Colourful Simplicial Depth

Deza, Antoine; Huang, Sui; Stephen, Tamon; Terlaky, Tamás

doi:10.1007/s00454-006-1233-3

Cited by 23 publications

(36 citation statements)

References 15 publications

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“…In the case b = 3, we have at least 15 − ℓ simplices, so ℓ = 2 or ℓ = 3, and in the case b = 4, we have 20 − 3ℓ so ℓ = 3. In summary, we need to rule out systems where the triple (ℓ, b, j) is one of (3, 4, 2), (3, 4, 1), (3,3,2), (2, 3, 2).…”

Section: Enumeration Detailsmentioning

confidence: 99%

A Note on Lower Bounds for Colourful Simplicial Depth

2013

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Abstract:The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d 2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.Keywords: colourful simplicial depth; Colourful Carathéodory Theorem; discrete geometry; polyhedra; combinatorial symmetry Classification: MSC 52A35, 05D05, 52B05A colourful configuration is the union of (d+1) sets, or colours,Without loss of generality we assume that the points in S ∪ {0} are in general position. We are interested in the colourful simplices formed by taking the convex hull of a set containing one point of each colour. The colourful simplicial depth problem is to find a colourful configuration, with each S i containing the origin 0 in the interior of its convex hull, minimizing the number of colourful simplices containing 0. We denote this minimum by µ(d). We take the simplices to be closed and remark that the minimum should be attained.

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Section: Enumeration Detailsmentioning

confidence: 99%

A Note on Lower Bounds for Colourful Simplicial Depth

2013

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“…Deza et al [DHST06] formalized the notion and considered bounds for the colourful depth of points in the intersection of the convex hulls of the colours. Among the recent work on colourful depth are proofs of the lower [Sar15] The monochrome simplicial depth can be computed by enumerating simplices, but in general dimension, it is quite challenging to compute it more efficiently [Alo06], [CO01], [FR05].…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Colourful Simplicial Depth and Medians in the Plane

Zasenko

Stephen

2016

Combinatorial Optimization and Applications

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Abstract. The colourful simplicial depth (CSD) of a point x ∈ R 2 relative to a configuration P = (P 1 , P 2 , . . . , P k ) of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point in R 2 , called a median, that maximizes colourful simplicial depth.For computing the colourful simplicial depth of x, our algorithm runs in time O (n log n + kn) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n 4 ). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O (n log n) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n 4 ) for finding a monochrome median.

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“…Deza et al [4] show that for d = 2 the smallest possible number of colorful simplices is 5, and for d = 3 this number is either 8 or 10. The following theorem shows that the number is 10.…”

Section: Introductionmentioning

confidence: 99%

Quadratically Many Colorful Simplices

Bárány¹,

Matoušek²

2007

SIAM J. Discrete Math.

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The colorful Carathéodory theorem asserts that if X 1 , X 2 , . . . , X d+1 are sets in R d , each containing the origin 0 in its convex hull, then exists a set S ⊆ X 1 ∪ · · · ∪ X d+1 with |S ∩ X i | = 1 for all i = 1, 2, . . . , d + 1 and 0 ∈ conv(S) (we call conv(S) a colorful covering simplex). Deza, Huang, Stephen and Terlaky proved that if the X i are in general position with respect to 0 (consequently, each X i has at least d + 1 points), then there are at least 2d colorful covering simplices, and they constructed an example with no more than d 2 + 1 such simplices. Under the same assumption, we show that there are at least 1 5 d(d + 1) colorful covering simplices, thus determining the order of magnitude. We also obtain a lower bound of 3d, which is better for small d, and in particular, together with a parity argument it settles the case d = 3, where the minimum possible number of colorful covering simplices is 10.

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Colourful Simplicial Depth

Cited by 23 publications

References 15 publications

A Note on Lower Bounds for Colourful Simplicial Depth

A Note on Lower Bounds for Colourful Simplicial Depth

Algorithms for Colourful Simplicial Depth and Medians in the Plane

Quadratically Many Colorful Simplices

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