Abstract
The use of realistic input models has gained popularity in the theory community. Assuming a realistic input model often precludes complicated hypothetical inputs, and the analysis yields bounds that better reflect the behaviour of algorithms in practice.
One of the most popular models for polygonal curves and objects is \(\lambda \)-low-density. To select the most efficient algorithm for a certain input, one often needs to approximate the \(\lambda \)-low-density value, or density for short. In this paper, we show that given a set of n objects in \(\mathbb {R}^2\), one can \((2+\varepsilon )\)-approximate the density value in \(O(n \log n + \lambda n/\varepsilon ^4)\) time.
Finally, we argue that some real-world trajectory data sets have small density values, warranting the recent development of specialised algorithms. This is done by computing approximate density values for 12 real-world trajectory data sets.
J. GudmundssonāFunded by the Australian Government through the Australian Research Council DP180102870.
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Notes
- 1.
The dependence onĀ \(\lambda \) is based on our own calculation, as it is not stated inĀ [6].
- 2.
Weisstein, Eric W. āIncircle.ā From MathWorldāA Wolfram Web Resource. https://mathworld.wolfram.com/Incircle.html.
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Gudmundsson, J., Huang, Z., Wong, S. (2024). Approximating theĀ \(\lambda \)-low-density Value. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14422. Springer, Cham. https://doi.org/10.1007/978-3-031-49190-0_5
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