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I/O-Efficient Path Traversal in Succinct Planar Graphs

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Abstract

We present a technique for representing bounded-degree planar graphs in a succinct fashion while permitting I/O-efficient traversal of paths. Using our representation, a graph with N vertices, (In this paper \(\lg {N}\) denotes \(\log _2{N}\)) each with an associated key of \(q= \mathrm {O}\left( \lg N\right) \) bits, can be stored in \(Nq+ \mathrm {O}\left( N\right) + \mathrm {o}\left( Nq\right) \) bits and traversing a path of length K takes \(\mathrm {O}\left( K / \lg B\right) \) I/Os, where B denotes the disk block size. By applying our construction to the dual of a terrain represented as a triangular irregular network, we can represent the terrain in the above space bounds and support path traversals on the terrain using \(\mathrm {O}\left( K / \lg B\right) \) I/Os, where K is the number of triangles visited by the path. This is useful for answering a number of queries on the terrain, such as reporting terrain profiles, trickle paths, and connected components.

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Notes

  1. Note that \(\log \left( {\begin{array}{c}N\\ R\end{array}}\right) + \mathrm {O}\left( N \log \log N / \log N\right) = \mathrm {o}\left( N\right) \) as long as \(R = \mathrm {o}\left( N\right) \).

  2. This ordering, as with the ordering used for assigning graph labels, is employed strictly for labeling purposes. It does not imply any arrangement of vertices within the structures used to represent the graph.

  3. \(\lg r_2\) bits would suffice; however it simplifies the analysis to use \(\lg N\) bits.

  4. We omit some minor details that are not relevant to triangles.

  5. All edges on the convex hull of \({\mathscr {T}}\) are considered wall edges.

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Acknowledgments

The authors wish to thank the School of Computer Science, Carleton University and the Natural Sciences and Engineering Research Council of Canada for financial assistance. The work was done while the second author was at the School of Computer Science, Carleton University, Canada.

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Authors and Affiliations

  1. School of Computer Science, Carleton University, 5302 HP, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Craig Dillabaugh & Anil Maheshwari

  2. Faculty of Computer Science, Dalhousie University, 6050 University Avenue, PO BOX 15000, Halifax, NS, B3H 4R2, Canada

    Meng He

  3. Faculty of Computer Science, Dalhousie University, 6050 University Ave, Halifax, NS, B3H 1W5, Canada

    Norbert Zeh

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  1. Craig Dillabaugh
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Correspondence to Craig Dillabaugh.

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Dillabaugh, C., He, M., Maheshwari, A. et al. I/O-Efficient Path Traversal in Succinct Planar Graphs. Algorithmica 77, 714–755 (2017). https://doi.org/10.1007/s00453-015-0086-7

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