Abstract
We present a data structure to maintain a set of intervals on the real line subject to fast insertions and deletions of the intervals, stabbing queries, and local updates. Intuitively, a local update replaces an interval by another one of roughly the same size and location. We investigate whether local updates can be implemented faster than a deletion followed by an insertion.
We present the first results for this problem for sets of possibly overlapping intervals. If the maximum depth of the overlap (a.k.a. ply) is bounded by a constant, our data structure performs insertions, deletions and stabbing queries in time \(O(\log {n})\), and local updates in time \(O(\log n/ \log \log n)\), where n is the number of intervals. We also analyze the dependence on the ply when it is not constant. Our results are adaptive: the times depend on the current ply at the time of each operation.
E. Khramtcova was partially supported by F.R.S.-FNRS, and by the SNF grant P2TIP2-168563 under the Early PostDoc Mobility program.
M. Löffler was partially supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.021.123 and 614.001.504.
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If an update of S violates this bounding box condition, B can easily be enlarged. Thus our assumption does not restrict the setting, but rather simplifies the description.
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Such leaves are not necessarily adjacent to C, as the adjacent ones might be empty.
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The pre-order traversal of a binary tree first visits the root, then it recursively visits the left subtree, and finally it recursively visits the right subtree.
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The post-order traversal of a binary tree first recursively visits the left subtree, then it recursively visits the right subtree, and finally it visits the root.
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References
Agarwal, P.K., Arge, L., Kaplan, H., Molad, E., Tarjan, R.E., Yi, K.: An optimal dynamic data structure for stabbing-semigroup queries. SIAM J. Comput. 41(1), 104–127 (2012)
Alstrup, S., Husfeldt, T., Rauhe, T.: Marked ancestor problems. In: 39th Annual Symposium on Foundations of Computer Science, pp. 534–543 (1998)
Arge, L., Vitter, J.S.: Optimal dynamic interval management in external memory. In: 37th Conference on Foundations of Computer Science, pp. 560–569 (1996)
Berg, M., Cheong, O., Kreveld, M., Overmars, M.: Computational Geometry - Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
Edelsbrunner, H.: Dynamic data structures for orthogonal intersection queries. Report F59. Technische Universität Graz (1980)
Kaplan, H., Molad, E., Tarjan, R.: Dynamic rectangular intersection with priorities. In: 35th ACM Symposium on Theory of Computing (STOC), pp. 639–648 (2003)
Löffler, M., Simons, J.A., Strash, D.: Dynamic planar point location with sub-logarithmic local updates. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 499–511. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40104-6_43
McCreight, E.M.: Efficient algorithms for enumerating intersecting intervals and rectangles. report csl-80-9. Technical report, Xerox Palo Alto Res. Center (1980)
McCreight, E.M.: Priority search trees. SIAM J. Comput. 14(2), 257–276 (1985)
Nekrich, Y.: Data structures with local update operations. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 138–147. Springer, Heidelberg (2008). doi:10.1007/978-3-540-69903-3_14
Nekrich, Y.: A dynamic stabbing-max data structure with sub-logarithmic query time. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 170–179. Springer, Heidelberg (2011). doi:10.1007/978-3-642-25591-5_19
Thorup, M.: Space efficient dynamic stabbing with fast queries. In: 35th ACM Symposium on Theory of Computing (STOC), pp. 649–658. ACM Press (2003)
Acknowledgements
We wish to thank Irina Kostitsyna for helpful discussions.
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Khramtcova, E., Löffler, M. (2017). Dynamic Stabbing Queries with Sub-logarithmic Local Updates for Overlapping Intervals. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_17
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