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Succinct Indices for Path Minimum, with Applications

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Abstract

In the path minimum problem, we preprocess a tree on n weighted nodes, such that given an arbitrary path, the node with the smallest weight along this path can be located. We design novel succinct indices for this problem under the indexing model, for which weights of nodes are read-only and can be accessed with ranks of nodes in the preorder traversal sequence of the input tree. We present

  • an index within O(m) bits of additional space that supports queries in \(O(\varvec{\alpha }(m, n))\) time and \(O(\varvec{\alpha }(m, n))\) accesses to the weights of nodes, for any integer \(m \ge n\); and

  • an index within \(2n + o(n)\) bits of additional space that supports queries in \(O(\varvec{\alpha }(n))\) time and \(O(\varvec{\alpha }(n))\) accesses to the weights of nodes.

Here \(\varvec{\alpha }(m, n)\) is the inverse-Ackermann function, and \(\varvec{\alpha }(n) = \varvec{\alpha }(n, n)\). These indices give us the first succinct data structures for the path minimum problem. Following the same approach, we also develop succinct data structures for semigroup path sum queries, for which a query asks for the sum of weights along a given query path. One of our data structures requires \(n\lg \sigma + 2n + o(n\lg \sigma )\) bits of space and \(O(\varvec{\alpha }(n))\) query time, where \(\sigma \) is the size of the semigroup. In the path reporting problem, queries ask for the nodes along a query path whose weights are within a two-sided query range. Using the succinct indices for path minimum queries, we achieve three different time/space tradeoffs for path reporting by designing

  • an O(n)-word data structure with \(O(\lg ^\epsilon n + occ \cdot \lg ^\epsilon n)\) query time;

  • an \(O(n\lg \lg n)\)-word data structure with \(O(\lg \lg n + occ \cdot \lg \lg n)\) query time; and

  • an \(O(n \lg ^\epsilon n)\)-word data structure with \(O(\lg \lg n + occ)\) query time.

Here occ is the number of nodes reported and \(\epsilon \) is an arbitrary constant between 0 and 1. These tradeoffs match the state of the art of two-dimensional orthogonal range reporting queries (Chan et al. 2011), which can be treated as a special case of path reporting queries. When the number of distinct weights is much smaller than n, we further improve both the query time and the space cost of these three results.

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Notes

  1. The function \((\lg ^{*})^{*}\) is the number of times \(\lg ^{*}\) must be iteratively applied before the result becomes less than or equal to 1. See Nivasch’s discussions [45] for more details.

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Authors and Affiliations

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

    Timothy M. Chan, J. Ian Munro & Gelin Zhou

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

    Meng He

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  1. Timothy M. Chan
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Correspondence to Gelin Zhou.

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The partial preliminary version of this article was published in Proceedings of the 22th Annual European Symposium on Algorithms (ESA 2014) [12]. This work was supported by NSERC and the Canada Research Chairs Program. Part of the first author’s work was done during his visit to the Department of Computer Science and Engineering, Hong Kong University of Science and Technology.

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Chan, T.M., He, M., Munro, J.I. et al. Succinct Indices for Path Minimum, with Applications. Algorithmica 78, 453–491 (2017). https://doi.org/10.1007/s00453-016-0170-7

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