J. L. Bentley and J. B. Saxe. Decomposable Searching Problems I: Static-to-Dynamic Transformations. Journal of Algorithms, 1(4):301--358, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... problems discussed here for instance, can the generalized orthogonal range searching problem in , for d 4, be solved with O(polylog(n) i) query time and O(n(log n) O(1) n) space; iii) developing general dynamization techniques for generalized problems, along the lines of, for instance, [5] for standard problems; iv) developing ecient solutions to generalized problems where the objects may be in time dependent motion; and (v) implementing and testing experimentally some of the solutions presented here. ....

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformations. Journal of Algorithms, 1:301-358, 1980.

.... main memory data structures: since it is easier to design data structures for static sets with good query time, there have been a number of efforts aimed at devising general methods for transforming static or semi dynamic data structures into dynamic ones for decomposable searching problems [9, 10, 11]. A searching problem is said to be decomposable if one can partition the input set into a set of disjoint subsets, perform the query on each subset independently and then easily compose the partial answers. The most useful geometric searching problems, like range searching, nearestneighbor ....

....Section 4 defines LR trees and presents the algorithms for queries and updates. Section 5 gives the performance study. Finally, Section 6 concludes our work. 2. A DYNAMIZATION TECHNIQUE FOR DECOMPOSABLE SEARCHING PROBLEMS In this section we will use the following notion of decomposability [9, 11]. DEFINITION 1. A searching problem P(q, S) on a set S with query q is called decomposable if and only if for any partition V = A,B ,S= A B, A =# of S and any query q , P(q, S) #(P(q, A) P(q, B) for some operator # computable in O(1) time. It follows promptly that, in order to ....

Bentley, J. L. and Saxe, J. B. (1980) Decomposable searching problems I: static-to-dynamic transformations. J. Algorithms, 1, 301--358.

....v. l v is visible if and only if its heightislessthany v . The query time is O( n) where n is the current size of S, since the number of ancestors visited all together is only O( n) Insertion. Assume wewant to insert a new segment s into T . We insert s into the appropriate O( n) nodes of T( [1]) In each of these nodes v,we insert s into the tree T v . Wenow need to update the boolean variables and the heights that are stored in the ancestors of v. This can be done quite easily bywalking along the path from v to the root. Consider the parent w of v. If c w is already true, then it ....

....the height of s, otherwise, y w does not change. We now move to the parent of w, etc. until we reach the root. The total time spent is n) where n is the current size of S. Deletion. Assume we need to delete the segment s from T . We delete it from each of the O( n) nodes in whichitisstored( [1]) Let v be one of these nodes. We first delete s from T v . Now, we need to update the variables stored with the nodes along the path from v to the root. Let w be the parentofv. If c w is false, then it of course remains false. Otherwise, if after removing s from v, v remains without any ....

J. Bentley and J. Saxe, Decomposable searching problems I: Static-to-dynamic transformation, J. Algorithms 1(198 301--358

....realized long ago [10] and while there have been many fundamental results in the area, our current knowledge is still limited. Dynamic data structures for all kinds of problems reducible to range searching [1] including linear convex programming, are known. A class of decomposable query problems [5] has been recognized as easy, for which simple general tricks are known. A useful technique has been devised to deal with problems of the form, which pair of objects minimizes a function [14] Yet, simple non convex minimization A preliminary version of this paper appeared in Proceedings of ....

....in Section 8, including improved time bounds for Klee s measure problem [26] in the case of 4 dimensional unit hypercubes, and for the minimum diameter spanning tree problem, which are of independent interest. 2 The strategy for semi online dynamization The most common dynamization strategy [5, 24, 25] is based on decomposing a set of objects into subsets, solving the problem on each subset, and combining the answers. An update a ects only a small number of subsets and thus can be eciently handled. Unfortunately, this simple approach is not viable for any of our problems, because they are not ....

[Article contains additional citation context not shown here]

J. Bentley and J. Saxe, Decomposable searching problems I: static-to-dynamic transformation, J. Algorithms, 1:301-358, 1980.

....construct the farthest point Voronoi diagram of each group. An update can be handled by rebuilding the Voronoi diagram of the a ected group in O( n log n) time, and a query can be performed in O( n log n) time by querying each group separately (by point location) and combining the answers [4]. By a similar approach, near O( n) time dynamic algorithms can also be obtained for the diameter and the smallest enclosing circle of a planar point set [19] although the task of combining the answers is not as self evident (the technical term for the required property is ....

....point q below all the lines, nd the closest line to q. The preprocessing time is O(n log n) and the query time is O(log n) 2 Formally, a query problem is decomposable if the answer on a set S 1 [S 2 can be computed from the answer on S 1 and the answer on S 2 in constant time. Bentley and Saxe [4] observed a simple binary counting method that in general turns any static data structure for a decomposable problem into an incremental data structure that supports insertions with the cost of an extra logarithmic factor. increasing functions. Then we can design a data structure for a set of ....

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301-358, 1980.

....data structure is rather complex to implement in practice. Note that the ECDF tree and the range tree are both static and internal memory structures. To dynamize a static data structure some standard techniques can be used [12] For example, the global rebuilding [24] or the logarithmic method [8]. To externalize an internal memory data structure, a widely used method is to augment it with block access capabilities [34] Range Sum for Data Cubes and Point Data. The data cube range sum problem addresses the following query: given a d dimensional array A and a query range q, nd the sum of ....

J. L. Bentley and N. B. Saxe, \Decomposable Searching Problems I: Static-to-Dynamic Transformations", J. of Alg. 1(4), 1980.

....time. 14 IV. Concluding remarks We finish this paper by giving some concluding remarks, and some ideas that are being worked out pmsenfiy. Many dynamic data suctures are obtained from static suctures by using one of the so called dynamizafion techniques (see Bentley [1] Bentley and Saxe [2], Overmars [7] Of course, these techniques can also be applied to shadow administrations. So the idea is the following. We are given a static data structure DS, that solves some searching problem. Now we make a shadow administration for this static slmcture. Then we apply a dynamization ....

J.L.Bentley and J.B.Saxe, Decomposable Searching Problems I: Static to Dynamic Transformations, J.of Algorithms _1 (1980),pp.301-358.

....A) and PR(x, B) in constant time. A number of techniques have been developed to design dynamic data structures for decomposable searching problems. It turns out that especially in the case where only insertions are performed efficient structures can be designed. See Bentley [1] Bentley and Saxe [2] or Overmars [9] Let PR be a decomposable searching problem, and let DS be a dynamic data structure solving PR. We consider the case in which only insertions are performed. Let S(n) be the size of the structure DS, and let Q(n) be the query time of DS. We assume that S(n) n and Q(n) are ....

....n glso be pefformM. For emple, we c rtrict oelv to a subclass of the decompoble searching problem, the sglled demsable eountin9 problem. Roughly speaking, a decomposable counting problem is a decotop.able segrcng pblem where the function h invert, that can also be computed in co.tang time (see [2,0]) An mple is the orthogond tahoe countino problem. Here we are given a set of points in the plane, and an aa parallel que rectangle, and we are raked how my points of are in the rtangle. For dompable co.ting proble we can dign a full dyc data stcture by intaining two structures. In one ....

J.L. Bentley and J.B. Saxe. Decomposable Searching Problems I: Static to Dynamic Transformations. J. of Algorithms 1 (1980), pp. 301-358.

....know the value of n, the total number of insert operations, in advance. Standard doubling techniques justify this assumption. Starting from the monotonic data structure presented in Sect. 2, we apply a general dynamization technique for decomposable search problems attributed to Bentley and Saxe [3]. The idea is that we divide the set of lines L into a partition based on the order the lines are inserted. More precisely every set C has a rank. If there are d sets of the same rank i, we merge them into one new set of rank i 1. Sets of rank 0 have size 1. We choose the parameter d = ....

J. Bentley and J. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1:301--358, 1980.

....k point set among m points. Then in the plane, we can maintain the k point set minimizing as points are inserted in time O(T (m) log n m log n) per insertion and space O(n log n S(m) Proof: We apply a standard dynamic to static reduction technique for decomposable searching problems [4] to the rectilinear nearest neighbor data structure of Lemma 3.3. Theorem 6.1. We can maintain the minimum measure k point set in the plane as points are inserted, with the following insertion times: O(k n) for perimeter, n) for L1 perimeter, O(k log k log n) for circumradius, ....

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. J. Algorithms, 1:301--358, 1980.

.... concepts the reader is referred to [BGH97] The results of this paper are based on kinetizing (i.e. maintaining under continuous point motion) a certain type of multidimensional range search tree (MDRS tree) Meh84] used to query the value of some specified function on subsets of the points [BS80] We apply this MDRS tree to obtain kinetic data structures for the closest pair among n moving points in IR , the closest pair among a collection of n moving objects, and approximate and exact MSTs for n moving points. Though the structures we propose are fully on line and work for arbitrary ....

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. J. Algorithms, 1:301--358, 1980.

.... Dynamically, there are at least three di#erent options: We can use a weight balanced range tree; the size bound increases to O(n log d 1 n) because of the need to rebuild canonical subsets in subtrees) Or we can handle deletions lazily and insertions using Bentley and Saxe s transformation [5], again at the expense of a logarithmic factor. Or we can adopt a randomized skip list version of range trees (e.g. see [25, Section 8.1] which seems to avoid the extra logarithmic factor. Theorem 2.2. Fix any constant d. The dynamic connectivity problem for boxes in IR d reduces to the ....

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301--358, 1980.

....a variant of the Overmars van Leeuwen structure. Their structure supports nd min queries in O(log n) time, worst case. The time for an n item make heap operation is O(n log n) and the total time for up to n deletions is O(n log n) Chan [11] applied the dynamization technique of Bentley and Saxe [10] and other ideas to the Hershberger SuriChazelle structure to obtain an implementation of parametric heaps that supports nd min in O(log n) worst case time and insert and delete in O(log 1 n) amortized time for an arbitrary positive . Very recently, Brodal and Jacob [3] have described an ....

.... and Chazelle structures, reducing their O(n log n) bound to O(n) while increasing the amortized time of delete to O(log n log log n) This trade o is advantagous for the next step, in which we add the ability to handle insertions by applying the dynamization technique of Bently and Saxe [10]. We use a simple binary counting scheme in which there are up to log n separate semi dynamic structures, each corresponding to a di erent power of two. Insertion is done by creating a new single node structure and then if necessary repeatedly combining structures (corresponding to a binary ....

Bentley J. and Saxe J. Decomposable searching problems I: Static-to-dynamic transformation". J. of algorithms, 1:301-358, 1980.

....structure is rather complex to implement in practice. Note that the ECDF tree and the range tree are both static and internal memory structures. To dynamize a static data structure some standard techniques can be used [CT92] For example, the global rebuilding ( Ove83] or the logarithmic method ([BS80]) To externalize an internal memory data structure, a widely used method is to augment it with block access capabilities [Vit01] Range Sum for Data Cubes. The data cube range sum problem addresses the following query: given a d dimensional array A and a query range q, find the sum of values of ....

J. L. Bentley and N. B. Saxe, "Decomposable Searching Problems I: Static-to-Dynamic Transformations", J. of Alg. 1(4), pp. 301-358, 1980.

....realized long ago [10] and while there have been many fundamental results in the area, our current knowledge is still limited. Dynamic data structures for all kinds of problems reducible to range searching [1] including linear convex programming, are known. A class of decomposable query problems [5] has been recognized as easy, for which simple general tricks are known. A useful technique has been devised to deal with problems of the form, which pair of objects minimizes a function [14] Yet, simple non convex minimization problems, like the width or smallest enclosing rectangle, or ....

....by known perturbation schemes. We mention some applications in Section 8, including an improved time bound for Klee s measure problem [25] in the case of 4 dimensional unit hypercubes, of independent interest. 2 The strategy for semi online dynamization The most common dynamization strategy [5, 23, 24] is based on decomposing a set of objects into subsets, solving the problem on each subset, and combining the answers. An update a ects only a small number of subsets and thus can be eciently handled. Unfortunately, this simple approach is not viable for any of our problems, because they are not ....

[Article contains additional citation context not shown here]

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301-358, 1980.

....construct the farthest point Voronoi diagram of each group. An update can be handled by rebuilding the Voronoi diagram of the a ected group in O( p n log n) time, and a query can be performed in O( p n log n) time by querying each group separately (by point location) and combining the answers [4]. By a similar approach, near O( p n) time dynamic algorithms can also be obtained for the diameter and the smallest enclosing circle of a planar point set [16] although the task of combining the answers is not as self evident (the technical term for the required property is ....

....a point q below all the lines, nd the closest line to q. The preprocessing time is O(n log n) and the query time is O(log n) Formally, a query problem is decomposable if the answer on a set S1 [ S2 can be computed from the answer on S1 and the answer on S2 in constant time. Bentley and Saxe [4] observed a simple binary counting method that in general turns any static data structure for a decomposable problem into an incremental data structure that supports insertions with the cost of an extra logarithmic factor. Lemma 2.2. Consider a decomposable query problem. Suppose we can ....

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301-358, 1980.

....geometric optimization. We also address some fundamental data structure issues raised by such optimization problems. In particular, many worst case geometric optimization algorithms use a variety of reductions from one problem to another, such as the static to dynamic reduction of Bentley and Saxe [9]; we examine methods for performing this sort of reduction in a way that preserves the average case behavior of the update sequence. 1 Our work also represents the first use of this type of average case analysis in dynamic graph algorithms. After this paper first appeared, Alberts and Rauch ....

....queries with orthogonal range searching. i.e. we wish to ask for the answer to a query such as a farthest neighbor problem, with the queried set being a subspace of the input within some rectangle or higher dimensional box given as part of the query. We consider any decomposable search problem [9] for which the answer for a given input set can be found quickly by combining answers in disjoint subsets of the input. We describe a solution for the one dimensional case, interval range searching; higher dimensional box range queries can be solved by iterating our construction once per ....

J. L. Bentley and J. Saxe. Decomposable searching problems I: staticto -dynamic transformation. J. Algorithms, 1:301--358, 1980.

....bounding boxes of the objects of C, and to use the fact that only a constant number of points are needed to stab all objects associated with each canonical subset of this tree. Our tree T can be maintained dynamically, using standard binary decomposition techniques of Bentley and Saxe [8]. If constructing a static version of the data structure takes time T (n) then inserting an object takes time O( T (n) n) log n) Since in our case T (n) O(n log 3 n) the cost of an insertion is O(log 4 n) Deletion of an object c is done as follows: We mark c as being deleted from each ....

.... (or O(log 2 n k) in the reporting mode) Since we repeat this O(log n) times, the total cost of a query is O(log 3 n) or O(log 3 n k) in the reporting mode) Insertion or deletion of a polygon can be done in time O(log 4 n) using the binary decomposition technique of Bentley and Saxe [8] mentioned above. If only deletions are required, then a deletion can be done in time O(log 3 n) We thus obtain: Theorem 2.5 Let C be a collection of n (fi; ffi) covered polygons in R 2 , each with O(1) edges. We can preprocess C in time O(n log 3 n) into a data structure of size O(n log ....

J. Bentley and J. Saxe, Decomposable searching problems I: Static-to-dynamic transformation, J. Algorithms 1 (1980), 301--358.

....queries with orthogonal range searching. i.e. we wish to ask for the answer to a query such as a farthest neighbor problem, with the queried set being a subspace of the input within some rectangle or higher dimensional box given as part of the query. We consider any decomposable search problem [7] for which the answer for a given input set can be found quickly by combining answers in disjoint subsets of the input. We describe a solution for the one dimensional case, interval range searching; higher dimensional box range queries can be solved by iterating our construction once per ....

J. L. Bentley and J. Saxe. Decomposable searching problems I: staticto -dynamic transformation. J. Algorithms, 1:301--358, 1980. 18

....geometric optimization. We also address some fundamental data structure issues raised by such optimization problems. In particular, many worst case geometric optimization algorithms use a variety of reductions from one problem to another, such as the static to dynamic reduction of Bentley and Saxe [8]; we examine methods for performing this sort of reduction in a way that preserves the average case behavior of the update sequence. 1.1 MotivationTo further motivate the average case analysis model, and point out some of its applications to geometric optimization, we first describe three ....

....queries with orthogonal range searching. i.e. we wish to ask for the answer to a query such as a farthest neighbor problem, with the queried set being a subspace of the input within some rectangle or higher dimensional box given as part of the query. We consider any decomposable search problem [8] for which the answer for a given input set can be found quickly by combining answers in disjoint subsets of the input. We describe a solution for one dimensional interval range searching; higher dimensional box range queries can be solved by iterating our construction once per dimension. Many ....

J. L. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301--358, 1980.

....log n) time. In order to apply Lemma 11, we need to specify the data structure DS . This data structure stores a set of points in R d , and supports nearestneighbor queries and insertions of points. We can obtain such a semi dynamic data structure by applying Bentley s logarithmic method, see [3, 4]. This technique transforms an arbitrary static data structure for nearest neighbor queries into one that also supports insertions of points. To be more speci c, let DS s be a static data structure storing a set of n points in R d , that supports nearest neighbor queries in Q s NN (n) time, ....

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Staticto -dynamic transformations. J. Algorithms, 1:301-358, 1980.

....log n) time. In order to apply Lemma 11, we need to specify the data structure DS . This data structure stores a set of points in R d , and supports nearestneighbor queries and insertions of points. We can obtain such a semi dynamic data structure by applying Bentley s logarithmic method, see [3, 4]. This technique transforms an arbitrary static data structure for nearest neighbor queries into one that also supports insertions of points. To be more specific, let DS s be a static data structure storing a set of n points in R d , that supports nearest neighbor queries in Q s NN (n) time, ....

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Staticto -dynamic transformations. J. Algorithms, 1:301--358, 1980.

....of the two influences, g( p q ) g(p q)org( p q ) g(q p) i.e. the division of the range 94 in two parts removes the complication of the absolute value from the cost function. Our algorithm for this problem can be viewed as a novel application of the Bentley Saxe dynamic to static reduction [10]: we perform two such reductions, in two di#erent orders, one for each type of eighth plane piece of the fragment point ranges. The di#ering order leaves the problem dynamic, but the reduction instead can be imagined as removing the vertical or horizontal boundaries of the pieces, leaving only the ....

....by this solution to solve the problem for the points on r, 3) compute the influence of the points above or on r, on the values of the points below r, and (4) recursively solve the problem below r. This divide and conquer technique is similar to the dynamic to static reduction of Bentley and Saxe [10]; it di#ers from the RNA structure algorithm of Aggarwal and Park [3] in that we divide only by rows, and not by columns. It does not seem possible to modify the algorithm of Aggarwal and Park to run in time depending on the sparsity of the problem, because at each level of their recursion they ....

J.L. Bentley and J.B. Saxe, Decomposable Searching Problems I: Static-toDynamic Transformation. J. Algorithms 1(4), December 1980, pp. 301--358.

....to perform insert delete operations on a data structure in less than P (n) n time, where P (n) is the preprocessing time of the data structure. If we allow only insertions (i.e. a point cannot be deleted from the structure) the static data structure can be modified, using standard techniques [44, 191, 208], so that a point can be inserted in time O(P (n) log n=n) and a query can be answered in time O(Q(n) log n) where Q(n) is the query time of the original static data structure. Roughly speaking, these techniques proceed as follows. Choose a parameter r 2 and set t = dlog r ne. Maintain a ....

J. L. Bentley and J. B. Saxe, Decomposable searching problems I: Static-to-dynamic transformation, J. Algorithms, 1 (1980), 301--358.

....Since the basic idea is the same, we only give a brief description of the structure. For Shallow cuttings December 2, 1995 Range Searching 21 the sake of simplicity, we describe a data structure that handles only deletions; insertions can then be handled using standard decomposition techniques [6, 15]. We need the following lemma, which is an easy consequence of a result by Chazelle et al. 18] but we give a somewhat simpler proof. Lemma 4.2 Let Xi be a cutting in R 3 consisting of t cells. Xi can be preprocessed in time O(t 3 ) into a data structure of size O(t 3 ) so that, for a ....

J. Bentley and J. Saxe, Decomposable searching problems I: Static-to-dynamic transformation, J. Algorithms 1 (1980), 301--358.

....know the value of n, the total number of insert operations, in advance. Standard doubling techniques justify this assumption. Starting from the monotonic data structure presented in Sect. 2, we apply a general dynamization technique for decomposable search problems attributed to Bentley and Saxe [3]. The idea is that we divide the set of lines L into a partition C based on the order the lines are inserted. More precisely every set C 2 C has a rank. If there are d sets of the same rank i, we merge them into one new set of rank i 1. Sets of rank 0 have size 1. We choose the parameter d = ....

J. Bentley and J. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1:301-358, 1980.

....using the technique of Agarwal and Matousek [6] Since the basic idea is the same, we only give a brief description of the structure. For the sake of simplicity, we describe a data structure that handles only deletions; insertions can then be handled using standard decomposition techniques [6, 15]. We need the following lemma, which is an easy consequence of a result by Chazelle et al. 18] but we give a somewhat simpler proof. Lemma 4.2 Let Xi be a cutting in R 3 consisting of t cells. Xi can be preprocessed in time O(t 3 ) into a data structure of size O(t 3 ) so that, for a ....

J. Bentley and J. Saxe, Decomposable searching problems I: Static-to-dynamic transformation, J. Algorithms 1 (1980), 301--358.

....published in [40] After this we consider a more general model, where the objects of the scene have different sizes. For the selection of important objects we first present an approximate solution which exploits the fact that the selection problem is a so called decomposable searching problem [15, 65, 35]. Then we show another method which uses geometric transformations to obtain a higher dimensional halfspace range searching problem and solve this problem with known algorithms. 1.2 Geometric spanners and weak spanners Geometric spanners have been studied intensively in the recent years. Let S ....

....searching problem and solve this problem with known algorithms. This approach was, for example, used in [46, 2, 9, 36, 83] for similar problems. 4.4. 1 Using circular range searching in size classes In this subsection we exploit that our search problem is a so called decomposable search problem [15, 65, 35]. We use a special decomposition which allows us to apply circular range searching algorithms for an approximated solution of the original searching problem. Decomposable search problems are search problems that satisfy the following conditions [35] Let S be a set of objects and let S be ....

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-todynamic transformation. J. Algorithms, 1:301--358, 1980.

....5 Dynamic Data Structures The previous section shows how solutions to a static linear programming query problem may be combined into the solution for the union of the constraints (intersection of the polytopes) with polylogarithmic overhead. This is reminiscent of decomposable searching problems [2, 6, 15]. In such a problem, we again ask query problems in some set; a problem is decomposable if the answers to queries in disjoint sets can be combined in constant time to give the answer in the union of the sets. If so, k sets can be combined in time O(k) In our case, we cannot simply combine the ....

J.L. Bentley and J.B. Saxe. Decomposable searching problems I: staticto -dynamic transformation. J. Algorithms 1 (1980) 301--358.

....for the computation of recurrence 8. Assuming that the cost function w is convex, we can compute recurrences 10 and 11 in time O(n m M log M#(M ) This time bound reduces to O(n m M log M) when w is concave. Our algorithm uses in a novel way an algorithmic technique devised by Bentley and Saxe [6], namely dynamic to static reduction. Matrix searching [1, 30] is also used. We remark that if the cost function is convex concave and simple, matrix searching can be replaced by the algorithm of Galil and Giancarlo [16] to obtain an O(n m M log M) algorithm both for simple convex and concave cost ....

J.L. Bentley and J.B. Saxe, Decomposable Searching Problems I: Static-to-Dynamic Transformation. J. Algorithms 1(4), December 1980, pp. 301--358.

....diameter problem mentioned earlier, and many similar geometric optimization problems. Dobkin and Suri call this problem that of finding minima of decomposable functions, however this terminology is unfortunate as the word decomposable has been used in the literature to mean something different [5]: a function f on sets is decomposable if whenever S is the disjoint union of S 1 and S 2 , f (S) can be computed easily from f (S 1 ) and f (S 2 ) We prefer to call this problem that of finding minima of binary functions. In any case no fully dynamic algorithm for such problems was previously ....

J. L. Bentley and J. Saxe. Decomposable searching problems I: static-todynamic transformation. J. Algorithms, 1:301--358, 1980.

....by National Science Foundation Grant CCR 91 06514. Department of Information and Computer Science, University of California, Irvine, CA 92717. # Department of Applied Mathematics, Charles University, Prague, and Freie Universitat Berlin. obtain fast dynamic algorithms for these special cases [5, 17]. Recently Dobkin and Suri extended these techniques to a semi online model in which insertions are completely arbitrary, but when an object is inserted we are told its deletion time [16] However, these techniques do not seem to extend to the general case where deletions are also arbitrary. In ....

J. Bentley and J. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. J. Algorithms 1 (1980) 301--358.

.... This diagram can be found in time O(n log n) using any of several algorithms [34, 47, 88] We then perform planar point location in this diagram; again, various algorithms are possible [49, 77, 84, 103] # We can then apply the static to dynamic transformation for decomposable search problems [15] to this data structure, producing an incremental data structure in which we can insert points to S and answer the same sorts of queries, both in time O(log 2 n) Theorem 4. We can maintain the Euclidean minimum spanning tree of a set of points in the plane, subject to point insertions only, in ....

J. L. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, vol. 1, 1980, pp. 301--358.

....to our data structures. Unfortunately the concave equivalent of lemma 5 is that, if a strip were to be added to the collection of strips from later diagonals, it would remain in one piece. There are algorithmic techniques that can reverse the order of strip insertion to comply with this lemma [3], but they would add another logarithmic factor to the time used for our algorithm. Instead we replace the partitions of the convex case with the following data structure. We maintain for each row and column of the matrix a complete binary tree, with the leaves labeled in order from 1 to n. Each ....

J.L. Bentley and J.B. Saxe, Decomposable Searching Problems I: Static-to-Dynamic Transformation. J. Algorithms 1(4), December 1980, pp. 301--358.

....computational geometry, dynamic programming, and other areas. Research on decomposable search problems has concentrated on transforming solutions of the static problem (in which V does not change) into data structures for the dynamic problem; this line of inquiry was initiated by Bentley and Saxe [1], who defined decomposable problems and described several such methods. Fix a decomposable search problem of interest. Suppose that, given a set with n points, we can build in time P (n) a data structure of size S(n) that can answer queries in time Q(n) The dynamic problem (with both insertions ....

....of P (n) simply by rebuilding the static data structure after each update. This technique uses O(S(n) space. By maintaining the sets in blocks of k points, and rebuilding a single block after each update, the problem can be solved with time bounds O(P (k) per update and O(nQ(k) k) per query [1]. The space is O(nS(k) k) which could be a significant reduction from O(S(n) if the static structure uses superlinear space. Bentley and Saxe described another technique, for insertions only. We maintain a collection of static problems with sizes distinct powers of two. Each query combines ....

[Article contains additional citation context not shown here]

J.L. Bentley and J.B. Saxe. Decomposable searching problems I: staticto -dynamic transformation. J. Algorithms 1 (1980) 301--358.

....and Suri [30] Over a sequence of deletions, the total structural change in L is linear (or if one thinks backwards, adding a line creates at most two new vertices) Since each advance can be charged to a distinct vertex of L, there are O(n) number of advances. 2 Dynamizing. Bentley and Saxe [8] described how a semi dynamic data structure for a decomposable problem implies a data structure supporting both insertions and deletions. This general dynamization technique uses a simple binary counting trick and slows down operations by a logarithmic factor. We observe that the technique is ....

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301--358, 1980.

....are at most a factor c larger than the corresponding quantities in S. 3. 1 Decomposable Query A search problem on a set of objects S is decomposable if for any partition (S 1 ; S 2 ) of S, the answer to the query on S can be obtained in constant time from the answers to queries on S 1 and S 2 [7, 15]. For example, the search for a point p in a point set S to minimize jjp Gamma qjj, where q is a given query point, is decomposable. For decomposable query questions, we can maintain a collection of static 1 It is a binary tree whose leaves store weighted items. Let w be the sum of all weights. ....

J.L.Bently, and J.Saxe. Decomposable Searching Problems I:Static to Dynamic Transformations. Journal of Algorithms 1, 1980,301-358.

....seen by a backwards argument: adding a line creates at most two new vertices and three new edges to the envelope) Dynamization of the deletion only structure. To incorporate insertions, we employ a general dynamization technique for decomposable search problems, attributed to Bentley and Saxe [6, 13, 28, 30]. This technique uses a simple binary counting trick and usually slows down update time by a logarithmic factor. To lower the effect of the slow down, we actually apply a b ary variant of the counting trick for a suitably chosen parameter b. One description is as follows. We maintain a partition ....

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301--358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable Searching Problems I: Static-to-Dynamic Transformations. Journal of Algorithms, 1(4):301--358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1(4):301--358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1(4):301--358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformations. J. Algorithms, 1:301-358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: staticto -dynamic transformation. J. Algorithms, 1:301--358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-todynamic transformation. J. Algorithms, 1:301-358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems i: Static-todynamic transformation. J. Algorithms, 1(4):301-358, 1980.

No context found.

J. L. Bentley and J. B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1(4):301--358, 1980.

No context found.

J. Bentley and J. Saxe. Decomposable searching problems I: static-to-dynamic transformation. J. Algorithms, 1:301-358, 1980.

No context found.

J. L. Bentley and N. B. Saxe, \Decomposable Searching Problems I: Static-to-Dynamic Transformations", J. of Alg. 1(4), 1980.

No context found.

J. L. Bentley and N. B. Saxe, "Decomposable Searching Problems I: Static-to-Dynamic Transformations", J. of Alg. 1(4), 1980.

No context found.

J. L. Bentley and J. B. Saxe, "Decomposable searching problems I: Staticto -dynamic transformation," J. Alg., no. 1, pp. 301--358, 1980.

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