Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
Functional approach to data structures and its use in multidimensional searching
SIAM Journal on Computing
Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation
Communications of the ACM
K-d trees for semidynamic point sets
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Proceedings of the eleventh annual symposium on Computational geometry
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
An optimal algorithm for approximate nearest neighbor searching
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
An Algorithm for Finding Best Matches in Logarithmic Expected Time
ACM Transactions on Mathematical Software (TOMS)
Multidimensional binary search trees used for associative searching
Communications of the ACM
Balanced Aspect Ratio Trees and Their Use for Drawing Very Large Graphs
GD '98 Proceedings of the 6th International Symposium on Graph Drawing
Balanced aspect ratio trees
Kinetic kd-Trees and Longest-Side kd-Trees
SIAM Journal on Computing
Balanced aspect ratio trees revisited
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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We show that a popular variant of the well known k-d tree data structure satisfies an important packing lemma. This variant is a binary spatial partitioning tree T defined on a set of n points in Rd, for fixed d ≥ 1, using the simple rule of splitting each node's hyper-rectangular region with a hyperplane that cuts the longest side. An interesting consequence of the packing lemma is that standard algorithms for performing approximate nearest-neighbor searching or range searching queries visit at most O(logd-1 n) nodes of such a tree T in the worst case. Traditionally, many variants of k-d trees have been empirically shown to exhibit polylogarithmic performance, and under certain restrictions in the data distribution some theoretical expected case results have been proven. This result, however, is the first one proving a worst-case polylogarithmic time bound for approximate geometric queries using the simple k-d tree data structure.