A simple and relatively efficient triangulation of then-cube

Discrete & Computational Geometry, Jun 2007

The only previously published triangulation of then-cube usingo(n!) simplices, due to Sallee, usesO(n−2n!) simplices. We point out a very simple method of achievingO(ρnn!) simplices, where ρ<1 is a constant.

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A simple and relatively efficient triangulation of then-cube

Discrete Comput Geom A Simple and Relatively Efficient Triangulation of the n-Cube* M a r k H a i m a n 0 0 Department of Mathematics, Massachusetts Institute of Technology , 77 Massachusetts Avenue,Cambridge, MA 02139 , USA The only previously published triangulation of the n-cube using o(n!) simplices, due to Sallee, uses O(n-2n!) simplices. We point out a very simple method of achieving O(p"n!)simplices, where p < 1 is a constant. - and hence Sallee uses at least O(n-2n!) simplices. (Readers of [ 7 ] should observe that the function g(n, m) of that paper is actually A(n - 1, m - 1), a fact not noted there.) The construction given below triangulates I n with O(p"n!) simplices, where p < 1. In fact, given a triangulation of I n into T(n) simplices for any particular value of n, we can take p = (T(n)/n!) 1/n. The smallest value for p obtainable from triangulations published to date is p = (13,248/40,320)1/a ~ 0.870 from Sallee's triangulation of I s. We remark that Todd [ 11 ] proposed the quantity R(n) = (T(n)/n!) 1In as a measure of a triangulation's efficiency. Previous constructions have lim,~ ~ R(n) = 1, whereas our results show that any value of R(n) achievable for one triangulation is achievable asymptotically. This is still far from R(n) = o(1), let alone the lower bound of R(n) = O(n- 1/2). 2. Construction Definition. A polyhedral decomposition of an n-dimensional polytope P is a union P = T1 u T 2 u . - . w TR of n-dimensional polytopes 7]i such that for all i,j the vertices of T/are vertices of P and Ti n Tj is a (possibly empty) face of both Ti and Tj. If each Ti is a simplex, then {Ti} is a triangulation of P. L e m m a 1. Every polyhedral decomposition of P can be refined to a triangulation. Proof. We require triangulations 0i of the T~ such that 0~ and 0j induce the same triangulation on T / n Tj considered as a face of T/and as a face of Tj. N o w there are well-known constructions [ 4 ], [ 6 ], [ 9 ], [ 12 ] whereby we associate to a total ordering ~ of the vertices of a polytope T a triangulation 0 in such a way that the triangulation induced on each face F G T is the one associated to the restriction of a to the vertices of F. Hence fixing any total ordering ~to of all the vertices of P and triangulating each T~in accordance with ao we obtain compatible triangulations 0~ as required. [] L e m m a 2 [ 1 ]. Every triangulation o f A k x A t uses exactly (k +/)!/k! l! simplices, where A, denotes an n-dimensional simplex. Proof. Realize A k in R k a s the convex hull of 0 and the unit coordinate vectors e ~ = ( 0 . . . . . 0, 1,0 . . . . . 0). Likewise A~. Then A k A,___R k+t has vertices 0, ei (I < i < k + / ) , and el + e~ (1 < i < k < j < k + / ) . Its volume is V(Ak)V(AI) = 1/k! l !. We claim every nondegenerate (k + / ) - s i m p l e x A spanned by vertices of A k A l has volume 1/(k +/)!. N o t e that there are affine-linear symmetries of A k x At acting transitively on the vertices. These preserve volume, so we can assume 0 is a vertex of A. Then _ (k + / ) ! v(A) is the determinant of the matrix M whose rows are the coordinates of the other vertices; we are to show that this determinant is _+1. If some el is a vertex of A, then expanding by minors on the corresponding row gives the result by induction. If not, then M is the edge-vertex incidence matrix of a (k, /)-bipartite graph with k + l edges. Having one too m a n y edges to be a tree, this graph must contain a cycle, necessarily even. But then M is singular, contrary to hypothesis. [ ] Theorem 1. Given a triangulation {$1 . . . . . S~} of a k-dimensional polytope P and a triangulation { TI . . . . . Tt} of an l-dimensional polytope Q, there exists a triangulation of P Q using s . t . ( k + l)!/k! l! simplices. Proof It is easy to see that {S, Tj} is a polyhedral decomposition of P Q. Refine it to a triangulation by Lemma 1. Each S~ T~ will then contain (k + l)!/k! 1! simplices by Lemma 2, establishing the result. [ ] Corollary 1. I f I" can be triangulated into T(n) simplices, then I k" can be trianyulated into [(kn)!/(n !)k] T(n)k = pk,(kn) ! simplices, where p = (T(n)/n !)1/,. Proof Immediate from Theorem 1 by induction on k. [ ] References 1. L.J. Billera , R. Cushman , and J. A. Sanders , The Stanley decomposition of the harmonic oscillator , Nederl. Akad. Wetensch. Proc. Ser. A 91 ( 1988 ), 375 - 393 . 2. R. W. Cottle , Minimal triangulations of the 4-cube , Discrete Math. 40 ( 1982 ), 25 - 29 . 3. C. Lee , Triangulating the d-cube, in Discrete Geometry and Convexity , J. E. Goodman , E. Lutwak , J. Malkevitch , and R. Pollack, eds., New York Academy of Sciences, New York ( 1985 ), pp. 205 - 21 I. 4. C. Lee , Some notes on triangulating polytopes, in Proc. 3 . Kolloquium iiber Diskrete Geometrie, Institut ffir Mathematik, Universit/it Salzburg ( 1985 ), pp. 173 - 181 . 5. P. S. Mara , Triangulations of the Cube, M. S. Thesis , Colorado State University ( 1972 ). 6. J. F. Sallee , A triangulation of the n-cube , Discrete Math. 40 ( 1982 ), 81 86 . 7. J. F. Sallee , Middle-cut triangulations of the n-cube, S I A M J. Algebraic Discrete Methods 5 , no. 3 ( 1984 ), 407 - 419 . 8. W. D. Smith , Polytope triangulations in d-space, improving Hadamard's inequality and maximal volumes of regular polytopes in hyperbolic d-space. Manuscript , Princeton, NJ ( September 1987 ). 9. R. P. Stanley , Decompositions of rational convex polytopes , Ann. Discrete Math. 6 ( 1980 ), 333 - 342 . 10. R. P. Stanley , Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA ( 1986 ). 11. M. J. Todd , The Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems , Vol. 124 , Springer-Verlag, Berlin ( 1976 ). 12. B. Von Hohenbalken , How To Simplicially Partition a Polytope, Research Paper No. 79 17 , Department of Economics, University of Alberta, Edmonton ( 1979 ).


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Mark Haiman. A simple and relatively efficient triangulation of then-cube, Discrete & Computational Geometry, 2007, 287-289, DOI: 10.1007/BF02574690