Filtering search: a new approach to query answering
SIAM Journal on Computing
Lower bounds for orthogonal range searching: I. The reporting case
Journal of the ACM (JACM)
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Discrete & Computational Geometry
Simplex range reporting on a pointer machine
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Space-Time Tradeoffs for Emptiness Queries
SIAM Journal on Computing
Efficient searching with linear constraints
Journal of Computer and System Sciences - Special issue on the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on principles of database systems
On a model of indexability and its bounds for range queries
Journal of the ACM (JACM)
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Lower bounds for intersection searching and fractional cascading in higher dimension
Journal of Computer and System Sciences - STOC 2001
Orthogonal Range Reporting in Three and Higher Dimensions
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the twenty-sixth annual symposium on Computational geometry
Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model
Proceedings of the twenty-eighth annual symposium on Computational geometry
Improved range searching lower bounds
Proceedings of the twenty-eighth annual symposium on Computational geometry
Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model
Proceedings of the twenty-eighth annual symposium on Computational geometry
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We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems. Simplex range reporting is the problem of storing n points in a data structure such that the $k$ points that lie inside a query simplex can be outputted efficiently. This is one of the fundamental and extensively studied problems in computational geometry. Currently, the best data structures for the problem achieve Q(n) + O(k) query time using ~O( (n / Q(n))d) space in which the ~O(.) notation either hides a polylogarithmic or an nε factor for any constant ε 0, (depending on the data structure and Q(n)). The best lower bound on this problem is due to Chazelle and Rosenberg who proved a space lower bound of Ω(nd-ε-dγ) for pointer machine data structures that can answer queries in O(nγ + k) time. For data structures with Q(n) + O(k) query time, we improve the space lower bound to Ω( (n/Q(n))d / 2O(√log Q(n))). Not only this reduces the overhead from polynomial to sub-polynomial, it also offers a smooth trade-off curve. For instance, for polylogarithmic values of Q(n), our lower bound is within a o(log n) factor of the conjectured trade-off curve. By a simple geometric transformation, we also improve the best lower bounds for the halfspace range reporting problem. Furthermore, we also study the external memory model and offer a new framework for proving lower bound in this model. For the first time we show that answering simplex range reporting queries with Q(n) + k/B I/Os requires Ω(B (n/(BQ(n)))d / 2O(√log Q(n)) space in which B is the block size.