Computational geometry: an introduction
Computational geometry: an introduction
An optimal-time algorithm for slope selection
SIAM Journal on Computing
Monitoring continuous band-join queries over dynamic data
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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We study space/time tradeoffs for querying some combinatorial structures. In the first, given an arrangement of n lines in general position in the plane, a query for a real number t asks about Rank(t), the number of vertices of the arrangement with x-coordinates ≤ t. We show that for K = O(n/logn), after a preprocessing step that uses space S = O(n2/(KlogK)) the query can be answered in time O(nlog K). The second query involves the Cartesian sum vectors a = (a1,...,an) and b = (b1,...,bn). For a given real t, it asks about Rank(t), the number of sums ai + bj which are ≤ t. We show that for some positive constant c and K ≤ c(log n)/(log logn), after a preprocessing step that uses space S = O(n2/K2), the query may be answered in time O((n/K)log K). Both results fit neatly between two obvious extremes.