M.L. Fredman. A lower bound on the complexity of orthogonal range queries. Journal of the ACM, 28:696-705, 1981  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for dynamic problems is the cell probe model described by Fredman and Saks [11] This model counts the number of memory cells read and written to by a program, without considering the complexity of the control flow of the program. Earlier work by Fredman and by Yao also uses essentially this model [12, 13, 43]. The Fredman and Saks paper gives upper and lower bounds for problems such as partial sums, list representation, and subset rank, on data structures with word size 1 and arbitrary word size. In some cases, the upper and lower bounds match, and give tight bounds. Most of the bounds are of the ....

Fredman, Michael L. A lower bound on the complexity of orthogonal range queries. Journal of the ACM (JACM) 28, 4 (1981), 696--705.

....semigroup (S; The problem is to answer a set of queries, where a query asks for the cumulative weight of some subset of the underlying elements. The case where elements are points in R d has been studied extensively under various types of queries. There are too many papers to cite; see [18, 38, 39, 7, 8, 4, 14, 9] for lower bounds and more references. Chazelle Rosenberg [12, 13] studied the case where the elements are packed into a d dimensional array and queries take the form of d rectangles (see also [38, 2] for d = 1. In [13] a tight lower bound of n m (m;n) semigroup operations is proved for ....

M. L. Fredman. A lower bound on the complexity of orthogonal range queries. J. ACM, 28(4):696--705, 1981.

....where the weights are drawn from some (commutative) semigroup (S; The problem is to answer a set of queries, where a query asks for the cumulative weight of some subset of the underlying elements. The case where elements are points in R d has been studied under various types of queries see [Fre81, Yao85, Cha89, Cha90, BCP93, CR96, Cha97] for lower bounds and references. Chazelle Rosenberg [CR89, CR91] studied the case where the elements are packed into a d dimensional array and queries take the form of d rectangles (see also [Yao82, AS87] for d = 1. In [CR91] a (tight) lower bound of n m (m;n) semigroup operations is ....

Michael L. Fredman. A lower bound on the complexity of orthogonal range queries. J. ACM, 28(4):696-705, 1981.

....additive semigroup other than the integers, nodes in the data structure can also be labeled with semigroup values, but these values can only be added. Most lower bounds, and a few upper bounds, are described in the so called semigroup arithmetic model, which was originally introduced by Fredman [127] and refined by Yao [258] In the semigroup arithmetic model, a data structure can be informally regarded as a set of precomputed partial sums in the underlying semigroup. The size of the data structure is the number of sums stored, and the query time is the minimum number of semigroup operations ....

....number of edges, a two dimensional range reporting query can be answered in O(log n k) time using linear space [64, 65] If the ranges are octants in R 3 , a rangereporting query can be answered in either O( k 1) log n) or O(log 2 n k) time using linear space [65] 3. 2 Lower bounds Fredman [125, 126, 127, 129] was the first to prove nontrivial lower bounds on orthogonal range searching, in a version of semigroup arithmetic model in which the points can be inserted and deleted dynamically. He showed that a mixed sequence of n insertions, deletions, and queries requires Omega Gamma n log d n) time. ....

M. L. Fredman, A lower bound on the complexity of orthogonal range queries, J. ACM, 28 (1981), 696--705.

....the semigroup ( max) and the Euclidean weight function; that is, w(p; q) is the Euclidean distance between p and q. Other problems have similar formulations. We believe that one can define a model for visibility type problems along the lines of the semigroup model of computation used by Fredman [12] and Chazelle [5, 6, 9] which has been used successfully to prove lower bounds on range searching problems. In particular, one needs to formalize the cost of computing the weight W . It seems reasonable that, in the absence of additional assumptions, computing W (T ) for an arbitrary subset T S ....

M. L. Fredman, A lower bound on the complexity of orthogonal range queries, J. ACM 28 (1981), 696--705.

....lie in s can be found efficiently. The problem comes under various guises, depending on whether one wants to enumerate the points or simply count them. All these variants can be unified into a common formulation by attaching weights to the points and endowing the weights with a semigroup structure [3, 4, 15, 16, 27]. If m is the amount of storage, a lower bound on the query time of Omega Gamma n=m 1=2 ) if d = 2, and Omega Gamma n= log n) m 1=d ) if d 2, was established by Chazelle [4] in the arithmetic model of Work by Herv e Bronnimann and Bernard Chazelle has been supported in part by NSF ....

....Geometry Center. y Department of Computer Science, Princeton University z Department of Computer Science, Princeton University x Mathematical Institute of the Hungarian Academy of Sciences and Courant Institute, New York University computation. The model, which is due to Fredman and Yao [3, 15, 16, 27], is general enough to encompass all known algorithms for the counting problem. Later, Chazelle, Sharir, and Welzl [9] provided a quasi matching upper bound (up to within a factor of n ffl , for any fixed ffl 0) More recently, Matousek [22] lowered the multiplicative factor to ....

Fredman, M.L. A lower bound on the complexity of orthogonal range queries, J. ACM, 28 (1981), 696--705.

....terms of variables) of the queries that are answered by the register i. In this case, we overestimate the number of variables per register. Summing over all possible registers, we obtain the equation 2. Using this general technique the authors prove lower bounds for a variety of problems. Fredman [43] considers range queries, and spherical queries in the dynamic setting [44] Fredman and Volper [46] consider the problem of partial match queries. They prove that the complexity of the problem is 1:226 d N , where the complexity of a query is defined as the total number of registers accessed by ....

M. L. Fredman. A lower bound on the complexity of orthogonal range queries. Journal of the ACM, 28(4):696--705, October 1981.

....problem is a nontrivial case of two dimensional range searching. We show in Section 3.2 that it is impossible to achieve optimal query time for this problem (O(log B n t=B) disk I Os) with only one copy of each object in secondary storage. For lower bounds on range searching in main memory, see [14] and [7] In Chapter 5, analyzing the hierarchy using the hierarchical decomposition of [37] and using techniques from the constraint indexing problem, we improve query I O time to O(log B n t=B log 2 B) using space O( n=B) log 2 c) pages. Amortized update I O time for the semi dynamic ....

....of data structures to perform external two dimensional range searching with performance comparable to that of B trees, we might try to prove that the problem is inherently harder than external one dimensional range search. Such bounds have been established for range searching in main memory by [7,14]. The following lemma is a preliminary result that shows that B tree like performance is probably not possible for two dimensional range search. We consider a simple grid of points and try to see if it is possible to place rectangles on this grid of points (i.e. tessellate the points) so that ....

M. L. Fredman, "A Lower Bound on the Complexity of Orthogonal Range Queries," J. ACM 28 (1981), 696--705.

....to justify the curse of dimensionality conjecture. That is, either the models with respect to which lower bounds have been established seem quite restricted or the bounds are quite weak. One nice example of a well structured model (for both dynamic and static data structure problems) is Fredman s [24, 25, 26] semi group model. The model is designed for searching problems (e.g. range queries) in 4 Using d bits per cell, it is possible to derive a one sided error implementation. which a semi group value is associated with each data point and one wants to retrieve the semi group sum of all data ....

M.L. Fredman. A lower bound on the complexity of orthogonal range queries. Journal of the ACM, 28:696-705, 1981

....problem is a nontrivial case of 2 dimensional range searching. We show in Section 2.2 that it is impossible to achieve optimal query time for this problem (O(log B n t=B) disk I O s) with only one copy of each object in secondary storage. For lower bounds on range searching in main memory, see [12] and [6] In Section 4, analyzing the hierarchy using the hierarchical decomposition of [33] and using techniques from the constraint indexing problem, we improve query I O time to O(log B n t=B log 2 B) using space O( n=B) log 2 c) pages. Amortized update I O time for the semi dynamic ....

....absence of data structures to perform external 2 dimensional range searching with performance comparable to that of B trees, we might try to prove that the problem is inherently harder than external 1 dimensional range search. Such bounds have been established for range searching in main memory by [6,12]. The following lemma is a preliminary result that shows that B tree like performance is probably not possible for 2 dimensional range search. We consider a simple grid of points and try to see if it is possible to place rectangles on this grid of points (i.e. tessellate the points) so that all ....

M. L. Fredman, "A Lower Bound on the Complexity of Orthogonal Range Queries," J. ACM 28 (1981), 696--705.

....an additive semigroup other than the integers, nodes in the data structure can also be labeled with semigroup values, but these values can only be added. Most lower bounds, and a few upper bounds, are described in the so called semigroup arithmetic model, which was originally introduced by Fredman [133] and refined by Yao [292] In the semigroup arithmetic model, a data structure can be informally regarded as a set of precomputed partial sums in the underlying semigroup. The size of the data structure is the number of sums stored, and the query time is the minimum number of semigroup operations ....

....of edges, a two dimensional range reporting query can be answered in O(log n k) time using linear space [67, 68] If the ranges are octants in R 3 , a range reporting query can be answered in either O( k 1) log n) or O(log 2 n k) time using linear space [68] 3. 2 Lower bounds Fredman [131, 132, 133, 135] was the first to prove nontrivial lower bounds on orthogonal range searching, in a version of semigroup arithmetic model in which the points can be inserted and deleted dynamically. He showed that a mixed sequence of n insertions, deletions, and queries requires Omega Gamma n log d n) time. ....

M. L. Fredman, A lower bound on the complexity of orthogonal range queries, J. ACM, 28 (1981), 696--705.

....two models is that on the pointer machine a memory cell can be accessed only through a series of pointers while in the RAM model any memory cell can be accessed in constant time. Most of the lower bounds will be given in the so called semigroup model , which was originally introduced by Fredman [61] and which is much weaker than the pointer machine or the RAM model. In the arithmetic model, a data structure is regarded as a set of precomputed sums in the underlying semigroup. The size of the data structure is the number of sums stored, and the query time is the number of semigroup operations ....

.... number of edges, a two dimensional range reporting query can be answered in O(log n k) time using linear space [35, 36] If the ranges are octants in R 3 , a range reporting query can be answered in either O( k 1) log n) or O(log 2 n k) time using linear space [36] LOWER BOUNDS Fredman [59, 60, 61] was the first to prove nontrivial lower bounds on orthogonal range searching, but he considered the framework in which the points were allowed to insert and delete dynamically. He showed that a mixed sequence of n insertions, deletions, and queries takes Omega Gamma n log d n) time. These ....

M. L. Fredman, A lower bound on the complexity of orthogonal range queries, J. ACM, 28 (1981), 696--705.

....n) of the divided k d tree [31] for query time complexity in dynamic environments. We also show that this structure is optimal when data is not replicated. Several results for the complexity of multidimensional range searching have been established in computational geometry literature. Fredman [12, 13, 14] analyzed the complexity of range searching in dynamic environments. Yao [32] analyzed it for semi dynamic environments. In static environments, Vaidya [30] established a storage space query time tradeoff that was later strengthened by Chazelle [6, 8] These analyses estimated the cost using the ....

M. L. Fredman. A lower bound on the complexity of orthogonal range queries. Journal of the ACM, 28:696--705, 1981.

....drawings of planar st graphs and general planar graphs use O(n) space, and support updates in O(log n) time and drawing queries in O(k log n) time. 9. 8 Lower Bounds General lower bound techniques for dynamic algorithms are discussed in [23,64] some relevant lower bounds are also given in [32,62,63,173]. 164] gives an upper bound that precisely matches the lower bounds of [63] and [173] the difference between these two is that they obtain the same quantitative lower bounds under different models of computation. 88] gives an upper bound that nearly matches the lower bound of [62] Fredman and ....

....updates in O(log n) time and drawing queries in O(k log n) time. 9. 8 Lower Bounds General lower bound techniques for dynamic algorithms are discussed in [23,64] some relevant lower bounds are also given in [32,62,63,173] 164] gives an upper bound that precisely matches the lower bounds of [63] and [173] the difference between these two is that they obtain the same quantitative lower bounds under different models of computation. 88] gives an upper bound that nearly matches the lower bound of [62] Fredman and Willard [65,174] explain how fusion trees refute many conjectures about ....

M.L. Fredman, "A Lower Bound on the Complexity of Orthogonal Range Queries," Journal of ACM 28 (1981), 696--706.

....problem is a nontrivial case of 2 dimensional range searching. We show in Section 2.2 that it is impossible to achieve optimal query time for this problem (O(log B n t=B) disk I O s) with only one copy of each object in secondary storage. For lower bounds on range searching in main memory, see [13] and [7] In Section 4, analyzing the hierarchy using the hierarchical decomposition of [34] we reduce the problem of indexing classes to a special case of external dynamic 2 dimensional range searching called 3 sided range searching. 3 sided range queries are a special case of 2 dimensional ....

....absence of data structures to perform external 2 dimensional range searching with performance comparable to that of B trees, we might try to prove that the problem is inherently harder than external 1 dimensional range search. Such bounds have been established for range searching in main memory by [7,13]. The following lemma is a preliminary result that shows that B tree like performance is probably not possible for 2 dimensional range search. We consider a simple grid of points and try to see if it is possible to place rectangles on this grid of points (i.e. tessellate the points) so that all ....

M. L. Fredman, "A Lower Bound on the Complexity of Orthogonal Range Queries," J. ACM 28 (1981), 696--705.

....to justify the curse of dimensionality conjecture. That is, either the models with respect to which lower bounds have been established seem quite restricted or the bounds are quite weak. One nice example of a well structured model (for both dynamic and static data structure problems) is Fredman s [23, 24, 25] semi group model. The model is designed for searching problems (e.g. range queries) in which a semi group value is associated with each data point and one wants to retrieve the semi group sum of all data points in some specified set (e.g. satisfying a partial match or more generally satisfying ....

M.L. Fredman. A lower bound on the complexity of orthogonal range queries. Journal of the ACM, 28:696-705, 1981

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M.L. Fredman. A lower bound on the complexity of orthogonal range queries. Journal of the ACM, 28:696-705, 1981

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