Abstract
Let G be a graph where each vertex is associated with a label. A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex v and a label \(\lambda \), returns a \((1+\varepsilon )\)-approximation of the distance from v to the closest vertex with label \(\lambda \) in G. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements. No such oracles were previously known.
For a full version of this paper, see https://arxiv.org/abs/1707.02414. This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 794/13).
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Notes
- 1.
We assume that a single comparison or addition of two numbers takes constant time.
- 2.
The discussion of \(\alpha \)-layered graphs in Sect. 2 refers to directed graphs, and hence also applies to undirected graphs.
- 3.
We assume that the endpoints of the intervals are vertices on Q, since otherwise once can add artificial vertices on Q without asymptotically changing the size of the graph.
- 4.
Formally, one needs to show that Lemma 1 holds for vertex-labeled oracles as well. See our full paper at https://arxiv.org/abs/1707.02414.
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We thank Paweł Gawrychowski and Oren Weimann for fruitful discussions.
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Laish, I., Mozes, S. (2018). Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs. In: Solis-Oba, R., Fleischer, R. (eds) Approximation and Online Algorithms. WAOA 2017. Lecture Notes in Computer Science(), vol 10787. Springer, Cham. https://doi.org/10.1007/978-3-319-89441-6_20
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