E. McCreight, "Priority search trees," SIAM J. Comput., vol. 14, pp. 257--276, May 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a balanced binary search tree, T c , in which the c colored points of S are stored in increasing x order. We maintain the colors themselves in a balanced search tree CT , and store with each color c in CT a pointer to T c . We also store the points of S in a balanced priority search tree (PST ) [28]. Recall that a PST on m points occupies O(m) space, supports insertions and deletions in O(log m) time, and can be used to report the k points lying inside a grounded query rectangle in O(log m k) time [28] Although this query is designed for query ranges of the form [l; r] 1; l] it can be ....

....to T c . We also store the points of S in a balanced priority search tree (PST ) 28] Recall that a PST on m points occupies O(m) space, supports insertions and deletions in O(log m) time, and can be used to report the k points lying inside a grounded query rectangle in O(log m k) time [28]. Although this query is designed for query ranges of the form [l; r] 1; l] it can be trivially modi ed to ignore the points on the upper edge of the range without a ecting its performance. Clearly, the space used by the entire data structure is O(n) where n = jSj. To answer a query q = ....

E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985.

....consider the case without any FDLs. We build a data structure, DSMin SV , to better organize all the void intervals. Our data structure is constructed by augmenting a balanced binary search tree (e.g. red black tree [10] which is can be viewed as a dynamic version of the priority search tree [11] supporting more restricted queries, and supports three major operations: burst query, interval insertion, and interval deletion. The data burst query takes a burst b as input, and outputs the Min SV interval, if such an interval exists. The interval insertion adds a new interval into the data ....

E. McCreight, "Priority search trees," SIAM J. Computing, vol. 14, No.2, pp. 257--276, 1985.

....concise way, compared to the original. It should be noted that one drawback of this technique is that it cannot be adapted for counting problems and its dynamization seems rather difficult to achieve. Another basic data structure is the priority search tree of Edelsbrunner and McCreight [Ede87, McC85] which will be maintained dynamically in the planar partition algorithm of chapter 5. For completeness, some proofs of these existing results are also presented. They are either new, or have been much simplified or adapted to be consistent with our terminology. 2.1 Basic Definitions 2.1.1 ....

....a variation of this problem, when q is a half range, i.e. one edge of q lies on a fixed line parallel to one of the axes, say the x axis. For example, let q be [q:x 1 ; q:x 2 ] Theta [c; q:y] where c is a fixed constant. For this special case, there is a classical data structure of McCreight [McC85] which solves this problem optimally with O(n) space, O(log n k) query time and O(n log n) preprocessing time, where k is the number of points enclosed in q. This data structure is called the priority search tree (PST) It is essentially a balanced search tree used in combination with a heap ....

E. M. McCreight. Priority search trees. SIAM J. Comput., 14:257--276, 1985.

....is: z C F: zi Yi for all 1 i D and zj yj for at least one 1 j D (8) It would be possible to use kd trees [26, 25] or range trees [27, 25] but these are both suited to querying F for elements which lie in bounded (hyper ) rectangles. Priority trees, developed by McCreight [28], are suited to rectangular queries in which the rectangle is unbounded on a single side. Sun and Steuer [29] describe an alternative data structure adapted for answering queries about domination and non domination; which has been extended recently by Mostaghim et al. 30] The dominates ....

....two dimensions. The points and the tree illustrated in figure 4 axe more akin to the general (D 2) case. Dominated trees also have applications to general sets (that is, sets whose elements axe not mutually non dominating) such as answering queries about enclosing rectangles; see, for example, [28, 31]. 5.3.1 Construction Construction of a dominated tree from M points F M = Ym m=l is described in Algorithm 2 and proceeds as follows. The first composite point Cl is constructed by finding the individual Ym with maximum first coordinate; this value forms the first coordinate of the composite ....

E.M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985.

.... attributes,# F ) S 4 = device, 600dpi,# F ) T 2 = printer, 1200dpi,# F ) T 3 = scanner, 1200dpi,# F ) T 1 = printer, 600dpi,# F ) T 4 = scanner, 600dpi,# F ) 2,2] 3,3] 0,0] printer [1,1] scanner [0,1] device [2,3] attributes [0,#) 4,#) address [4,4] A1 [5,5] A2 [x,x] A Figure 3. Excerpts of the graph (#,match # ) above (a) and (F,match F ) below (b) in a service brokering scenario. Note, that the definition of a mapping to intervals is trivial, if the graph is a tree as in (b) However, Figure 3(b) suggests that it is no problem to map fields to ....

....However, there are some problems when implementing this strategy. The program units that compute # # (T) need to know about the servers tuple domains. Furthermore, the computation has to verify for every q, whether the intersection of I # (# F ) T) and # q is empty. If priority search trees [5] are used, # # (T) takes O( log p) in computation complexity. s 1 1 0 12345 1 2 3 4 Figure 5. Distribution strategies based on tuple domains for five servers s 1 , s 5 . The tuples and # F are the same as in Figure 4. However, the distribution strategy based on tuple ....

McCreight, E. M.: Priority Search Trees, SIAM J. Computing 14, Pages 257-276 (1985)

....of the segment trees) as associated structure. The conditions of Lemma 2 can namely also be stated as (py, LDOo(p) y) E ] HDOo(q) y: qv] x [q: oo] Hence we could use a structure for range queries as associated structure and, because the second range is halfinfinite, a priority search tree ([6]) suffices. This not only reduces the update time of the associated structure to O(log n) but it also uses only linear space. This would result in a structure with a query time of O(log 2 n) and an update time (for P only) of O(log n) using O(nlogn) storage. 3.2 A dynamic structure In the ....

....can only be stored on their search paths. Thus we have some sort of priority queue on the coordinates of the points, implemented as a heap, with the restriction that a point can only be stored at nodes where it is maximal. Thus the structure is related to the priority search tree of McCreight[6]. Now given two vertical lines l: x = Ix and v: x = vx, the points in P ,r that are maximal with respect to Ot,r can be determined as follows: Consider the leaves 7 and 7t where the search paths to l and v end. Clearly the points with x coordinate between l and v must be stored in the leaves ....

McCreight, E.M., Priority search trees, SIAM J. Computing 14 (1985), pp.257- 276.

....shortest paths algorithm and an efficient implementation of the first fit heuristics for the bin packing problem. Second, we describe a simple implementation technique for the abstract data type. The standard implementation of priority search queues, McCreight s priority search trees [14], combines binary search trees and heaps. Unfortunately, balanced search trees and heaps do not go well together. Rotations that are typically used to maintain balance destroy the heap property and restoring the property takes (h) time where h is the height of the tree. Consequently, in order to ....

....the heap structured tree of Figure 2. This transformation usually involves additional matches. In our example, Erik has to play with Mary to determine the second best player of the first half of the tournament. Pursuing this idea further leads to a data structure known as a priority search tree [14]. We will come back to this data structure in Sections and 8. An alternative possibility, which we will investigate in this section, is to label each internal node with the loser of the match, instead of the winner, and to drop the external nodes altogether. If we additionally place the ....

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E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257 276, May 1985.

....trees is available from http: www. dcs. ex. ac. uk academic s fevers on moea (10) 15 It would be possible to use kd trees [17, 16] or range trees [18, 16] but these are both suited to querying F for elements which lie in bounded (hyper ) rectangles. Priority trees, developed by McCreight [19], are suited to rectangular queries in which the rectangle is unbounded on a single side. As noted in section 2, the dominates relation imposes a partial order on individuals. However, since the elements of F are mutually non dominating, this relation cannot be used directly to construct, for ....

....Y2. With more than two objectives this is no longer the case and the points illustrated in figure 6 are more akin to the general case. The dominated tree also has applications to general sets (that is, not non dominated sets) such as answering queries about enclosing rectangles; see, for example, [19, 20]. 16 Further details regarding the construction and update of dominated trees can be found in Appendix A. 6 EXPERIMENTS In order to evaluate the efficiency of E SPEA, results of a comparison with SPEA are presented in this section. Deb has proposed a number of test objective functions for ....

E.M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985. 45

....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 memory. These include [BlGa, BlGb, ....

E. M. McCreight, "Priority Search Trees," SIAM Journal of Computing 14(2) (1985), 257-- 276.

....is no y order requirement for interior bitings. We can select speci vertex in current front for new biting. If we select the vertex with the lgest y vue, then we can use the priority search tree to report all points in a three sided rectgle ( the top side of the rectgle is open ) As showed in [8], the priority sech tree c be built in O(n log n) time, the report time is O(log n k) d the space requirement is O(n) More importantly, the priority search tree c be updated in O(logn) time per deletion of a vertex d insertion of a vertex. Notice that there are only a constt intersections to ....

Edward M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-270, 1985.

....similar results for multidimensional arrays and set valued attributes. Leveraging our results on indexability, we solve the long standing open problem of optimal three sided range search in external memory. Our access method, the EPS tree, is an adaptation of McCreight s priority search tree [McC85] to external memory. The EPS tree achieves optimal search performance. We also develop update techniques with asymptotically optimal amortized and worst case costs. We perform the first thorough experimental evaluation of an asymptotically optimal access method for multi dimensional range ....

....in fully externalizing the interval tree, with optimal query I O, space, and worst case dynamic update cost. 2.1. 3 Priority search tree Virtually all planar orthogonal range search problems, can be solved optimally in main memory, by techniques based on the priority search tree of McCreight [McC85] We will present this data structure in some detail, in Ch. 6. This data structure solves optimally the dynamic case of three sided queries: organize a set of planar points, so that the points contained in a region of the form [x 1 , x 2 ] #, y] can be retrieved e#ciently. Because of its ....

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E.M. McCreight. Priority search trees. SIAM Journal of Computing, 14(2):257--276, 1985.

....blocks of external memory. Here T denotes the number of points reported. The structure supports insertions only in O(log B N (log B N) B) I Os amortized. A simpler static structure with the same bounds was described by Ramaswamy [36] In internal memory, the priority search tree of McCreight [31] can be used to answer more general queries than diagonal corner queries, namely 3 sided range queries, and a number of attempts have been made at externalizing this structure [15, 27, 38] The structure by Icking et al. 27] uses optimal space but answers queries in O(log 2 N T=B) I Os. The ....

E. McCreight. Priority search trees. SIAM Journal of Computing, 14(2):257-276, 1985.

....and partial sums: Lower bounds for range searching problems in the plane are known for structured or algebraic models [13, 22, 39, 42] We extend these to the stronger cell probe model. priority search trees: We show that Willard s RAM improvement [40] of McCreight s classic data structure [29] is optimal. union nd problems: Gabow and Tarjan [26] showed that a version of the union nd problem (static tree union nd) is solvable in constant amortised time per operation, provably easier than the general problem. We show that with respect to single operation worst case bounds, the ....

....it holds in the cell probe model, so it makes minimal assumptions on the computational model and none on the data structure. The upper bound for this problem is slightly larger, amortised O(log n log log n) using fractional cascading [31] yet for dominance queries it is O(log n= log log n) using [29, 40], matching our lower bound. The traditional, algebraic models used for proving lower bounds for range queries do not provide bounds for the emptiness problem, since free answers to the emptiness query are built into the model [13, p. 44] Recently, a handful of papers have established the bound ....

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J.C. McCreight. Priority search trees. SIAM J. Comput., 14:256-276, 1985.

....worst case performance when objects are crowded we make use of a well known line sweep technique for computing rectangle intersections. Objects and vision areas are approximated by bounding squares. Square intersections are detected in a left to right sweep by maintaining two priority search trees [4] which contain the current set of vertical intervals of sight and object squares. Whenever a new square appears in the sweep its vertical projection is checked for intersections with the other type of squares and then added to the respective set of intervals. At exit of a square its vertical ....

E.M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257-276, May 1985.

....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 research issues: i) How can ....

.... 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 performance. The closest work to ....

E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257-276, May 1985.

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E. McCreight, "Priority search trees," SIAM J. Comput., vol. 14, pp. 257--276, May 1985.

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Edward M. McCreight. Priority search trees. Tech Report CSL-81-5, Xerox PARC, 1981.

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E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257--276, 1985. 13

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E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257--276, May 1985.

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E. M. McCreight, Priority search trees, SIAM J. Comput. 14 (1985), pp. 257-276.

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E. M. McCreight, Priority search trees, SIAM J. Comput. 14 (1985), pp. 257-276.

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E. M. McCreight. Priority search trees. Siam J. on Computing, 14:257-276, 1985.

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Edward M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257-276, 1985.

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E. M. McCreight, Priority search trees, SIAM J. Comput., 14 (1985), pp. 257--276.

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McCreight, E.M., Priority search trees, SIAM J. Computing 14 (1985), 257- 276.

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