D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. J. ACM, 32:597--617, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....point p i S, the sum of distances from p i to its k nearest neighbors, namely, the points of S which are contained in R i . In order to compute e#ciently the sums of distances in all the squares R i , we apply the orthogonal range searching algorithm for weighted points of Willard and Lueker [15] which is defined as follows. Given n weighted points in d space and a query d rectangle Q, compute the accumulated weight of the points in Q. The data structure in [15] is of size O(n log n) it can be constructed in time O(n log n) and a range query can be answered in time O(log n) We ....

....of distances in all the squares R i , we apply the orthogonal range searching algorithm for weighted points of Willard and Lueker [15] which is defined as follows. Given n weighted points in d space and a query d rectangle Q, compute the accumulated weight of the points in Q. The data structure in [15] is of size O(n log n) it can be constructed in time O(n log n) and a range query can be answered in time O(log n) We show how to apply their data structure and algorithm to our problem. Let q S be the point for which we want to compute the sum of distances from it to its k nearest ....

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D.E. Willard, G.S. Lueker, "Adding range restriction capability to dynamic data structures", in J. ACM, 32(1985), pp. 597--617. 10

....points of S that are contained in any query interval can be determined in O(log n) resp. O(log n= log log n) time. The dynamic counting problem using the same basic two level tree structure as in the rst solution for the static counting problem. However, T is now a BB( tree [32] and the auxiliary structure, D(t) at each node t of T is a balanced binary search tree where the points are stored at the leaves in left to right order by non decreasing y coordinate. To facilitate the querying, each node v of D(t) stores a count of the number of points in its subtree. Given a ....

....a non leaf, then we query the structures at v using the NE quadrant and the NW quadrant derived from q (i.e. the quadrants with corners at (a; c) and (b; c) respectively) and then combine the answers. Updates on T are performed using the amortized case updating strategy for BB ( trees [32]. The correctness of the method should be clear. The space and query time bounds follow from Lemma 1.2. Since the amortized insertion time of the quadrant searching structure is O(log n) the insertion in the BB( tree takes amortized time O(log n) 32] To solve the problem for general query ....

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D.E. Willard and G.S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32:597-617, 1985.

....prefix sum cube P d1 d2 measure value d1 d2 0 0 3 0 1 7 0 2 2 0 3 3 . 4 2 3 4 3 3 Data set D Figure 1: The original array and prefix sum array A number of highly sophisticated aggregation techniques for sparse data have been proposed for computational geometry applications [8, 7, 23]. However, typically the storage overhead is super linear, e.g. O(N log d Gamma1 N) for a data set of size N , which is infeasible for large multidimensional data sets in data warehousing applications. Also, since the data structures are fairly involved, they are rarely used in practice. ....

D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, 1985.

....is not effective if requests are not accessing sectors in order or if there is a large gap between groups of target sectors. There is an inherent dimensionality curse in computing range aggregates. The best known algorithms whose runtime is provably sub linear in the size of the data set (e.g. [7, 38]) have polylogarithmic query cost and storage overhead. However, for dimensionalities d 9 a polylogarithmic cost is practically worse than a linear cost. Let n denote the number of data points, i.e. tuples in the data warehouse s table. Then for d = 9 the polylogarithmic value log n is only ....

D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, 1985.

....Lue82, Wil85] which requires O(n log n) memory space and O(d n log n) preprocessing time, one can obtain a polylogarithmic query time of O(log n) for the d dimensional range searching problem (see e.g. PS85] p. 85) Dynamization of the range searching problem can be found in [LW82, WL85] In the following, we shall consider the two dimensional version of the problem and present those solutions which have logarithmic query time performance. The question of whether there is an O(n) space, O(log n k) query time algorithm for solving this problem is still open. 2.3.1 Priority ....

D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. J. ACM, 32:597--617, 1985.

....amortized time, and all k colors of intervals containing a query point can be reported in O(log n k) amortized time. 3. 2 Two dimensional structure To obtain a dynamic data structure for the two dimensional colored segment intersection problem, we apply a so called range restriction, see e.g. [25, 27]. Basically, this comes down to maintaining a balanced binary tree on the y coordinates of the segments, and every node stores a data structure as described above. The effect of the performance is a multiplicative factor of log n in the update and query time and the space requirements. Hence, we ....

Willard, D. E., and G. S. Lueker, Adding Range Restriction Capability to Dynamic Data Structures, J. ACM 32 (1985), pp. 597--617. 35

....the equal block method. This technique applies to all data struc tures for decomposable searching problems, as described above. We will also show that, using concatenable data structures, range restrictions can be sAded to query problems. Unlike previous methods for adding range restrictions (see [14, 15]) our method also works for static data structures (and, in fact, makes them dynamic) Some applications of the results we obtain are the following: A concatenable interval tree that supports stabbing queries in O(v log n k) time (where k is the output size) insertions, deletions, splits and ....

....amortized; s; v) O(log v vr(r f( v) CT; T; N) O(log N T( f(V) r(N) o(f(iv) M, r f(r) Proof. From the lemmas 4, 5, 6, 7 and 8. Being able to split and concatenate data structures has an interesting application. In Scholten and Overmars [14] and Willard and Lueker [15], the problem of adding range restrictions to searching problems is considered. Assume we have some query problem on a set S of objects. We add to each object p in S some value vp. The 11 query is extended with two values a and b and we ask to answer the query over only those objects p in S for ....

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Willard, D. E., and G. S. Lueker, Adding range restriction capability to dynamic data structures, J. ACM $ (1985), pp. 597-617. 19

....asks for all points that lie in some range ( b] b] i.e. those points p = pl, p) with a pi b and . and a p, b. Many data structures have been proposed for solving the range searching prob lem. Here we will use the range tree (also called e.g. super B tree) See e.g. [2,3,5,6]. We will briefly describe the structure here. A 1 dimensional range tree is a simple binary tree storing the keys in sorted order in its leaves. To perform searching efficiently we link the leaves in a double linked list. To perform a query with a range [a. b] we search with a in the tree to ....

Willard, D.E., and G.S. Lueker, Adding range restriction capability to dynamic data structures, J. A CM 32 (1985), 597-617. 8

.... k) The storage the above structure uses is O(n log n) and it can be built in O(n log 2 n) time. Both bounds follow immediately from the corresponding bounds of the asso ciated structures stated in Lemma 6. Dynamization of the structure can be accomplished using the techniques de scribed in [13]. In [13] Willard and Lueker describe how the rebuilding that is necessary when the main structure is out of balance can be done little by little during several updates, instead of all in one time. Their techniques lead to an update time of O(U(n)log n) where U(n) denotes the time needed for an ....

....storage the above structure uses is O(n log n) and it can be built in O(n log 2 n) time. Both bounds follow immediately from the corresponding bounds of the asso ciated structures stated in Lemma 6. Dynamization of the structure can be accomplished using the techniques de scribed in [13] In [13], Willard and Lueker describe how the rebuilding that is necessary when the main structure is out of balance can be done little by little during several updates, instead of all in one time. Their techniques lead to an update time of O(U(n)log n) where U(n) denotes the time needed for an update in ....

Willard, D.E., and G.S. Lueker, Adding range restriction capability to dynamic data structures, Journal of the ACM32 (1985), pp. 597-617. 22

.... a point and then reinserting it with a new weight) By relaxing the equal sized partitioning in the recursive definition of T to a balanced partitioning and using a careful rotation scheme, the above structure can be maintained and queried within O(t f log n) worst case time per operation [WL85] Another way to dynamize T is to do some local and global rebuilding after every few operations, which gives an amortized bound of O(t f log per operation [Meh84, Ove83] or to use randomized search trees [AS89] and obtain the same bounds in expectation. 4 2.2 Kinetizing the tree T We now ....

D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. J. ACM, 32:597--617, 1985.

....tree, built over the points in u s subtree, and ordered by the y coordinate. Extension to higher dimensions is done recursively. As described, the range tree over a d dimensional point set, requires space O(n log n) and can answer range queries in time O(log n t) Willard and Lueker [WL85] develop a comprehensive set of techniques for adding range search capability to dynamic data structures. Also, trade o#s between space and time are possible. Their techniques explore the theory of range trees. 2.1.5 Filtering search One of the most elegant techniques in range searching is that ....

D.E. Willard and G.S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, 1985.

....memory it can, for example, be used to convert amortized bounds to worst case bounds (Fixing B to a constant in our result yields an internal memory interval tree with worst case update bounds. It can also be used as a (simpler) alternative to the rather complicated structure developed in [41] in order to add range restriction capabilities to internal memory dynamic data structures. It seems possible to use the techniques in [41] to remove the amortization from the update bound of the Even though the external priority search tree [8] solves a more general problem than the external ....

....internal memory interval tree with worst case update bounds. It can also be used as a (simpler) alternative to the rather complicated structure developed in [41] in order to add range restriction capabilities to internal memory dynamic data structures. It seems possible to use the techniques in [41] to remove the amortization from the update bound of the Even though the external priority search tree [8] solves a more general problem than the external interval tree, it cannot be used as an alternative to the external interval tree in the point location structures. internal interval tree, ....

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D. Willard and G. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597-617, 1985. 21

....sex, weight, salary etc. A typical orthogonal range query is of the form find all males of age between 30 and 40 years with an income between 20,000 and 40,000 . The orthogonal range searching problem has numerous applications and has been studied extensively for the last decades, see e.g. [1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 20, 22, 24, 25, 26, 27, 30, 31, 40, 41, 42, 43, 45, 46, 47]. Willard [43] gives a comprehensive list of references on the subject and gives applications to the theory of databases. For surveys see, e.g. the survey by Agarwal [1] and the books by Mehlhorn [27] and Preparate and Shamos [31] In this paper we consider various orthogonal range searching ....

....reporting in R 3 . Accepting a penalty for each reported point Chazelle [14] gave a data structure with query time O(log 2 n k log log(4n k) and using space O(n log n log log n) or query time O(log 2 n k log # (2n k) and using space O(n log n) Using a method of Willard and Lueker [46], the above bounds can be extended, for any fixed d, to d dimensional range reporting, for d # 4, with a penalty of a factor O(log d 3 n) in space and query time (excluding the term involving k) We show how the above bounds can be extended for any fixed d, to d dimensional range reporting, ....

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D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, July 1985.

....query problem in a growing list of timestamped leaves. 3. 1 Computing the latest occurrence Using balanced binary search trees one can solve the dynamic range maximum minimum problem in time O(log i) per range maximum or minimum query, where there are i entries in the list (see e.g. [16]) However, it was demonstrated [1, 17] that the range minimum problem on a changing list can not be solved in the cell probe model with amortized constant time per query. This seems to bode ill for our time stamped indexing problem. Indeed, for the latest occurrence query we do not know how to ....

D. Willard and G. Lueker. Adding range restriction capability to dynamic data structures. J. ACM, 32:597-617, 1985.

....author was at BRICS. etc. A typical orthogonal range query is of the form find all males of age between 30 and 40 years with an income between 20,000 and 40,000 . The orthogonal range searching problem has numerous applications and has been studied extensively for the last decades, see e.g. [1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 20, 22, 24, 25, 26, 27, 30, 31, 40, 41, 42, 43, 45, 46, 47]. Willard [43] gives a comprehensive list of references on the subject and gives applications to the theory of databases. For surveys see, e.g. the survey by Agarwal [1] and the books by Mehlhorn [27] and Preparate and Shamos [31] In this paper we consider various orthogonal range searching ....

....in R 3 . Accepting a penalty for each reported point Chazelle [14] gave a data structure with query time O(log 2 n k log log(4n=k) and using space O(n log n log log n) or query time O(log 2 n k log (2n=k) and using space O(n log n) Using a method of Willard and Lueker [46], the above bounds can be extended, for any fixed d, to d dimensional range reporting, for d 4, with a penalty of a factor O(log d 3 n) in space and query time (excluding the term involving k) We show how the above bounds can be extended for any fixed d, to d dimensional range reporting, for ....

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D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, July 1985.

....easy way, and thus remove the amortization from the bounds. Actually, it turns out that the weight balanced B tree can be used as a simple alternative to existing complicated ways of removing amortization from internalmemory data structures. For example it can be used in the structure described in [136] for adding range restriction capabilities to dynamic data structures, and by fixing B to a constant in the external interval tree one obtains an internal memory interval tree with worst case update bounds. Even though it is not reported in the literature it seems possible to use the techniques in ....

....for adding range restriction capabilities to dynamic data structures, and by fixing B to a constant in the external interval tree one obtains an internal memory interval tree with worst case update bounds. Even though it is not reported in the literature it seems possible to use the techniques in [136] to remove the amortization from the internal memory interval tree, but our method seems much simpler. The main property of the weight balance B tree that allows relatively easy removal of amortization, is that rebalancing is done by splitting of nodes instead of rotations as in the BB[ff] tree ....

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D. Willard and G. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, 1985.

....for choosing between different implementations of the same data type. In section 4 we describe such a mechanism based on multiple inheritance. It will be available in the next version of LEDA. Augmented trees are abundant in computational geometry. Examples of augmented trees are range trees [3, 21], segment trees [4] interval trees [8] and dynamization of order decomposable searching problems [17] Asymptotically efficient realizations of augmented trees are usually based on weight balanced trees [16] in the case of range and segment trees implementations are available in LEDA. Skip ....

....which a balanced tree scheme underlying an augmented tree must fulfill and in the second part we show that skip lists are a particularly elegant and efficient way to satisfy them. 5.1 Requirements Augmented trees are abundant in computational geometry. Examples of augmented trees are range trees [3, 21], segment trees [4] interval trees [8, 7] and dynamization of order decomposable searching problems [17] An account of these data structures can also be found in sections VII.2.2, VIII.5.1.3. VIII.5.1.1. and VII.1.3 of [11] We will use 2 dimensional range trees as our running example. Let S ....

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Willard, D.E., Lueker, G.S., Adding Range Restriction Capability to Dynamic Data Structures Journal of the ACM, Vol.32, No.3, 1985, 597-617. 13

....external memory as well as internal memory data structures. For example fixing B to a constant in our result yields an internal memory interval tree with worst case update bounds. Our B tree structure can also be used as a (simpler) alternative to the rather complicated structure developed in [44] in order to add range restriction capabilities to internal memory dynamic data structures. It seems possible to use the techniques in [44] to remove the amortization from the update bound of the internal interval tree, but our method is much simpler. Finally, in Section 5 we discuss how to use ....

....interval tree with worst case update bounds. Our B tree structure can also be used as a (simpler) alternative to the rather complicated structure developed in [44] in order to add range restriction capabilities to internal memory dynamic data structures. It seems possible to use the techniques in [44] to remove the amortization from the update bound of the internal interval tree, but our method is much simpler. Finally, in Section 5 we discuss how to use the ideas behind our external interval tree to develop an external version of the segment tree with space bound O( N=B) log B N ) This ....

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D. Willard and G. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597--617, 1985. 24

....rather powerful property of data structures based on decomposition schemes (described in Section 2) is that they can be cascaded together to answer more complex queries, at the increase of a logarithmic factor in their performance. This property has been implicitly used for a long time, see e.g. [108, 173, 176, 256, 225]. The real power of the cascading property was first observed by Dobkin and Edelsbrunner [100] who used this property to answer several complex geometric queries. Since their result, several papers have exploited and extended this property to solve numerous geometric searching problems; see [10, ....

....structure for d dimensional simplex range searching can be constructed by cascading d 1 halfspace range searching structures, since a d simplex is an intersection of at most d 1 halfspaces. Multi level data structures were also proposed for range restriction, introduced by Willard and Lueker [256] and Scholten and Overmars [225] The following theorem states a general result for multi level data structures. Theorem 5.1. Let S; P 1 ; P 2 ; D 1 ; D 2 be as defined above, and let r be a parameter. Suppose the size and query time of each of the data structures are at most S(n) and ....

D. E. Willard and G. S. Lueker, Adding range restriction capability to dynamic data structures, J. ACM, 32 (1985), 597--617.

....) per operation. Their data structure takes space O(n 1 # ) for d = 2, and has higher space bounds in higher dimensions. We note that for rectilinear (L 1 and L # ) metrics, orthogonal range query data structures let us solve the post office problem in time O(log d n) per operation [16, 17, 28]. 3 Ordered Nearest Neighbors Suppose we are given a bichromatic set S of red and blue points. We define a bichromatic ordered nearest neighbor path to be a path produced by the following sequence of operations. We first choose p 1 arbitrarily. Then we successively extend this path by one more ....

D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. J. Assoc. Comput. Mach., 32:597--617, 1985. 14

....range searching problem (with arbitrary coordinates) to one in which all the coordinates have value in the range [1: n] This reduction only adds a factor of d log n to the query time. Therefore, using the p range tree and the d dimensional range searching structure of Willard and Leuker [11] we get the following bounds for answering general d dimensional queries. Theorem 4.3 Given a set of n points in d dimensional space (d 3) there exists a main memory data structure to store the points using O(n log d Gamma1 n) storage such that orthogonal range queries can be performed in ....

D. E. Willard and G. S. Leuker, "Adding Range Restriction Capability to Dynamic Data Structures," Journal of the ACM 32 (1985), 597--617.

....in O(log n) time, for any positive constant ffl. At this moment, however, no dynamic data structures are known for this problem. In this paper, we first show that for the L1 metric, the nearest neighbor problem can be solved efficiently. To be more precise, we show that the range tree (see [12, 14, 15, 25]) can be used to solve this problem. As a result, we solve the L1 neighbor problem with a query time of O( log n) D Gamma1 log log n) and an amortized update time of O( log n) D Gamma1 log log n) using O(n(log n) D Gamma1 ) space. For the static version of this problem, the query time is ....

....reduces the size of a dynamic closest pair structure. Applying this transformation repeatedly to the structure of Section 6 gives the new results mentioned in Table 1. Some concluding remarks are given in Section 8. 2. Range trees. In this section, we recall the definition of a range tree. See [12, 13, 15, 25]. The coordinates of a point p in IR D are denoted by p i , 1 i D. Moreover, we denote by p 0 the point (p 2 ; pD ) in IR D Gamma1 . If S is a set of points in IR D , then we define S 0 : fp 0 : p 2 Sg. We note that S 0 is to be considered as a multiset, i.e. elements may ....

D. E. Willard and G. S. Lueker, Adding range restriction capability to dynamic data structures, J. ACM, 32 (1985), pp. 597--617.

....point p i 2 S, the sum of distances from p i to its k nearest neighbors, namely, the points of S which are contained in R i . In order to compute efficiently the sums of distances in all the squares R i , we apply the orthogonal range searching algorithm for weighted points of Willard and Lueker [15] which is defined as follows. Given n weighted points in d space and a query d rectangle Q, compute the accumulated weight of the points in Q. The data structure in [15] is of size O(n log d Gamma1 n) it can be constructed in time O(n log d Gamma1 n) and a range query can be answered in time ....

....of distances in all the squares R i , we apply the orthogonal range searching algorithm for weighted points of Willard and Lueker [15] which is defined as follows. Given n weighted points in d space and a query d rectangle Q, compute the accumulated weight of the points in Q. The data structure in [15] is of size O(n log d Gamma1 n) it can be constructed in time O(n log d Gamma1 n) and a range query can be answered in time O(log d n) We show how to apply their data structure and algorithm to our problem. Let q 2 S be the point for which we want to compute the sum of distances from it ....

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D.E. Willard, G.S. Lueker, "Adding range restriction capability to dynamic data structures", in J. ACM, 32(1985), pp. 597--617.

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D. E. Willard and G. S. Lueker. Adding range restriction capability to dynamic data structures. J. ACM, 32:597--617, 1985.

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