Weight Balancing on Boundaries and Skeletons

Barba, Luis; Cheong, Otfried; Carufel, Jean-Lou De; Dobbins, Michael Gene; Fleischer, Rudolf; Kawamura, Akitoshi; Korman, Matias; Okamoto, Yoshio; Pach, János; Tang, Yuan; Tokuyama, Takeshi; Verdonschot, Sander; Wang, Tianhao

doi:10.1145/2582112.2582142

Cited by 3 publications

(11 citation statements)

References 10 publications

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“…We can show similarly that for even k ≥ 3, G(k) is on the segment OG (3). The proof is thus completed.…”

Section: Thus We Know Inductively That Every G(k) Is On the Line G(2supporting

confidence: 53%

“…Teramoto et al [2] studied the insertion of points into the unit square in R d in such a way that the Euclidean distance between any pair of points becomes as uniform as possible. Recently, Barba et al [3] considered the problem that given a set of weights, a closed connected region, and a target position, we are asked to place the weights on the boundary of the region so that the center of mass lies at the target.…”

Section: Related Workmentioning

confidence: 99%

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Online Weight Balancing on the Unit Circle

Fujiwara

Fujito

2014

Lecture Notes in Computer Science

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SUMMARYWe consider a problem as follows: Given unit weights arriving in an online manner with the total cardinality unknown, upon each arrival we decide where to place it on the unit circle in R 2 . The objective is to set the center of mass of the placed weights as close to the origin as possible. We apply competitive analysis defining the competitive difference as a performance measure. We first present an optimal strategy for placing unit weights which achieves a competitive difference of 1 5 . We next consider a variant in which the destination of each weight must be chosen from a set of positions that equally divide the unit circle. We give a simple strategy whose competitive difference is 0.35. Moreover, in the offline setting, several conditions for the center of mass to lie at the origin are derived. key words: online algorithm, competitive analysis, computational geometry, online optimization

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“…We can show similarly that for even k ≥ 3, G(k) is on the segment OG (3). The proof is thus completed.…”

Section: Thus We Know Inductively That Every G(k) Is On the Line G(2supporting

confidence: 53%

Section: Related Workmentioning

confidence: 99%

Online Weight Balancing on the Unit Circle

Fujiwara

Fujito

2014

Lecture Notes in Computer Science

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“…Does Theorem 1 hold for the weighted barycenter with fixed but unequal weights? The theorem fails to generalize for the 1-skeleton of a triangular prism in 3 with weights 1, 1, 1 + , but may generalize when more weighted points are allowed [3]. Note that the theorem implies an analogous statement for weights and skeletons of distinct dimension that result from sequential partitions of d into equal parts.…”

mentioning

confidence: 92%

“…The conjecture was originally motivated by the engineering problem of determining how counterweights can be attached to some 3-dimensional object to adjust its center of mass to reduce vibrations when the object moves [1] [3]. Theorem 1 is loosely related to several results in topology and combinatorial geometry.…”

mentioning

confidence: 99%

A point in a $$nd$$ n d -polytope is the barycenter of $$n$$ n points in its $$d$$ d -faces

Dobbins

2014

Invent. math.

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ABSTRACT. Using equivariant topology, we prove that it is always possible to find n points in the d -dimensional faces of a n d -dimensional convex polytope P so that their center of mass is a target point in P. Equivalently, the n -fold Minkowski sum of a n d -polytope's d -skeleton is that polytope scaled by n . This verifies a conjecture by Takeshi Tokuyama.The goal of this article is to prove the following theorem, recently conjectured by Takeshi Tokuyama for the 1-skeleton (see Acknowledgments), which to the author's knowledge, was the first time the question had been considered. The conjecture was originally motivated by the engineering problem of determining how counterweights can be attached to some 3-dimensional object to adjust its center of mass to reduce vibrations when the object movesTheorem 1. For any n d -polytope P and for any point p ∈ P, there are pointsTheorem 1 is loosely related to several results in topology and combinatorial geometry. One version of the Borsuk-Ulam Theorem states that any continuous map ψ from the sphere d to d attains the same value at an antipodal pair ∃x . ψ(−x ) = ψ(x ). This has numerous applications, a collection of which is presented in [7]. The barycenter of antipodal points is the origin, so Theorem 1 for n = 2, d = 1 is just the Borsuk-Ulam Theorem for d = 1 where ψ is the radial distance of a convex polygon in polar coordinates. Gromov's Waist Theorem may be viewed as a further refinement of the Borsuk-Ulam Theorem that says a continuous map from a sphere to a Euclidean space of lower dimension is constant on a large subset of correspondingly higher dimension distributed widely around the sphere [6]. Intuitively, we may view Theorem 1 for n = 2 as saying such a subset for the radial distance of a polytope includes an antipodal pair of a correspondingly lower dimensional subspace, the d -skeleton. Munkholm gave a generalization of the Borsuk-Ulam Theorem to cyclic actions of prime order on the sphere that says an equivariant map must be constant on a large subspace of orbits in the quotient space [8]. We may view Theorem 1 for n prime as being related in an analogous way. After seeing the proof, we will make this connection more explicit.In another sense, Theorem 1 is analogous to Carathéodory's Theorem. Carathéodory's Theorem states that the convex hull of a set X ⊂ d is the union of convex hulls of all subsets of d + 1 points in X . Letting X be the 0-skeleton (vertices) of a polytope, this means any point in a d -polytope is a weighted barycenter of d + 1 vertices. Bárány and Karasev recently gave conditions on a set for its convex hull to be the union of convex hulls of a smaller number of points [2]. Here, we similarly reduce the number of points, but we further require any point in P to be the exact barycenter and we only consider P as the convex hull of a skeleton. Theorem 1 also 2010 Mathematics Subject Classification. 51M04, 51M20, 52B11, 55N91.

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“…Our weight assignment problem is closely related to the weight balancing problem which was studied by Barba et al [4]. The input consists of a simple polygon, a target point inside the polygon, and a set of weights.…”

Section: Introductionmentioning

confidence: 99%

Assigning weights to minimize the covering radius in the plane

Ahn

2019

Computational Geometry

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Given a set P of n points in the plane and a multiset W of k weights with k ≤ n, we assign each weight in W to a distinct point in P to minimize the maximum weighted distance from the weighted center of P to any point in P . In this paper, we give two algorithms which take O(k 2 n 2 log 3 n) time and O(k 5 n log 3 k + kn log 3 n) time, respectively. For a constant k, the second algorithm takes only O(n log 3 n) time, which is near linear.

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Weight Balancing on Boundaries and Skeletons

Cited by 3 publications

References 10 publications

Online Weight Balancing on the Unit Circle

Online Weight Balancing on the Unit Circle

A point in a $$nd$$ n d -polytope is the barycenter of $$n$$ n points in its $$d$$ d -faces

Assigning weights to minimize the covering radius in the plane

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