Rotation distance, triangulations, and hyperbolic geometry
Authors:
Daniel D. Sleator, Robert E. Tarjan and William P. Thurston
Journal:
J. Amer. Math. Soc. 1 (1988), 647681
MSC:
Primary 68P05; Secondary 05C05, 05C35, 51M10, 68R10
MathSciNet review:
928904
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Abstract: A rotation in a binary tree is a local restructuring that changes the tree into another tree. Rotations are useful in the design of treebased data structures. The rotation distance between a pair of trees is the minimum number of rotations needed to convert one tree into the other. In this paper we establish a tight bound of on the maximum rotation distance between two node trees for all large . The hard and novel part of the proof is the lower bound, which makes use of volumetric arguments in hyperbolic space. Our proof also gives a tight bound on the minimum number of tetrahedra needed to dissect a polyhedron in the worst case and reveals connections among binary trees, triangulations, polyhedra, and hyperbolic geometry.
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 J. Pallo, On rotation distance in the lattice of binary trees, Inform. Process. Lett. 25 (1987), 369373. MR 905781
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 D. D. Sleator, R. E. Tarjan, and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, Proc. 18th Annual ACM Symposium on Theory of Computing, May 2830, 1986, Berkeley, CA, Association for Computing Machinery, NY, pp. 122135. MR 904133 (89a:68045)
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 W. P. Thurston and J. R. Weeks, The mathematics of threedimensional manifolds, Scientific American 251 (1984), 108120.
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 W. P. Thurston, The geometry and topology of threedimensional manifolds, Princeton Univ., Dept. of Math., 1979.
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 W. T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956), 99116. MR 0081471 (18:408e)
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 K. Wagner, Bemerkungen zum Vierfarbenproblem, J. Deutschen Math.Verein. 46 (1936), 2632.
 [15]
 H. Whitney, A theorem on graphs, Ann. of Math. 32 (1931), 378390. MR 1503003
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Additional Information
DOI:
http://dx.doi.org/10.1090/S08940347198809289044
PII:
S 08940347(1988)09289044
Article copyright:
© Copyright 1988
American Mathematical Society
