Abstract
In the limited-workspace model, we assume that the input of size n lies in a random access read-only memory. The output has to be reported sequentially, and it cannot be accessed or modified. In addition, there is a read-write workspace of O(s) words, where \(s \in \{1, \dots , n\}\) is a given parameter. In a time-space trade-off, we are interested in how the running time of an algorithm improves as s varies from 1 to n.
We present a time-space trade-off for computing the Euclidean minimum spanning tree (\({{\mathrm{EMST}}}\)) of a set V of n sites in the plane. We present an algorithm that computes \({{\mathrm{EMST}}}(V)\) using \(O(n^3\log s /s^2)\) time and O(s) words of workspace. Our algorithm uses the fact that \({{\mathrm{EMST}}}(V)\) is a subgraph of the bounded-degree relative neighborhood graph of V, and applies Kruskal’s MST algorithm on it. To achieve this with limited workspace, we introduce a compact representation of planar graphs, called an s-net which allows us to manipulate its component structure during the execution of the algorithm.
B. Banyassady and W. Mulzer were supported in part by DFG project MU/3501/2. L. Barba was supported by the ETH Postdoctoral Fellowship.
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Notes
- 1.
Naturally, if \(i + s - 1 > m\), we report the edges \(e_i, \dots , e_m\).
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Acknowledgments
This work was initiated at the Fields Workshop on Discrete and Computational Geometry, held July 31–August 04, 2017, at Carleton university. The authors would like to thank them and all the participants of the workshop for inspiring discussions and for providing a great research atmosphere.
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Banyassady, B., Barba, L., Mulzer, W. (2018). Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_9
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