Drawing Hamiltonian Cycles with No Large Angles

Dumitrescu, Adrian; Pach, János; Tóth, Géza

doi:10.1007/978-3-642-11805-0_3

“…The improvement is based on Lemma 2.4 below. This lemma -in fact a stronger version, stating that any two opposite cones defined by the three concurrent lines contain the same number of points-has been proved for even n by Dumitrescu et al [11]. Our proof of Lemma 2.4 is similar to their proof.…”

Section: (I)supporting

confidence: 57%

“…We give it because we also need it for odd n, and because we will need an understanding of the proof to describe our algorithm for computing the concurrent triple in the lemma. Our algorithm will run in O(n log 2 n) time, a significant improvement over the O(n 4/3 log 1+ε n) running time obtained (for the case of even n) by Dumitrescu et al [11].…”

Section: (I)mentioning

confidence: 71%

On beta-Plurality Points in Spatial Voting Games

Aronov¹,

Berg²,

Gudmundsson³

et al. 2020

Preprint

0

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Let V be a set of n points in R d , called voters. A point p ∈ R d is a plurality point for V when the following holds: for every q ∈ R d the number of voters closer to p than to q is at least the number of voters closer to q than to p. Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q. For most voter sets a plurality point does not exist. We therefore introduce the concept of β-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to p (but not to q) is scaled by a factor β, for some constant 0 < β 1. We investigate the existence and computation of β-plurality points, and obtain the following results. Define β * d := sup{β : any finite multiset V in R d admits a β-plurality point}. We prove that β * 2 = √ 3/2, and that 1/ √ d β * d √ 3/2 for all d 3. Define β(V ) := sup{β : V admits a β-plurality point}. We present an algorithm that, given a voter set V in R d , computes an (1 − ε) • β(V ) plurality point in time O( n 2 ε 3d−2 • log n ε d−1 • log 2 1 ε ).

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“…The next lemma shows that this is indeed the case. The lemma-in fact, a stronger version, stating that any two opposite cones defined by the three concurrent lines contain the same number of pointshas been proved for even n by Dumitrescu et al [11]. Our proof of Lemma 2.5 is similar to their proof.…”

Section: Proof Consider a Votersupporting

confidence: 55%

“…We give it because we also need it for odd n, and because we will need an understanding of the proof to describe our algorithm for computing the concurrent triple in the lemma. Our algorithm will run in O (n log n) time, a significant improvement over the O (n 4/3 log 1+ε n) running time obtained (for the case of even n) by Dumitrescu et al [11]. Lemma 2.5.…”

Section: Proof Consider a Votermentioning

confidence: 80%

On β-Plurality Points in Spatial Voting Games

Aronov

¹

,

Berg

²

,

Gudmundsson

³

et al. 2021

ACM Trans. Algorithms

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Let V be a set of n points in mathcal R d , called voters . A point p ∈ mathcal R d is a plurality point for V when the following holds: For every q ∈ mathcal R d , the number of voters closer to p than to q is at least the number of voters closer to q than to p . Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q . For most voter sets, a plurality point does not exist. We therefore introduce the concept of β-plurality points , which are defined similarly to regular plurality points, except that the distance of each voter to p (but not to q ) is scaled by a factor β , for some constant 0< β ⩽ 1. We investigate the existence and computation of β -plurality points and obtain the following results. • Define β * d := {β : any finite multiset V in mathcal R d admits a β-plurality point. We prove that β * d = √3/2, and that 1/√ d ⩽ β * d ⩽ √ 3/2 for all d ⩾ 3. • Define β ( p, V ) := sup {β : p is a β -plurality point for V }. Given a voter set V in mathcal R 2 , we provide an algorithm that runs in O ( n log n ) time and computes a point p such that β ( p , V ) ⩾ β * b . Moreover, for d ⩾ 2, we can compute a point p with β ( p , V ) ⩾ 1/√ d in O ( n ) time. • Define β ( V ) := sup { β : V admits a β -plurality point}. We present an algorithm that, given a voter set V in mathcal R d , computes an ((1-ɛ)ċ β ( V ))-plurality point in time O n 2 ɛ 3d-2 ċ log n ɛ d-1 ċ log 2 1ɛ).

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“…Note that the ζ-median and the (−ζ)-median of V are the same, so that L V,ζ and L V,−ζ are the same line, but with opposite directions. These considerations lead to the following useful observation, first proved in this precise form in [21, Lemma 3] (see also [22,Lemma 2] and [14,Lemma 2]). Lemma 8.2.…”

Section: Equilateral Triangles In the Planementioning

confidence: 91%

On The Number of Similar Instances of a Pattern in a Finite Set

Ábrego¹,

Fernández-Merchant²,

Katz³

et al. 2015

Preprint

0

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New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an n-point subset of the plane is shown to be no more than ⌊(4n − 1)(n − 1)/18⌋. The number of k-term arithmetic progressions that lie within an n-point subset of the line is shown to be at most (n − r)(n + r − k + 1)/(2k − 2), where r is the remainder when n is divided by k − 1. This upper bound is achieved when the n points themselves form an arithmetic progression, but for some values of k and n, it can also be achieved for other configurations of the n points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

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Drawing Hamiltonian Cycles with No Large Angles

Cited by 5 publications

References 12 publications

On beta-Plurality Points in Spatial Voting Games

On beta-Plurality Points in Spatial Voting Games

On β-Plurality Points in Spatial Voting Games

On The Number of Similar Instances of a Pattern in a Finite Set

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