Abstract
This paper considers the problem of finding minimal length tree networks on the unit sphere Φ of a given point set (V) where distance is measured along great circular arcs. The related problems of finding a Steiner Minimal TreeSMT(V) and of finding a Minimum Spanning TreeMST(V) are treated through a simplicial decomposition technique based on computing the Delaunay TriangulationDT(V) and the Voronoi DiagramVD(V) of the given point set.O(N logN) algorithms for computingDT(V),VD(V), andMST(V) as well as anO(N logN) heuristic for finding a sub-optimalSMT(V) solution are presented, together with experimental results for randomly distributed points on Φ.
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References
A.A. Aly, D.C. Kay and D.W. Litwhiler, Location dominance on spherical surfaces, Oper. Res. 27(1979)972–981.
M. Bern and R. Graham, The shortest-network problem, Sci. Amer. 260(1989)84–89.
J.E. Beasley, A greedy heuristic for the Euclidean and rectilinear Steiner problem, Working Paper, Department of Management Science, Imperial College, London (1989).
S.K. Chang, The generation of minimal trees with a Steiner topology, J. ACM 19(1972)699–711.
D. Cheriton and R.E. Tarjan, Finding minimal spanning trees, SIAM J. Comput. 5(1976)724–742.
F.R.K. Chung and F.K. Hwang, A lower bound for the Steiner tree problem, SIAM J. Appl. Math. 34(1976)27–36.
F.R.K. Chung and R.L. Graham, Steiner trees for ladders, Ann. Discr. Math. 2(1978)173–200.
E.J. Cockayne and Z.A. Melzak, Steiner's problem for set terminals, J. Appl. Math. 26(1968)213–218.
E.J. Cockayne and Z.A. Melzak, Euclidean constructability in graph minimization problems, Math. Mag. 42(1969)206–208.
E.J. Cockayne, On Fermat's problem on the surface of a sphere, Math. Mag. (Sept–Oct, 1972) 216–219.
D.R. Courant and H. Robbins,What is Mathematics? (Oxford University Press, New York, 1941).
D. Dobkin and P. Thurston, The geometry of circles: Voronoi diagrams, Moebius transformations, convex hulls, fortunes algorithm, the cut locus and parameterization of shape, unpublished notes for a course, Department of Computer Science, Princeton University (1987).
J.D.H. Donnay, Spherical trigonometry, in:Encyclopedia of Mathematics (Interscience, New York, 1945).
Z. Drezner and G.O. Wesolowsky, Facility location on a sphere, J. Oper. Res. Soc. 29(1979)997–1004.
D.-Z. Du, F.K. Hwang and J.F. Weng, Steiner minimal trees on zig-zag lines, Trans. Amer. Math. Soc. 278(1982)149–156.
M.R. Garey, R.L. Graham and D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math. 32(1977)835–859.
M.R. Garey and D.S. Johnson,Computers and Intractability; A Guide to the Theory of NP-completeness (Freeman, San Francisco, 1979).
R.L. Graham and F.K. Hwang, Remarks on Steiner minimal trees, Bull. Inst. Math. Acad. Sinica 4(1976)177–182.
M.G. Greening, Solution to problem E2233, Amer. Math. Monthly 4(1971)303–304.
E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16(1968)1–29.
L. Guibas and J. Stolfi, Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams, ACM Trans. Graphics 4(1985)74–123.
D. Hilbert and S. Cohn-Vossen,Geometry and the Imagination (Chelsea Publ., New York, 1962).
F.W. Hwang and D. Richards, Steiner tree problems (1989), unpublished manuscript.
D.W. Litwhiler, Steiner's problem and Fagnano's result on the sphere, Math. Progr. 18(1980)286–290.
R.F. Love, J.G. Morris and G.O. Wesolowsky,Facilities Location (North-Holland, Amsterdam, 1988).
Z.A. Melzak, On the problem of Steiner, Can. Math. Bull. 4(1961)143–148.
R. Motwhani and P. Raghavan, Geometry on the sphere: Deterministic and probabilistic computations (1987), unpublished manuscript.
F. Preparata and M.I. Shamos,Computational Geometry (Springer, 1985).
R.C. Prim, Shortest connecting networks and some generalizations, BSTJ 36(1957)1389–11401.
M.I. Shamos, Computational geometry, Ph.D. Thesis, Yale University (1978).
J. MacGregor Smith and J.S. Liebman, Steiner trees, Steiner circuits, and the interference problem in building design, Eng. Opt. 4(1979)15–36.
J. MacGregor Smith, D.T. Lee and J.S. Liebman, AnO(N logN) heuristic for Steiner minimal tree problems on the Euclidean metric, Networks 11(1981)23–39.
P.M. Vaidya, Minimum spanning trees ink-dimensional space, SIAM J. Comput. 17(1988)572–582.
G.O. Wesolowsky, Location problems on a sphere, Regional Sci. Urban Econ. 12(1982)495–508.
P. Winter, Steiner problem in networks: A survey, Networks 17(1987)129–167.
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Dolan, J., Weiss, R. & MacGregor Smith, J. Minimal length tree networks on the unit sphere. Ann Oper Res 33, 501–535 (1991). https://doi.org/10.1007/BF02067239
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DOI: https://doi.org/10.1007/BF02067239