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Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs

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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 λ, returns a (1 + ε)-approximation of the distance from v to the closest vertex with label λ 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.

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Notes

  1. We assume that a single comparison or addition of two numbers takes constant time.

  2. The lemma applies to any graph (not necessarily planar). The size of a graph H is defined as |H| = |V (H)| + |E(H)|. For planar graphs |H| = O(|V (H)|).

  3. The discussion of α-layered graphs in Section 2 refers to directed graphs, and hence also applies to undirected graphs.

  4. We assume that the endpoints of the intervals are vertices on Q, since otherwise one can add artificial vertices on Q without asymptotically changing the size of the graph.

  5. Note that if one is only interested in reporting distances, and is not interested in being able to also trace back the shortest path, then the doubly-linked list can be replaced with a counter for the number of vertices represented by each ID in Lr(q, λ).

  6. Formally, one needs to show that Lemma 1 holds for vertex-labeled oracles as well. We refer the reader to the detailed proof due to Mozes and Skop [16, Section 5.1].

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Acknowledgements

We thank Paweł Gawrychowski and Oren Weimann for fruitful discussions.

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  1. Efi Arazi School of Computer Science, The Interdisciplinary Center Herzliya, Herzliya, Israel

    Itay Laish & Shay Mozes

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  1. Itay Laish
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Correspondence to Shay Mozes.

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An extended abstract of this work was presented in the 15th Workshop on Approximation and Online Algorithms (WAOA 2017), held in Vienna, Austria, September 2017.

This article is part of the Topical Collection on Special Issue on Approximation and Online Algorithms (2017)

This research was supported by the ISRAEL SCIENCE FOUNDATION (grant Nos. 794/13, 592/17).

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Laish, I., Mozes, S. Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs. Theory Comput Syst 63, 1849–1874 (2019). https://doi.org/10.1007/s00224-019-09949-5

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