J.L. Bentley. Algorithms for Klee's Rectangle Problems. Technical Report, Computer Science Department, Carnegie-Mellon University, Pittsburgh, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a sequence of examples they demonstrated the validity of their approach. The examples mentioned in [14, Section 2] are: orthog3 onal line segment intersection reporting, the all nearest neighbors problem [21] the 3D maxima problem [17] computing the measure of a set of axis parallel rectangles [9], computing the visibility of a set of line segments from a point [4] batched orthogonal range queries, and reporting pairwise intersections of axis parallel rectangles. We investigate if the distribution sweeping approach can be adapted to the cache oblivious model, and answer this in the ....

J. L. Bentley. Algorithms for Klee's rectangle problems. CarnegieMellon University, Pittsburgh, Penn., Department of Computer Science, unpublished notes, 1977.

....of n intervals, find the length of their union. He gave an O(n log n) time solution and asked if this was optimal. This generated considerable interest in the problem, and shortly after, Fredman and Weide [4] proved that f2(n log n) is a lower bound under the usual model of computation. Bentley [2] considered the natural extension to d dimensional space where we ask for the dimensional measure of a set of rectangles. He showed that the O(n log n) bound holds for d = 2 as well, and for d 2, the result generalizes to an upper bound of O(r d 1 logn) Thus the results axe optimal for d = ....

....Cs THEN INCR(TOTs) Ms: measure of Cs ELSIF box paxtially covers Cs THEN Insert(box,Ism(5) nsert(box, s(5) IF TOTs 0 THEN Ms: measure of Cs ELSE END The routine is called as Insert(box,root) The deletion routine will be similar. Note the similarity with the methods of Bentley[2] and van Leeuwen and Wood[7] for the 1 and 2 dimensional case. It is immediately cleax that the amount of time 4 required depends on the number of nodes visited and the amount of time required for computing the measure at the leaves. In the sequel of this paper we will show that partition trees ....

Bentley, J.L., Algorithms for Klee's rectangle problem, Unpublished notes, Dept. of Computer Science, CMU, 1977.

....shown in [KRV] that 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 ....

....data structure segment tree. The segment tree is an elegant data structure that is used to answer stabbing queries on a collection of intervals. Before we discuss the use of path caching in this context we give a brief description of the segment tree; a more complete treatment can be found in [Ben]. For ease of exposition we will assume that none of the input intervals share any endpoints. To build a segment tree on a set of n intervals we first build a binary search tree T on the 2n endpoints of the intervals. The endpoints e 1 ; e 2 ; Delta Delta Delta ; e k are stored at the leaves ....

J. L. Bentley, "Algorithms for Klee's Rectangle Problems," Dept. of Computer Science, Carnegie Mellon Univ. unpublished notes, 1977.

....the algorithm the empty regions for all of the cells have to be overlayed and a position with the minimum number of empty regions found. This could be done by computing an arrangement of rectillinear polygons, but this would take n ) time and space. Instead, we perform a vertical plane sweep [1, 4] of the kernel from top to bottom. There are two types of events during the sweep, the addition of an interval at it s maximum height and the deletion of an interval at it s minimum height. At each point in time we would like to keep track of a point that is contained in the least number of ....

J. L. Bentley. Algorithms for Klee's rectangle problems. Canegie-Mellon University, Department of Computer Science, unpublished notes, 1977.

....technique [9] to achieve IO e#ciency. The algorithms in the second category requires additional index structures. The algorithm in [20] uses join indices [24] that are actually partially precomputed join result. R # trees [4] are used in the algorithms of [7, 11] 1] extended segment trees [5] for external memory to improve IO performance. 22] used a combination of a space filling curve and filter trees that extends the idea of quad trees (see [21] In [25] we developed interval B trees that combine segment trees and B trees and an algorithm that uses interval B trees ....

J. L. Bentley. Algorithms for Klee's rectangle problems. Technical report, CarnegieMellon University, 1977.

....Tree As attempts to solve EPD optimally using the buffer tree or distribution sweeping fail we are led to other approaches. The solution derived in [GIS] is based on an external version of the segment tree developed in [Buffer] In this section we describe this structure. The segment tree [23, 108] is a well known dynamic data structure used to store a set of segments in one dimension, such that given a query point all segments containing the point can be found efficiently. Such queries are normally called stabbing queries. In internal memory 20 oe 0 oe 1 oe 2 oe 3 oe 4 p m=4 slabs oe i ....

....problems like two dimensional or higher dimensional range searching. The problem of two dimensional range searching both in main and external memory has been the subject of much research, and many elegant data structures like the range tree [24] the priority search tree [89] the segment tree [23], and the interval tree [53, 54] have been proposed for internal memory twodimensional range searching and its special cases. These structures are not efficient when mapped to external memory. However, the practical need for I O support has led to the development of a large number of external data ....

[Article contains additional citation context not shown here]

J. L. Bentley. Algorithms for klee's rectangle problems. Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....more general problems like 2 dimensional or higher dimensional range searching. The problem of 2 dimensional range searching in both main and external memory has been the subject of much research. Many elegant data structures like the range tree [7] the priority search tree [27] the segment tree [6], and the interval tree [14, 15] have been proposed for use in main memory for 2 dimensional range searching and its special cases (see [11] for a detailed survey) Most of these structures are not efficient when mapped to external memory. However, the practical need for I O support has led to the ....

....log function, that is, the number of times one must apply log to get below 2. It should be mentioned that the p range tree can be extended to answer general 2 dimensional queries, and that very recently a static structure for 3 dimensional queries has been developed in [43] The segment tree [6] can also be used to solve the stabbing query problem, but even in internal memory it uses more than linear space. Some attempts have been made to externalizing this structure [8, 35] and they all use O( N=B) log 2 N) blocks of external memory. The best of them [35] is static and answers queries ....

J. L. Bentley. Algorithms for klee's rectangle problems. Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....the algorithm the empty regions for all of the cells have to be overlayed and a position with the minimum number of empty regions found. This could be done by computing an arrangement of rectillinear polygons, but this would take n 2 ) time and space. Instead, we perform a vertical plane sweep [1, 4] of the kernel from top to bottom. There are two types of events during the sweep, the addition of an interval at it s maximum height and the deletion of an interval at it s minimum height. At each point in time we would like to keep track of a point that is contained in the least number of ....

J. L. Bentley. Algorithms for Klee's rectangle problems. Canegie-Mellon University, Department of Computer Science, unpublished notes, 1977.

....with the new rectangle is performed, and are removed after the sweepline has passed over them. Many optimal and suboptimal dynamic data structures for intervals have been proposed; important examples are the interval tree [Ede83] the priority search tree [McC85] and the segment tree [Ben77] 4.2 The Square Root Rule In most implementations of plane sweeping algorithms, the maximum amount of memory ever needed is determined by the maximum number of rectangles that are intersected by a single horizontal line. For most real life input data this number, which we will refer to as ....

J. L. Bentley. Algorithms for Klee's rectangle problems. Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....Segment Tree As attempts to solve EPD optimally using the bu#er tree or distribution sweeping fail we are led to other approaches. The solution derived in [GIS] is based on an external version of the segment tree developed in [Bu#er] In this section we describe this structure. The segment tree [23, 108] is a well known dynamic data structure used to store a set of segments in one dimension, such that given a query point all segments containing the point can be found e#ciently. Such queries are normally called stabbing queries. In internal memory 20 #0 #1 #2 #3 #4 # m 4 slabs # i # m 4 nodes ....

....problems like two dimensional or higher dimensional range searching. The problem of two dimensional range searching both in main and external memory has been the subject of much research, and many elegant data structures like the range tree [24] the priority search tree [89] the segment tree [23], and the interval tree [53, 54] have been proposed for internal memory twodimensional range searching and its special cases. These structures are not e#cient when mapped to external memory. However, the practical need for I O support has led to the development of a large number of external data ....

[Article contains additional citation context not shown here]

J. L. Bentley. Algorithms for klee's rectangle problems. Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....more general problems 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 ....

J. L. Bentley, "Algorithms for Klee's Rectangle Problems," Dept. of Computer Science, Carnegie Mellon Univ. unpublished notes, 1977. 10

....on a given line, see [AV96, Vit98] Since a large number of efficient main memory data structures has been designed for this problem and related issues, the trend has been in the past decade to adapt these techniques to multidimensional access methods. The range tree[Ben80] the segment tree[Ben77], the interval tree [Ede83] and the priority searchtree[Cre85] are elegant main memory structures for 2 dimensional range searching. The above theoretical work is justified by the three fields of application mentionned above: moving objects, network applications and constraint databases. Network ....

J. L. Bentley. Algorithms for Klee's rectangle problems. Technical report, Department of Computer Science, Carnegie Mellon Univ., 1977.

....with the new rectangle is performed, and are removed after the sweepline has passed over them. Many optimal and suboptimal dynamic data structures for intervals have been proposed; important examples are the interval tree [Ede83] the priority search tree [McC85] and the segment tree [Ben77] 4.2 The Square Root Rule In most implementations of plane sweeping algorithms, the maximum amount of memory ever needed is determined by the maximum number of rectangles that are intersected by a single horizontal line. For most real life input data this number, which we will refer to as ....

J. L. Bentley. Algorithms for Klee's rectangle problems. Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....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 ....

J. L. Bentley, "Algorithms for Klee's Rectangle Problems," Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....more general problems like 2 dimensional or higher dimensional range searching. The problem of 2dimensional range searching in both main and external memory has been the subject of much research. Many elegant data structures like the range tree [7] the priority search tree [27] the segment tree [6], and the interval tree [14, 15] have been proposed for use in main memory for 2 dimensional range searching and its special cases (see [11] for a detailed survey) Most of these structures are not efficient when mapped to external memory. However, the practical need for I O support has led to the ....

....log # function, that is, the number of times one must apply log # to get below 2. It should be mentioned that the p range tree can be extended to answer general 2 dimensional queries, and that very recently a static structure for 3 dimensional queries has been developed in [43] The segment tree [6] can also be used to solve the stabbing query problem, but even in internal memory it uses more than linear space. Some attempts have been Space (blocks) Query I O bound Update I O bound Priority search tree [21] O(N B) O(log 2 N T B) XP tree [8] O(N B) O(log B N T ) Metablock tree [23] ....

J. L. Bentley. Algorithms for klee's rectangle problems. Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....and its special cases (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 ....

....(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 as well. There have ....

[Article contains additional citation context not shown here]

J. L. Bentley, "Algorithms for Klee's Rectangle Problems," Dept. of Computer Science, Carnegie Mellon Univ., unpublished notes, 1977.

....Finally, we show some applications of this technique. This work has been supported in part by FONDECYT grants 1940271 and 1950622. 1 Introduction In many practical applications the problem of manipulating segments arises under different forms. Typical examples are computational geometry [14, 11, 6, 3], temporal databases [7, 2] database models with constraints [9, 8] and structured text search [13, 10] Because of this situation, the problem of manipulating a set of segments has been extensively studied [14, 5] All these approaches focus on single operations, in which a single segment is ....

J. Bentley. Algorithms for Klee's rectangle problems. Dept. of Computer Science, CarnegieMellon Univ. Unpublished notes., 1977.

....(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 as well. There have ....

J. L. Bentley, "Algorithms for Klee's Rectangle Problems," Dept. of Computer Science, Carnegie Mellon Univ. unpublished notes, 1977.

No context found.

J.L. Bentley. Algorithms for Klee's Rectangle Problems. Technical Report, Computer Science Department, Carnegie-Mellon University, Pittsburgh, 1977.

No context found.

J. L. Bentley. Algorithms for Klee's rectangle problems. Carnegie-Mellon University, Pittsburgh, Penn., Department of Computer Science, unpublished notes, 1977.

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J.L. Bentley. Algorithms for Klee's Rectangle Problems. Technical Report, Computer Science Department, Carnegie-Mellon University, Pittsburgh, 1977.

No context found.

J.L. Bentley. Algorithms for Klee's Rectangle Problems. Technical Report, Computer Science Department, Carnegie-Mellon University, Pittsburgh, 1977.

No context found.

J. L. Bentley. Algorithms for Klee's rectangle problems. Carnegie-Mellon Univ., Penn., Dept. of Comp. Sci. Unpublished notes, 1977.

No context found.

J.L. Bentley. Algorithms for Klee's Rectangle Problems. Technical Report, Computer Science Department, Carnegie-Mellon University, Pittsburgh, 1977.

No context found.

J.L. Bentley. Algorithms for Klee's rectangle problems. Carnegie-Mellon Univ., Penn., Dept. of Comp. Sci. Unpublished notes, 1977.

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