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25
The quadtree and related hierarchical data structures
 ACM Computing Surveys
, 1984
"... A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics ..."
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Cited by 421 (11 self)
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A tutorial survey is presented of the quadtree and related hierarchical data structures. They are based on the principle of recursive decomposition. The emphasis is on the representation of data used in applications in image processing, computer graphics, geographic information systems, and robotics. There is a greater emphasis on region data (i.e., twodimensional shapes) and to a lesser extent on point, curvilinear, and threedimensional data. A number of operations in which such data structures find use are examined in greater detail.
Balanced Aspect Ratio Trees: Combining the Advantages of kd Trees and Octrees
"... Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded as ..."
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Cited by 55 (8 self)
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Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded aspect ratio). While previous hierarchical data structures, such as kd trees, quadtrees, octrees, fairsplit trees, and balanced box decompositions can guarantee some of these properties, we know of no previous data structure that combines alI of these properties simultaneously. The BAR tree data structure has numerous applications ranging from solving several geometric searching problems in fixed dimensional space to aiding in the visualization of graphs and threedimensional worlds.
Efficient Nearest Neighbor Searching for Motion Planning
, 2002
"... We present and implement an efficient algorithm for performing nearestneighbor queries in topological spaces that usually arise in the context of motion planning. Our approach extends the Kd treebased ANN algorithm, which was developed by Arya and Mount for Euclidean spaces. We argue the correctne ..."
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Cited by 32 (5 self)
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We present and implement an efficient algorithm for performing nearestneighbor queries in topological spaces that usually arise in the context of motion planning. Our approach extends the Kd treebased ANN algorithm, which was developed by Arya and Mount for Euclidean spaces. We argue the correctness of the algorithm and illustrate its efficiency through computed examples. We have applied the algorithm to both probabilistic roadmaps (PRMs) and Rapidlyexploring Random Trees (RRTs). Substantial performance improvements are shown for motion planning examples.
Optimal Dynamic Range Searching in Nonreplicating Index Structures
 In Proc. International Conference on Database Theory, LNCS 1540
, 1997
"... We consider the problem of dynamic range searching in tree structures that do not replicate data. We propose a new dynamic structure, called the Otree, that achieves a query time complexity of O(n (d\Gamma1)=d ) on n ddimensional points and an amortized insertion/deletion time complexity of O(l ..."
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Cited by 26 (2 self)
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We consider the problem of dynamic range searching in tree structures that do not replicate data. We propose a new dynamic structure, called the Otree, that achieves a query time complexity of O(n (d\Gamma1)=d ) on n ddimensional points and an amortized insertion/deletion time complexity of O(log n). We show that this structure is optimal when data is not replicated. In addition to optimal query and insertion/deletion times, the Otree also supports exact match queries in worstcase logarithmic time. 1 Introduction Given a set S of ddimensional points, a range query q is specified by d 1dimensional intervals [q s i ; q e i ], one for each dimension i, and retrieves all points p = (p 1 ; p 2 ; : : : p d ) in S such that h8i 2 f1; : : : ; dg : q s i p i q e i i. This type of searching in multidimensional space has important applications in geographic information systems, image databases, and computer graphics. Several structures such as the range trees [3], Prange trees [29...
Efficient CrossTrees for External Memory
, 1998
"... . We describe efficient methods for organizing and maintaining large multidimensional data sets in external memory. This is particular important as access to external memory is currently several order of magnitudes slower than access to main memory, and current technology advances are likely to make ..."
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Cited by 21 (1 self)
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. We describe efficient methods for organizing and maintaining large multidimensional data sets in external memory. This is particular important as access to external memory is currently several order of magnitudes slower than access to main memory, and current technology advances are likely to make this gap even wider. We focus particularly on multidimensional data sets which must be kept simultaneously sorted under several total orderings: these orderings may be defined by the user, and may also be changed dynamically by the user throughout the lifetime of the data structures, according to the application at hand. Besides standard insertions and deletions of data, our proposed solution can perform efficiently split and concatenate operations on the whole data sets according to any ordering. This allows the user: (1) to dynamically rearrange any ordering of a segment of data, in a time that is faster than recomputing the new ordering from scratch; (2) to efficiently answer queries rel...
Improving Partial Rebuilding by Using Simple Balance Criteria
"... Some new classes of balanced trees, defined by very simple balance criteria, are introduced. Those trees can be maintained by partial rebuilding at lower update cost than previously used weightbalanced trees. The used balance criteria also allow us to maintain a balanced tree without any balance in ..."
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Cited by 21 (4 self)
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Some new classes of balanced trees, defined by very simple balance criteria, are introduced. Those trees can be maintained by partial rebuilding at lower update cost than previously used weightbalanced trees. The used balance criteria also allow us to maintain a balanced tree without any balance information stored in the nodes.
General balanced trees
 Journal of Algorithms
, 1999
"... We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the ..."
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Cited by 19 (0 self)
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We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the nodes. Thus, in the case when amortized bounds are sufficient, there is no need for sophisticated balance criteria. The maintenance algorithms use partial rebuilding. This is important for certain applications and has previously been used with weightbalanced trees. We show that the amortized cost incurred by general balanced trees is lower than what has been shown for weightbalanced trees. � 1999 Academic Press 1.
Discovering interesting holes in data
 In Proceedings of IJCAI
, 1997
"... Current machine learning and discovery techniques focus on discovering rules or regularities that exist in data. An important aspect of the research that has been ignored in the past is the learning or discovering of interesting holes in the database. If we view each case in the database as a point ..."
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Cited by 15 (2 self)
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Current machine learning and discovery techniques focus on discovering rules or regularities that exist in data. An important aspect of the research that has been ignored in the past is the learning or discovering of interesting holes in the database. If we view each case in the database as a point in a itdimensional space, then a hole is simply a region in the space that contains no data point. Clearly, not every hole is interesting. Some holes are obvious because it is known that certain value combinations are not possible. Some holes exist because there are insufficient cases in the database. However, in some situations, empty regions do carry important information. For instance, they could warn us about some missing value combinations that are either not known before or are unexpected. Knowing these missing value combinations may lead to significant discoveries. In this paper, we propose an algorithm to discover holes in databases. 1
Randomized KDimensional Binary Search Trees
, 1998
"... This paper introduces randomized Kdimensional binary search trees (randomized Kdtrees), a variant of Kdimensional binary trees. This data structure allows the efficient maintenance of multidimensional records for any sequence of insertions and deletions; and thus, is fully dynamic. We show that ..."
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Cited by 12 (3 self)
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This paper introduces randomized Kdimensional binary search trees (randomized Kdtrees), a variant of Kdimensional binary trees. This data structure allows the efficient maintenance of multidimensional records for any sequence of insertions and deletions; and thus, is fully dynamic. We show that several types of associative queries are efficiently supported by randomized Kdtrees. In particular, a randomized Kdtree with n records answers exact match queries in expected O(log n) time. Partial match queries are answered in expected O(n 1\Gammaf (s=K) ) time, when s out of K attributes are specified, with 0 ! f(s=K) ! 1 a real valued function of s=K). Nearest neighbor queries are answered online in expected O(log n) time. Our randomized algorithms guarantee that their expected time bounds hold irrespective of the order and number of insertions and deletions. Keywords: Randomized Algorithms, Multidimensional Data Structures, Kdtrees, Associative Queries, Multidimensional Diction...
Efficient Splitting and Merging Algorithms for Order Decomposable Problems
, 1997
"... Let S be a set whose items are sorted with respect to d ? 1 total orders OE 1 ; : : : ; OE d , and which is subject to dynamic operations, such as insertions of a single item, deletions of a single item, split and concatenate operations performed according to any chosen order OE i (1 i d). This g ..."
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Cited by 11 (2 self)
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Let S be a set whose items are sorted with respect to d ? 1 total orders OE 1 ; : : : ; OE d , and which is subject to dynamic operations, such as insertions of a single item, deletions of a single item, split and concatenate operations performed according to any chosen order OE i (1 i d). This generalizes to dimension d ? 1 the notion of concatenable data structures, such as the 23trees, which support splits and concatenates under a single total order. The main contribution of this paper is a general and novel technique for solving order decomposable problems on S, which yields new and efficient concatenable data structures for dimension d ? 1. By using our technique we maintain S with the following time bounds: O(log n) for the insertion or the deletion of a single item, where n is the number of items currently in S; O(n 1\Gamma1=d ) for splits and concatenates along any order, and for rectangular range queries. The space required is O(n). We provide several applications of ...