Abstract
The quality of an embedding Φ: V ↦ℝ2 of a graph G = (V, E) into the Euclidean plane is the ratio of max {u, v} ∈ E ||Φ(u) − Φ(v)||2 to \(\min_{\{u, v\} \not\in E} ||\Phi(u) - \Phi(v)||_2\). Given a unit disk graph G = (V, E), we seek algorithms to compute an embedding Φ: V ↦ℝ2 of best (smallest) quality. This paper presents a simple, combinatorial algorithm for computing a O(log2.5 n)-quality 2-dimensional embedding of a given unit disk graph. Note that G comes with no associated geometric information. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in ℝO(1). Our results extend to unit ball graphs (UBGs) in fixed dimensional Euclidean space. Constructing a “growth-restricted approximation” of the given unit disk graph lies at the core of our algorithm. This approach allows us to bypass the standard and costly technique of solving a linear program with exponentially many “spreading constraints.” As a side effect of our construction, we get a constant-factor approximation to the minimum clique cover problem on UBGs, described without geometry. Our problem is a version of the well known localization problem in wireless networks.
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Pemmaraju, S.V., Pirwani, I.A. (2007). Good Quality Virtual Realization of Unit Ball Graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_29
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DOI: https://doi.org/10.1007/978-3-540-75520-3_29
Publisher Name: Springer, Berlin, Heidelberg
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