Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees

  • Bahareh BanyassadyEmail author
  • Luis Barba
  • Wolfgang Mulzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


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.


Euclidean minimum spanning tree Relative neighborhood graph Time-space trade-off Limited workspace model Kruskal’s algorithm 



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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany
  2. 2.ETH ZurichZurichSwitzerland

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