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 two 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(\({\varvec{n}}^{4/3}\)). 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’s formula for finding the roots of a cubic equation. We believe that our technique has the potential to derive improved bounds for the number of halving edges.
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Solanki, N., Chauhan, P., Pal, M. (2022). Rederiving the Upper Bound for Halving Edges Using Cardano’s Formula. In: Sharma, T.K., Ahn, C.W., Verma, O.P., Panigrahi, B.K. (eds) Soft Computing: Theories and Applications. Advances in Intelligent Systems and Computing, vol 1380. Springer, Singapore. https://doi.org/10.1007/978-981-16-1740-9_21
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DOI: https://doi.org/10.1007/978-981-16-1740-9_21
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