Abstract
We show how to efficiently maintain a minimum piercing set for a set S of intervals on the line, under insertions and deletions to/from S. A linear-size dynamic data structure is presented, which enables us to compute a new minimum piercing set following an insertion or deletion in time O(c(S) log ∣S∣), where c(S) is the size of the new minimum piercing set. We also show how to maintain a piercing set for S of size at most (1+ε)c(S), for 0 < ε≤ 1, in Ō(log∣S∣/ε) amortized time per update. We then apply these results to obtain efficient (sometimes improved) solutions to the following three problems: (i) the shooter location problem, (ii) computing a minimum piercing set for arcs on a circle, and (iii) dynamically maintaining a box cover for a d-dimensional point set.
Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and by an Intel research grant.
Supported by the Pacific Institute for Mathematical Studies and by the NSERC research grant.
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Katz, M.J., Nielsen, F., Segal, M. (2000). Maintenance of a Piercing Set for Intervals with Applications. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, SH. (eds) Algorithms and Computation. ISAAC 2000. Lecture Notes in Computer Science, vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40996-3_47
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DOI: https://doi.org/10.1007/3-540-40996-3_47
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