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arXiv · 1802.03730

Rederiving the Upper Bound for Halving Edges using Cardano's Formula

Pintu Chauhan·Manjish Pal·Napendra Solanki
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Abstract

In this paper we rederive an old upper bound on the number of halving edges present in the halving graph of an arbitrary set of $n$ points in 2-dimensions which are placed in general position. We provide a different analysis of an identity discovered by Andrejak et al, to rederive this upper bound of $O(n^{4/3})$. In the original paper of Andrejak et al. the proof is based on a naive analysis whereas in this paper we obtain the same upper bound by tightening the analysis thereby opening a new door to derive these upper bounds using the identity. Our analysis is based on a result of Cardano for finding the roots of a cubic equation. We believe that our technique has the potential to derive improved bounds on the number of halving edges.

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Pintu Chauhan, Manjish Pal, Napendra Solanki. 2018-02-11. Rederiving the Upper Bound for Halving Edges using Cardano's Formula. https://arxiv.org/abs/1802.03730

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