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, 2001

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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy.

### An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries ∗

"... Let S be a set of n intervals in R, and let (S,+) be any commutative semigroup. We assign a weight ω(s) ∈ S to each interval in S. For a point x ∈ R, let S(x) ⊆ S be the set of intervals that contain x. Given a point q ∈ R, the stabbing-semigroup query asks for ω(s). We propose a linear-size dynam ..."

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Let S be a set of n intervals in R, and let (S,+) be any commutative semigroup. We assign a weight ω(s) ∈ S to each interval in S. For a point x ∈ R, let S(x) ⊆ S be the set of intervals that contain x. Given a point q ∈ R, the stabbing-semigroup query asks for ω(s). We propose a linear-size dynamic data structure, under the pointermachine model, that answers queries in worst-caseO(logn) time, and supports both insertions and deletions of intervals in amortized O(logn) time. It is the first data structure that attains the optimal O(logn) bound for all three operations. Furthermore, our structure can easily be adapted to external memory, where we obtain a linear-size structure that answers queries and supports updates inO(logB n) I/Os, where B is the disk block size. For the restricted case of nested family of intervals (every pair of intervals are either disjoint or one contains the other), we present a simpler solution based on dynamic trees. computing ∑ s∈S(q) 1