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)
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Lower bounds for off-line range searching
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Handbook of discrete and computational geometry
Algorithms for three-dimensional dominance searching in linear space
Information Processing Letters
New data structures for orthogonal range searching
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Orthogonal Range Reporting in Three and Higher Dimensions
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
New results on two-dimensional orthogonal range aggregation in external memory
Proceedings of the thirtieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Orthogonal range searching on the RAM, revisited
Proceedings of the twenty-seventh annual symposium on Computational geometry
Persistent predecessor search and orthogonal point location on the word RAM
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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
Succinct indices for range queries with applications to orthogonal range maxima
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
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Orthogonal range reporting is the problem of storing a set of n points in d-dimensional space, such that the k points in an axis-orthogonal query box can be reported efficiently. While the 2-d version of the problem was completely characterized in the pointer machine model more than two decades ago, this is not the case in higher dimensions. In this paper we provide a space optimal pointer machine data structure for 3-d orthogonal range reporting that answers queries in O(log n + k) time. Thus we settle the complexity of the problem in 3-d. We use this result to obtain improved structures in higher dimensions, namely structures with a log n/ log log n factor increase in space and query time per dimension. Thus for d e 3 we obtain a structure that both uses optimal O(n(log n/ log log n)d--1) space and answers queries in the best known query bound O(log n(log n/ log log n)d--3 + k). Furthermore, we show that any data structure for the d-dimensional orthogonal range reporting problem in the pointer machine model of computation that uses S(n) space must spend Ω((log n/ log(S(n)/n))⌊d/2⌋--1) time to answer queries. Thus, if S(n)/n is poly-logarithmic, then the query time is at least Ω((log n/ log log n)⌊d/2⌋--1). This is the first known non-trivial higher dimensional orthogonal range reporting query lower bound and it has two important implications. First, it shows that the query bound increases with dimension. Second, in combination with our upper bounds it shows that the optimal query bound increases from Θ(log n + k) to Ω((log n/ log log n)2 + k) somewhere between three and six dimensions. Finally, we show that our techniques also lead to improved structures for point location in rectilinear subdivisions, that is, the problem of storing a set of n disjoint d-dimensional axis-orthogonal rectangles, such that the rectangle containing a query point q can be found efficiently.