spacesProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998206(65-74)Online publication date: 13-Jun-2011
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Diameter bounds for geometric distance-regular graphs

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Diameter bounds for geometric distance-regular graphs

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Upper bounds on Roman domination numbers of graphs

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