H. Edelsbrunner, "A new Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the key component of dynamic interval management is answering stabbing queries. Given a set of input intervals, to answer a stabbing query for a point q we have to report all intervals that intersect q. Elegant solutions exist for this problem in main memory. The segment tree [Ben] interval tree [Edea, Edeb], and the priority search tree [McC] can all solve this problem well. Of these, the priority search tree solves a slightly more general problem (3 sided queries) with optimal query and update times and uses optimal storage. Many algorithms have been presented to solve this problem in secondary ....

H. Edelsbrunner, "A new Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....(distance queries) or nding pairs of objects that intersect or are within a certain distance from each other (join queries) In this paper we concentrate on intersection queries. Typical spatial data structures, supporting these queries for in memory and disk based data, include interval trees [Ede83a, Ede83b], priority search trees [McC85] quadtree based structures, and the R tree and its variants [Gut84, BKSS90] Surveys of the various spatial data structures presented in the literature can be found in [GG98] and [Sam89] In recent years, a signi cant amount of research has been done to address two ....

....good tree arrangements. 8 Related Work R trees are the basis of our work as they are suited for both point and rectangular data, and are suitable for dynamic contexts (where insertions and deletions are interleaved with intersection queries) Other spatial data structures such as interval trees [Ede83a, Ede83b] and priority search tree [McC85] are specialized for intersection queries, and have better asymptotic behaviour. But because of the more complex index structures they are more dicult to update, and hence not as useful in a dynamic context. They are also more dicult to tune for good cache ....

H. Edelsbrunner. A new approach to rectangle intersections: Part I. International Journal of Computer Mathematics, 13(3-4):209-219, 1983.

....(distance queries) or finding pairs of objects that intersect or are within a certain distance from each other (join queries) In this paper we concentrate on intersection queries. Typical spatial data structures, supporting these queries for in memory and diskbased data, include interval trees [3, 4], priority search trees [11] quadtree based structures, and the R tree and its variants [9, 1] Surveys of the various spatial data structures presented in the literature can be found in [6] and [16] In recent years, a significant amount of research has been done to address two research issues: ....

....good tree arrangements. 8 Related Work R trees are the basis of our work as they are suited for both point and rectangular data, and are suitable for dynamic contexts (where insertions and deletions are interleaved with intersection queries) Other spatial data structures such as interval trees [3, 4] and priority search tree [11] are specialized for intersection queries, and have better asymptotic behaviour. But because of the more complex index structures they are more difficult to update, and hence not as useful in a dynamic context. They are also more difficult to tune for good cache ....

H. Edelsbrunner. A new approach to rectangle intersections: Part I. International Journal of Computer Mathematics, 13(3--4):209--219, 1983.

....like two and higher dimensional range searching. The problem of 2 dimensional range searching in both main memory and secondary memory has been the subject of much research. Many elegant data structures like the range tree [3] priority search tree [22] segment tree [2] and interval tree [12,13] have been proposed for use in main memory for 2 dimensional range searching and its special cases (see [7] for a detailed survey) Most of these algorithms are not efficient when mapped to secondary storage. However, the practical need for good I O support has led to the development of a large ....

H. Edelsbrunner, "A new Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....intersection is the problem where a data structure is built on an input set of intervals and then queried (or updated) in an on line fashion. That is, the results of a query have to be returned before the next query is processed. This problem has been extensively studied both in main memory [5,7,8,17] and secondary storage [4,14,16,19,25] In particular, 4] recently resolved the open problem of whether it is possible to build a dynamic, worst case optimal data structure for this problem. However, directly applying the on line intersection algorithm to this problem results in a running time of ....

H. Edelsbrunner, "A New Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....(see [9] for a detailed survey) Most of these algorithms cannot easily be mapped efficiently to secondary storage. We present a general technique called path caching that can be used to map many main memory data structures like the priority search tree [27] segment tree [3] and interval tree [12,13] efficiently to secondary storage. These data structures are relevant and important to the indexing problems we are considering because of the following reasons: 1) Dynamic interval management can be solved efficiently in main memory by all the three data structures mentioned above; and (2) ....

....extensively in the literature (see [9] As mentioned before, the best in core bounds have been achieved using the priority search tree of [27] yielding O(n) space, dynamic query time O(log 2 n t) and update time O(log 2 n) which are all optimal. Other data structures like the Interval Tree [12,13], and Segment Tree [3] can also solve the interval management problem optimally in core, with respect to the query time. Among these, the priority search tree does the best because it solves the interval management problem in optimal time and space, and provides an optimal worst case update time ....

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H. Edelsbrunner, "A New Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....extensively in the literature (see [7] As mentioned before, the best in core bounds have been achieved using the priority search tree of [24] yielding O(n) space, dynamic query time O(log 2 n t) and update time O(log 2 n) which are all optimal. Other data structures like the Interval Tree [10,11], and Segment Tree [2] can also solve the interval management problem optimally in core, with respect to the query time. Among these, the priority search tree does the best because it solves the interval management problem in optimal time and space, and provides an optimal worst case update time ....

H. Edelsbrunner, "A new Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....example, a(i; n) Theta(log n) for i = 1 and a(i; n) Theta(log n) for i = 2. This method is based on a new union copy data structure for sets, which generalizes the well known union find structure. 5. 3 Interval Tree Here we present the interval tree, a data structure due to Edelsbrunner [55], which is called a 1 fold rectangle tree in the original paper. Let X be a set of N points on a line, and S a set of n segments with endpoints in X . An interval tree T for X and S is a data structure that supports the following operations: ffl report all the segments of S that contain a query ....

....of r (case (b) and orthogonal segment intersection queries (cases (c) and (d) r r r r r r r r (a) b) c) d) Figure 7: Four mutually exclusive cases for rectangle intersection. The d dimensional orthogonal rectangle intersection queries can be computed using a d fold rectangle tree [54,55], which is based on the interval tree. The d fold rectangle tree uses O(n log d Gamma1 n) space and supports queries in time O(log 2d Gamma1 n k) Off line updates, in which the elements to be inserted and deleted belong to a fixed set of size n, are supported in O(log d n) time. 5.5 Ray ....

H. Edelsbrunner, "A New Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....the key component of dynamic interval management is answering stabbing queries. Given a set of input intervals, to answer a stabbing query for a point q we have to report all intervals that intersect q. Elegant solutions exist for this problem in main memory. The segment tree [Ben] interval tree [Edea, Edeb], and the priority search tree [McC] can all solve this problem well. Of these, the priority search tree solves a slightly more general problem (3 sided queries) with optimal query and update times and uses optimal storage. Many algorithms have been presented to solve this problem in secondary ....

H. Edelsbrunner, "A new Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....extensively in the literature (see [8] As mentioned before, the best in core bounds have been achieved using the priority search tree of [25] yielding O(n) space, dynamic query time O(log 2 n t) and update time O(log 2 n) which are all optimal. Other data structures like the Interval Tree [11,12], and Segment Tree [3] can also solve the interval management problem optimally in core, with respect to the query time. Among these, the priority search tree does the best because it solves the interval management problem in optimal time and space, and provides an optimal worst case update time ....

H. Edelsbrunner, "A New Approach to Rectangle Intersections, Part I," Int. J. Computer Mathematics 13 (1983), 209--219.

....boundaries indicated by parentheses. Open intervals have one boundary at positive or negative infinity, and points have both boundaries equal. Examples of intervals are [17,19) 12,12] inf,22] Another data structure that can be used to process stabbing queries is the interval tree [3, 4]. Unfortunately, as with the segment tree, all the intervals must be known in advance to construct an interval tree. A data structure that can index intervals dynamically is the R tree [6] R trees are a multi dimensional extension of B trees in which each tree node contains a set of possibly ....

H. Edelsbrunner. A new approach to rectangle intersections: Part I. International Journal of Computer Mathematics, 13(3-4):209--219, 1983.

....query. A query takes O(logn) time. The segment tree works well in a static environment, but is not adequate when it is necessary to dynamically add and delete intervals in the tree while processing queries. Another data structure that can be used to process stabbing queries is the interval tree [Ede83a, Ede83b]. Unfortunately, as with the segment tree, all the intervals must be known in advance to construct an interval tree. A data structure that can index intervals dynamically is the R tree [Gut84] R trees are a multidimensional extension of B trees in which each tree node contains a set of possibly ....

H. Edelsbrunner. A new approach to rectangle intersections: Part I. International Journal of Computer Mathematics, 13(3-4):209--219, 1983.

....query takes O(log n) time. The segment tree works well in a static environment, but is not adequate when it is necessary to dynamically add and delete intervals, as it is in an active database predicate index. Another data structure that can be used to process stabbing queries is the interval tree [8, 9]. Unfortunately, as with the segment tree, all the intervals must be known in advance to construct an interval tree. R trees can index intervals dynamically [15] Subtrees of each R tree index node contain only data that lies within a containing rectangle in the index node. Since rectangles in ....

....discussed later, the boolean operators and and or are conditional. In other words, in evaluation of p and q, if p is false, then q is never evaluated. Similarly, in evaluation of p or q, if p is true, then q is never evaluated. Comments are indicated by a as in C . a. 2,17] b. 17,20] c. [8,12] d. 7,7] e. inf,17) Example intervals: 20 b a a d c c e N U L L H e a d e r inf 2 7 7 8 12 17 a, e b c e a, e e a e Figure 4: Example of an interval skip list for intervals shown. procedure findIntervals(K,L,S) x : L.header; S : OE Step down to bottom level. i : maxLevel while i 0 and ....

H. Edelsbrunner. A new approach to rectangle intersections: Part I. International Journal of Computer Mathematics, 13(3-4):209--219, 1983.

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