Abstract
We consider distance queries in vertex labeled planar graphs. For any fixed \(0 < \epsilon \le 1/2\) we show how to preprocess an undirected planar graph with vertex labels and edge lengths to answer queries of the following form. Given a vertex u and a label \(\lambda \) return a \((1+\epsilon )\)-approximation of the distance between u and its closest vertex with label \(\lambda \). The query time of our data structure is \(O(\lg \lg {n} + \epsilon ^{-1})\), where n is the number of vertices. The space and preprocessing time of our data structure are nearly linear. We give a similar data structure for directed planar graphs with slightly worse performance. The best prior result for the undirected case has similar space and preprocessing bounds, but exponentially slower query time. No nontrivial results were previously considered for the directed case.
This work was partially supported by Israel Science Foundation grant 794/13 and by the Israeli ministry of absorption.
A full version of this paper, including figures, can be found in http://arxiv.org/abs/1504.04690.
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Notes
- 1.
Thorup’s treatment [11] of the undirected step does not contain the full details. See the full version of this paper for ellaboration.
- 2.
The term quasi-\(\epsilon \)-cover is not used by Thorup. He uses \(\epsilon \)-covers for this notion.
- 3.
Klein showed this lemma for \(\epsilon \)-covering sets, while Thorup showed a similar lemma using a different notion of \(\epsilon \)-covering sets.
- 4.
In [11] a thinning procedure is given only for the directed case, and it is claimed that quasi-\(\epsilon \)-covering sets can be thinned. We believe this is not correct. See the full version. Instead, we give here a thinning procedure for \(\epsilon \)-covering sets.
- 5.
We refer to the vertices of \(\mathcal T\) as nodes to distinguish them from the vertices of G.
- 6.
These connections are only required for the efficient construction.
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Mozes, S., Skop, E.E. (2015). Efficient Vertex-Label Distance Oracles for Planar Graphs. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_9
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