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Dynamic Path Queries in Linear Space

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Abstract

In the path reporting problem, we preprocess a tree on n nodes each of which is assigned a weight, such that given an arbitrary path and a weight range, we can report the nodes whose weights are within the range. We consider this problem in dynamic settings, and propose the first non-trivial linear-space solution that supports path reporting in \(O((\lg n{/}\lg \lg n)^2 + occ \lg n{/}\lg \lg n)\) time, where occ is the output size, and the insertion and deletion of a node of an arbitrary degree in \(O(\lg ^{2+\epsilon } n)\) amortized time, for any constant \(\epsilon \in (0, 1)\). Obvious solutions based on directly dynamizing solutions to the static version of this problem all require \(\Omega ((\lg n{/}\lg \lg n)^2)\) time for each node reported, and thus our query time is much faster. We also design data structures that support path counting and path reporting queries in \(O((\lg n{/}\lg \lg n)^2)\) time, and insertions and deletions in \(O((\lg n{/}\lg \lg n)^2)\) amortized time. This matches the best known results for dynamic two-dimensional range counting (He and Munro in Comput Geom 47(2):268–281, 2014) and range selection (He et al., in: Proceedings of the 22nd international symposium on algorithms and computation, ISAAC, Yokohama, Japan, 2011), which can be viewed as special cases of path counting and path selection.

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Notes

  1. We use \(\lg \) to denote the base-2 logarithm.

  2. A generalized version of this operation was referred to as the method-lookup problem in previous work [28], where the authors considered static trees and multi-labeled nodes. Incidentally, their approach is similar to tree extraction but requires more space.

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Authors and Affiliations

  1. Faculty of Computer Science, Dalhousie University, Halifax, Canada

    Meng He

  2. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    J. Ian Munro & Gelin Zhou

Authors
  1. Meng He
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  2. J. Ian Munro
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  3. Gelin Zhou
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Correspondence to Gelin Zhou.

Additional information

The preliminary partial version of this article was published in the Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC 2014) [23]. This work was supported by NSERC and the Canada Research Chairs Program.

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He, M., Munro, J.I. & Zhou, G. Dynamic Path Queries in Linear Space. Algorithmica 80, 3728–3765 (2018). https://doi.org/10.1007/s00453-018-0413-x

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