M.L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Computing, 10:1-10, 1981  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to databases; it corresponds to the generalized select operator of relational algebra, or, as it is sometimes called, # selection. The basic problem statement can be further elaborated, in various ways that we will examine next. One generalization of the problem, introduced by Fredman [Fre80, Fre81] is as follows: let (S, be a semigroup, that is, let be an associative and commutative operation over S. Also, let f be a mapping from D to S. For a given range r R, we wish to compute the semigroup sum x#I#p(x,r) f(x) This generalization can model the computation of range ....

....geometric range search problems, there is no solution with both optimal time and space complexity. This state of things was observed early on, with many researchers calling for either improved upper bounds, or lower bounds that would resolve the issue. Some early lower bounds by Fredman [Fre80, Fre81] Yao [Yao82] and Vaidya [Vai89] indicated that indeed, optimal query time O(log n t) with linear space was impossible for many problems. However, it was the work of Chazelle [Cha95, Cha90a, Cha90b] that provided tight lower bounds for orthogonal range search, in various memory models (for main ....

M.L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM Journal of Computing, 10:1--10, 1981.

....Reif and Tate provided two general techniques for the design of ecient dynamic algebraic algorithms. They also presented lower bounds and time space trade o s for several problems. Apart from Reif and Tate s work, we also meet dynamic algebraic problems in the literature on the prefix sum problem [Fre82, Fre81, Yao85, HF93, FS89, BAG91]; the speci c case of f i (x) P i j=1 x i for i = 1; n. The aim of this paper is to present three techniques for showing lower bounds for dynamic algebraic problems. We use them to show lower bounds on the worst case time complexity per operation for several natural problems where ....

M.L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10:1-10, 1981.

....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 ....

M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM Journal on Computing, 10(1):1--10, 1981.

....and Tate provided two general techniques for the design of efficient dynamic algebraic algorithms. They also presented lower bounds and time space trade offs for several problems. Apart from Reif and Tate s work, we also meet dynamic algebraic problems in the literature on the prefix sum problem [8, 7, 28, 13, 9, 2]; the specific case of f i (x) # i j=1 x i for i = 1, n. The aim of this paper is to present three techniques for showing lower bounds for dynamic algebraic problems. We use them to show lower bounds on the worst case time complexity per operation for several natural problems where ....

M.L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10:1--10, 1981.

....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 ....

M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM Journal on Computing, 10(1):1--10, 1981.

....n weighted points in the plane and n halfplanes, we consider the classical halfplane range searching problem, which is to compute the sum of the weights of the points within each of the given regions. If subtractions are not allowed (the semigroup model) the problem is almost completely solved [7, 11, 15]; see also [9, 16, 17, 19] for surveys of the vast literature on the subject. In the (commutative) group model, where subtractions are allowed, there is little evidence that any power should be gained beyond polylog speedups, but proving it has been elusive. In fact, in that model no superlinear ....

M. L. FREDMAN, Lower bounds on the complexity of some optimal data structures, SIAM J. Comput., 10 (1981), pp. 1--10.

....range reporting data structures do not fit in the framework, they nevertheless admit a tradeoff. In particular, a halfspace reporting query in R d can be answered in O( n polylog n) m 1=bd=2c k) using O(m) space. Geometric Range Searching and Its Relatives 27 4. 4 Lower bounds Fredman [128] showed that a sequence of n insertions, deletions, and halfplane queries on a set of points in the plane requires Omega Gamma n 4=3 ) time, under the semigroup model. His technique, however, does not extend to static data structures. In a series of papers, Chazelle has proved nontrivial lower ....

M. L. Fredman, Lower bounds on the complexity of some optimal data structures, SIAM J. Comput., 10 (1981), 1--10.

....and query time. These are the rst such lower bounds for any range searching problem in any model of computation; earlier models do not even de ne preprocessing time. #### ######## ######## Most geometric range searching lower bounds are presented in the Fredman Yao semigroup arithmetic model [33, 56]. In this model, the # This research was done at the Computer Science Division, U. C. Berkeley, and at the Center for Geometric Computing, Department of Computer Science, Duke University, with the support of a Graduate Assistance in Areas of National Need fellowship, NSF grant DMS 9627683, and U. ....

.... by requiring only an approximation of the correct output; see, for example, 7, 24, 37] SPACE TIME TRADEOFFS FOR EMPTINESS QUERIES 5 #### ########### We begin by reviewing the de nition of the semigroup arithmetic model, originally introduced by Fredman to study dynamic range searching problems [33], and later re ned for the static setting by Yao [56] A semigroup (## ) is a set # equipped with an associative addition operator : # # # # #. A semigroup is commutative if the equation # # = # # is true for all ## # # #. A linear form is a sum of variables over the semigroup, where ....

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M. L. Fredman, Lower bounds on the complexity of some optimal data structures, SIAM J. Comput., 10 (1981), pp. 1-10.

....bounds of Omega n 2d= d 1) log d= d 1) n) and Omega n 2d= d 1) log 5=2;fl n) in the online and offline cases, respectively, where fl 0isasmall constant that depends on d. For related results, see also [6, 12] These lower bounds hold in the Fredman Yao semigroup arithmetic model [25]. In this model, points are given arbitrary weights from an additive semigroup, and the complexity of the algorithm is given by the number of additions required to calculate the total weight of the points in each range. Unfortunately, the semigroup model is inappropriate for studying Hopcroft s ....

....cover and the running time of the divide and conquer algorithm is readily apparent in this case. In Section 4, we generalize this connection to higher dimensions. 3.2 Two Dimensions To derivelower bounds for 2 (n# m) and i 2 (n# m) we use the following combinatorial result of Erdos. See [25]or[17, p.112] for proofs. Lemma 3.2 (Erdos) For all n and m, there is a set of n points and m lines in the plane with Omega n n 2=3 m 2=3 m) incidentpairs.Thus, I 2 (n# m) Omega (n n 2=3 m 2=3 m) Fredman [25] uses Erdos construction to provelower bounds for dynamic range ....

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M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10:1--10, 1981.

....in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require Omega Gamma n 1=3 Delta time (modulo a logarithmic factor) 1 Introduction A considerable amount of attention has been given to simplex range searching in the last few years [1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 19, 21, 22, 26, 28, 29]. This is the problem of preprocessing a set P of n points in Euclidean d space so that, given an arbitrary query simplex s, which points 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 ....

....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 ....

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Fredman, M.L. Lower bounds on the complexity of some optimal data structures, SIAM J. Comput. 10 (1981), 1--10.

....and query time. These are the rst such lower bounds for any range searching problem in any model of computation; earlier models do not even de ne preprocessing time. 1.1. Previous Results. Most geometric range searching lower bounds are presented in the Fredman Yao semigroup arithmetic model [33, 56]. In this model, the This research was done at the Computer Science Division, U. C. Berkeley, and at the Center for Geometric Computing, Department of Computer Science, Duke University, with the support of a Graduate Assistance in Areas of National Need fellowship, NSF grant DMS 9627683, and U. ....

....by requiring only an approximation of the correct output; see, for example, 7, 24, 37] SPACE TIME TRADEOFFS FOR EMPTINESS QUERIES 5 2.1. De nitions. We begin by reviewing the de nition of the semigroup arithmetic model, originally introduced by Fredman to study dynamic range searching problems [33], and later re ned for the static setting by Yao [56] A semigroup (S; is a set S equipped with an associative addition operator : S S S. A semigroup is commutative if the equation x y = y x is true for all x; y 2 S. A linear form is a sum of variables over the semigroup, where each ....

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M. L. Fredman, Lower bounds on the complexity of some optimal data structures, SIAM J. Comput., 10 (1981), pp. 1-10.

....set P t Theta Q t contains two problem instances that have distinct answers, meaning that no point can serve as an approximate nearest neighbor for both instances. 8. 2 The semi group model A well structured model for the study of lower bounds for search problems is Fredman s semi group model [20, 45, 44]. A semi group S is defined as a set closed under an associative addition operation. 33 (Sometimes, we may assume that the operation is also commutative) In this model, a semi group value is associated with each data point. Let U = fv 1 ; v 2 ; v n g be the set of semi group values defined ....

....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 a sequence of N update and query operations. This implies that there ....

M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM Journal on Computing, 10(1):1--10, 1981.

....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. Lower bounds on the complexity of some optimal data structures. SIAM J. Computing, 10:1-10, 1981

....and Tate provided two general techniques for the design of efficient dynamic algebraic algorithms. They also presented lower bounds and time space trade offs for several problems. Apart from Reif and Tate s work, we also meet dynamic algebraic problems in the literature on the prefix sum problem [5, 6, 25, 11, 7, 2]; the specific case of f i (x) P i j=1 x i for i = 1: n. The aim of this paper is to present three techniques for showing lower bounds for dynamic algebraic problems. We use them to show lower bounds on the worst case time complexity per operation for several natural problems where Reif and ....

M.L Fredman. Lower bounds on the complexity of some optimal data structures. SIAM Journal on Computing, 1981.

....(iv) Finally, it is not essential for D 1 or D 2 to be tree based data structures. It is sufficient to have an efficient, r convergent decomposition scheme with a partial order on the canonical subsets, where each canonical subset satisfies a property similar to (P1) 4. 4 Lower bounds Fredman [134] showed that a sequence of n insertions, deletions, and halfplane queries on a set of points in the plane requires Omega Gamma n 4=3 ) time, in the semigroup model. His technique, however, does not extend to static data structures. In a series of papers, Chazelle has proved nontrivial lower ....

M. L. Fredman, Lower bounds on the complexity of some optimal data structures, SIAM J. Comput., 10 (1981), 1--10.

....variables storing values in S. Each variable stores the sum of values associated with some subset of I. A query is answered by summing a subset of these variables. An update is accomplished by recomputing the values of the appropriate variables. This class has been used by Fredman and others [F80, BFK81, F81, F81b] for analysis of query problems. If the values associated with the interval are from a commutative group (i.e. inverses exist) then there is a data structure which enables both update and query algorithms of time complexity O(log n) F79] For the semigroup problem, the use of range trees enables ....

Fredman, Michael L., Lower bounds on the complexity of some optimal data structures. SIAM Journal on Computing, 10, 1981, 1-10.

....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. Lower bounds on the complexity of some optimal data structures. SIAM Journal of Computing, 10:1--10, 1981.

.... neighbor queries [2] linear programming queries [23, 7] depth ordering [5] collision detection [11] and output sensitive convex hull construction [23, 8] Most previous range searching lower bounds are presented in the so called semigroup arithmetic model, originally introduced by Fredman [20] and later refined by Yao [31] In this model, the points are given weights from an additive semigroup, and the goal of a range query is to determine the total weight of the points in a query region. A data structure in this model can be informally regarded as a set of precomputed partial sums in ....

....dimensions, we easily observe that the clusters in an optimal storage scheme must consist of maximal colinear subsets of the set of points. The two dimensional lower bound follows from a construction of n points and s 1 lines with Omega (n 2=3 s 2=3 ) incidences discovered by Erdos; see [20] or [28, p. 177] The higher dimensional results follow from a natural generalization of the Erdos construction to higher dimensions [19, Lemmas 3.5 and 3.10] We omit further details from this extended abstract. 3 Partition Graphs A partition graph is a directed acyclic (multi )graph, with one ....

M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput. 10:1-- 10, 1981.

....are known in advance, Chazelle establishes a slightly weaker bound of Omega Gamma n 4=3 = log 4=3 n) 9] although an Omega Gamma n 4=3 ) lower bound follows easily from the Erdos construction using Chazelle s methods. Both lower bounds hold in the Fredman Yao semigroup arithmetic model [19], in which we give the points arbitrary weights from a semigroup, and count the number of arithmetic operations required to calculate the answer. Unfortunately, this model is inappropriate for studying Hopcroft s problem. If there are no incidences, the we perform no additions; conversely, if we ....

....size is twice that of the original minor. 2 3.2 Two Dimensions Let I(P; H) denote the number of incident pointhyperplane pairs between a set of points P and a set of hyperplanes H. To derive lower bounds for 2 (n; m) and i 2 (n; m) we use the following combinatorial result of Erdos. See [19] or [13, p.112] for proofs. Lemma 3.2 (Erdos) For all n and m, there is a set of n points and m lines in the plane with Omega Gamma n n 2=3 m 2=3 m) incident pairs. Fredman [19] uses Erdos construction to prove lower bounds for dynamic range query data structures in the plane. 5 ....

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M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10:1-- 10, 1981.

....query time; that is, we prove lower bounds on the size of the data structure as a function of its worst case query time, or vice versa. We also prove tradeoffs between preprocessing time and query time. Most range searching lower bounds are presented in the Fredman Yao semigroup arithmetic model [31, 54]. In this model, the points are given weights from a semigroup, and the goal of a range query is to determine the total weight of the points in a query region. A data structure in this model can be informally regarded as a set of precomputed partial sums in the underlying semigroup. The size of ....

....emptiness queries in both the online and offline settings. Finally, in Section 9, we offer our conclusions. 2 Semigroup Arithmetic 2. 1 Definitions We begin by reviewing the definition of the semigroup arithmetic model, originally introduced by Fredman to study dynamic range searching problems [31], and later refined for the static setting by Yao [54] A semigroup (S; is a set S equipped with an associative addition operator : S Theta S S. A semigroup is commutative if the equation x y = y x is true for all x; y 2 S. A linear form is a sum of variables over the semigroup, where ....

[Article contains additional citation context not shown here]

M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput. 10:1--10, 1981.

....[41] Better lower bounds are known for a few of these problems in less powerful models, but these models are inappropriate in more general settings. Chazelle [14, 13] has proven a number of lower bounds for online and offline range counting problems in the Fredman Yao semigroup arithmetic model [27]. In this model, the points are given arbitrary weights from a fixed semigroup, and the complexity of a problem is given by the worst case number of semigroup additions required to 3 Formally, the reductions in [29] and [41] are quasilinear time many one reductions, and the reductions we ....

....line of any other point Best known upper bound: O(n 4=3 2 O(log n) 36] A number of different results suggest that the true complexity of Hopcroft s problem is Theta(n 4=3 ) Erdos has constructed a set of n points and n lines with Omega Gamma n 4=3 ) point line incidences. See [27]. The existence of such a configuration implies that any algorithm that solves the reporting version of Hopcroft s problem must take time Omega Gamma n 4=3 ) in the worst case. Lower bounds due to Fredman [27] and Chazelle [13] imply that any algorithm that solves the online counting version ....

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M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10:1--10, 1981.

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M.L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Computing, 10:1-10, 1981

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Michael L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10(1):1--10, 1981.

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M. L. Fredman. Lower bounds on the complexity of some optimal data structures. SIAM J. Comput., 10:1--10, 1981.

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