A general approach to d-dimensional geometric queries

@inproceedings{Yao1985AGA,
  title={A general approach to d-dimensional geometric queries},
  author={Andrew Chi-Chih Yao and F. Frances Yao},
  booktitle={STOC '85},
  year={1985}
}
It is shown that any bounded region in <italic>E</italic><supscrpt><italic>d</italic></supscrpt> can be divided into 2<supscrpt><italic>d</italic></supscrpt> subregions of equal volume in such a way that no hyperplane in <italic>E</italic><supscrpt><italic>d</italic></supscrpt> can intersect all 2<supscrpt><italic>d</italic></supscrpt> of the subregions. This theorem provides the basis of a data structure scheme for organizing <italic>n</italic> points in <italic>d</italic> dimensions. Under… 
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References

SHOWING 1-10 OF 19 REFERENCES
On k-hulls and related problems
TLDR
Efficient computation of the 'cut' guaranteed by the classical 'Ham Sandwich theorem', faster preprocessing time for polygon retrieval, and theoretical improvements to a problem of intersecting lines and points posed by Hopcroft.
A 3-space partition and its applications
TLDR
It is shown that one can always find three planes that divide S into eight open regions, of which no seven together contain more than n points, which gives rise to a data structure, what is called an octant-tree, for representing any point set in 3-space.
On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems
  • A. Yao
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1982
TLDR
By employing a subroutine that solves the post office problem, it is shown that, for fixed k $\geq$ 3, such a minimum spanning tree can be found in time O($n^{2-a(k)} {(log n)}^{1-a (k)}$), where a(k) = $2^{-(k+1)}$.
Lower Bounds on the Complexity of Some Optimal Data Structures
TLDR
A technique is presented for deriving lower bounds on the complexity of optimal data structures which permit insertions and deletions of records, and queries of the form query (Region) where value (r) lies in a commutative semi-group S, and $\Gamma $ denotes a set of regions of the space of possible keys.
Partitioning Point Sets in 4 Dimensions
  • R. Cole
  • Computer Science, Mathematics
    ICALP
  • 1985
TLDR
It is proved there exists a parallel planes partition of any set of n points in 4 dimensions that yields a data structure for the half-space retrieval problem in 3 dimensions that has linear size and achieves a sublinear query time.
Space Searching for Intersecting Objects
A Decision Method For Elementary Algebra And Geometry
By a decision method for a class K of sentence (or other expressions) is meant a method by means of which, given any sentence θ, one can always decide in a finite number of steps whether θ is in K;
Multidimensional Searching Problems
Classic binary search is extended to multidimensional search problems. This extension yields efficient algorithms for a number of tasks such as a secondary searching problem of Knuth, region location
On Construrting minimum spanning trors in kdimensional spares and related problems
  • SIAM J . on Computing
  • 1982
...
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