## K-D Trees Are Better when Cut on the Longest Side (2000)

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Venue: | IN LNCS 1879, ESA 2000 |

Citations: | 11 - 2 self |

### BibTeX

@INPROCEEDINGS{Dickerson00k-dtrees,

author = {Matthew Dickerson and Christian A. Duncan and Michael T. Goodrich},

title = {K-D Trees Are Better when Cut on the Longest Side},

booktitle = {IN LNCS 1879, ESA 2000},

year = {2000},

pages = {179--190},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

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### Abstract

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 IR d, 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(log d−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.

### Citations

1211 |
Multidimensional binary search trees used for associative searching
- Bentley
- 1975
(Show Context)
Citation Context ...tion). Notice that i + j ≤ d. More importantly, for any region in L belonging to class C(i, j) weknowi ≥ 1 (by the definition of L). Also, note that the root of T is in the class C(d, 0).s(1) (1) (2) =-=(3)-=- K-D Trees Are Better when Cut on the Longest Side 185 (2) (1) (2) (a) (b) (c) (d) (e) Fig. 1. The various possibilities for the class C in the plane. The hypercube H is shown in outline while a regio... |

818 | An optimal algorithm for approximate nearest neighbor searching in fixed dimensions
- Arya, Mount, et al.
- 1998
(Show Context)
Citation Context ...eve logarithmic query time for approximate nearest-neighbor searching, although the trees they define do not necessarily have O(log n) depth. The balanced box-decomposition (BBD) trees of Arya et al. =-=[2,1]-=-, on the other hand, have O(log n) depth and have regions with good aspect ratio, and achieve logarithmic-time performance for approximate nearest-neighbor and range searching. The BBD tree deviates f... |

638 | An algorithm for finding best matches in logarithmic expected time
- Friedman, Bentley, et al.
- 1977
(Show Context)
Citation Context ... to determine the cut directions. In the original paper by Bentley [3], the heuristic was to simply alternate between the d possible directions in a round-robin fashion. Friedman, Bentley, and Finkel =-=[12]-=- study an alternate definition, where the cut is defined perpendicular to the direction with maximum spread, thatis,the direction where the difference between coordinate values in that direction in S ... |

591 | Fibonacci heaps and their uses in improved network optimization algorithms - Fredman, Tarjan |

249 | A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields
- Callahan, Kosaraju
- 1995
(Show Context)
Citation Context ...e strict k-d tree approach, there have been several data structures developed for efficiently performing approximate nearest-neighbor searching and range search queries. For example, fair-split trees =-=[5]-=-, defined by Callahan and Kosaraju, achieve logarithmic query time for approximate nearest-neighbor searching, although the trees they define do not necessarily have O(log n) depth. The balanced box-d... |

133 | A functional approach to data structures and its use in multidimensional searching
- Chazelle
- 1988
(Show Context)
Citation Context ...tersect the square without being at a corner, and this intersection must contain a point of S This result, although surprising, is still not better than the range searching data structure of Chazelle =-=[6]-=-, which performs exact queries in the plane in O(log n) timeandO(n) space regardless of the aspect ratio of the query. It is simply interesting to note that such exact queries are achievable using a w... |

96 |
K-d trees for semidynamic point sets
- Bentley
- 1990
(Show Context)
Citation Context ...icular to the longest side of the bounding box B. This heuristic is often used in practice, since it is so simple to implement and tends to mimic the behavior of the maximum-spread heuristic. Bentley =-=[4]-=- reports on experimental results for k-d trees defined using the maximum-spread heuristic, showing that for a variety of input distributions this variation performs remarkably well for nearest-neighbo... |

86 | Approximate range searching
- Arya, Mount
(Show Context)
Citation Context ...eve logarithmic query time for approximate nearest-neighbor searching, although the trees they define do not necessarily have O(log n) depth. The balanced box-decomposition (BBD) trees of Arya et al. =-=[2,1]-=-, on the other hand, have O(log n) depth and have regions with good aspect ratio, and achieve logarithmic-time performance for approximate nearest-neighbor and range searching. The BBD tree deviates f... |

82 | Relaxed heaps: An alternative to Fibonacci heaps with applications to parellel computation - Driscoll, Gabow, et al. |

55 | Balanced aspect ratio trees: Combining the advantages of k-d trees and octrees - Duncan, Goodrich, et al. |

37 |
Worst-case analysis for region and partial region searches in multidimensional binary search trees and balanced quad trees
- Lee, Wong
- 1977
(Show Context)
Citation Context ...owever, or even approximate nearest-neighbor searching. Still, for range searching queries, where one wishes to count (or report) the points inside a given axis-parallel hyper-rectangle, Lee and Wong =-=[13]-=- show that the standard k-d tree structure can be used to achieve a worst-case query time of Θ(n (d−1)/d ) (plus output size in the reporting case). Silva-Filho [16] shows that this bound even holds i... |

31 |
Dynamic MultiDimensional Data structures Based on Quad- and k-D Trees
- Overmars, Leeuwen
- 1982
(Show Context)
Citation Context ...inly not a theoretical improvement on the BBD tree structure [2,1] or the BAR tree structure [9,10], this result helps explain why longest-side k-d trees perform well in practice. Several researchers =-=[4,15,19]-=- have studied the dynamic behavior of k-d trees, where items can be inserted and/or removed. A natural open question, then, is whether one can show if a natural dynamic variant of the k-d tree fits in... |

20 | Balanced aspect ratio trees and their use for drawing very large graphs
- DUNCAN, GOODRICH, et al.
- 1998
(Show Context)
Citation Context ...neighbor and range searching. The BBD tree deviates from the k-d tree approach by introducing holes into the middle of regions. The balanced-aspect ratio (BAR) trees of Duncan, Goodrich, and Kobourov =-=[9,10]-=- achieve similar bounds to those of Arya et al., but do so using simple hyperplane cuts at each internal node. However, their BSP trees are not strictly k-d trees as the cuts need not be axis-aligned ... |

14 |
Divided k-d trees
- Kreveld, Overmars
- 1991
(Show Context)
Citation Context ...inly not a theoretical improvement on the BBD tree structure [2,1] or the BAR tree structure [9,10], this result helps explain why longest-side k-d trees perform well in practice. Several researchers =-=[4,15,19]-=- have studied the dynamic behavior of k-d trees, where items can be inserted and/or removed. A natural open question, then, is whether one can show if a natural dynamic variant of the k-d tree fits in... |

13 |
Balanced Aspect Ratio Trees
- Duncan
- 1999
(Show Context)
Citation Context ...ds of fair-split trees [5], BBD trees [2,1], and BARs182 Matthew Dickerson et al. trees [9,10], they are nonetheless intriguing for a number of reasons. For example, many empirical results (e.g., see =-=[4,8,12]-=-) show that k-d trees defined with the maximum-spread and longest-side splitting rules perform well in practice, but no worst-case theoretical evidence has been given to support these observations. Ou... |

10 | Its okay to be skinny, if your friends are fat
- Maneewongvatana, Mount
- 1999
(Show Context)
Citation Context ...r it turns out occasional skinny objects are acceptable as long as there are not too many of them and their neighbors are not skinny. Indeed, a recent paper on this topic by Maneewongvatana and Mount =-=[14]-=- can be viewed as turning the Bombeck quote around by stating, “It’s okay to be skinny, if your friends are fat.” ⋆ This research partially supported by NSF grant CCR-9732300 and ARO grant DAAH04-96-1... |

9 |
Refinements to Nearest-Neighbor Searching in
- Sproull
- 1991
(Show Context)
Citation Context ...chieve O(log n) expected query times for approximate nearest-neighbor and range searching. Silva-Filho [17] studies the choosing of a cutting hyperplane based on probabilistic considerations. Sproull =-=[18]-=-considers several other heuristics for k-d trees in practice. One standard simplification is to use the following splitting rule: The Longest Side Rule: Choose a splitting hyperplane perpendicular to ... |

4 |
Filho. Optimal choice of discriminators in a balanced k-d binary search tree
- Silva
- 1981
(Show Context)
Citation Context ...ow that for data distributions with bounded density, k-d trees defined using this heuristic can achieve O(log n) expected query times for approximate nearest-neighbor and range searching. Silva-Filho =-=[17]-=- studies the choosing of a cutting hyperplane based on probabilistic considerations. Sproull [18]considers several other heuristics for k-d trees in practice. One standard simplification is to use the... |

3 |
Average case analysis of region search in balanced k-d trees
- Filho
- 1979
(Show Context)
Citation Context ...llel hyper-rectangle, Lee and Wong [13] show that the standard k-d tree structure can be used to achieve a worst-case query time of Θ(n (d−1)/d ) (plus output size in the reporting case). Silva-Filho =-=[16]-=- shows that this bound even holds in the average case for range searching in the standard k-d tree structure. Deviating from the strict k-d tree approach, there have been several data structures devel... |