J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Step 1: how to construct the concave chains second option. We now offer a different method for step 1 that also achieves O(n log n) expected time and has the advantage of being generalizable to 3 d (as we will see in Section 3) This second option is based on Matousek s shallow cutting lemma [37]. Ramos [46] building on [2, 14] has given a corresponding randomized construction in 3 d, which will be useful to us. We restate the 3 d result below in a form we find convenient. Lemma 2.1 Given n lower halfspaces in IR , the ( k) level can be covered by O(n=k) cells, each intersecting ....

....O(k) bounding planes. Moreover, the cells form the vertical decomposition of the region underneath some concave surface 0 of size O(n=k) The cells, the list of planes intersecting each cell, and 0 can all be constructed in O(n log n) expected time. Proof: Matousek s shallow cutting lemma [37] and Ramos algorithm [46] give a set of cells (tetrahedra) satisfying the first statement in O(n log n) expected time. To get the second statement, we need to fiddle with the construction. Say the maximum number of planes intersecting a cell is less than bk. Remove a cell from if one of its ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....data structure had query cost O(n log 4 3 t) for d = 2. This was improved by a series of papers, most notably by the seminal result of Haussler and Welzl [HW87] who derived a cost of O(n 1 d(d 1) # t) Vari20 ous improvements followed, culminating with the result of Matousek [Mat92] who obtained O(n 1 1 #d 2# polylog(n) t) query cost, still with linear space. These techniques were adapted to external memory by the works of Agarwal et al. AAE 98] Kollios, Gunopoulos and Tsotras [KGT99] and Agarwal, Arge and Erickson [AAE00] Another series of techniques began ....

....the concept of arrangements. For the planar case, their technique achieves optimal time O(log n t) with linear space. However, generalizing the approach to higher dimensions, introduces non linear space, typically exponential to the problem dimension. For example, the extension of Matousek [Mat92] can answer halfspace queries in in time O(log n t) but requires O(n #d 2# polylog(n) space. However, the technique is useful at least in low dimensions, and has been externalized by Agarwal et al. AAE 98] 2.1.8 Lower bounds In many geometric range search problems, there is no ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....of a hyperplane can be done in time O(dn) with a trivial algorithm. This is repeated O(c log n=c) times over the entire algorithm. For fixed d, the whole process can be done in expected time O(n 1 ffi (nc) 1= 1 fl= 1 ffi) using sophisticated data structures for half plane range queries [Mat92]. Finally, note that the expected number of oracle calls (half plane emptiness tests) is O(fc log(n=c) where f is the number of facets of conv(R) This can be avoided when L is a polytope, see below. 4 Particular measures In this section we describe several implementations of find bad facet ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....of a fixed degree polynomial) can be reported in O(m k) time. 1 It can also determine, in O(m ) time, whether any point of P lies below a query curve. A point can be inserted or deleted into from P in O(log 2 m) time. Although our approach is closely based on Matousek s algorithm [22] for reporting points that lie below a query line, a number of technical difficulties have to be overcome to extend this algorithm to the case of algebraic curves. A similar data structure also works for pseudolines that are extensions of circular arcs; see below for a formal definition of such ....

....S is a family Xi = f(S 1 ; 4 1 ) S u ; 4 u )g, where S 1 ; S u form a partition of S, 4 i is a pseudo trapezoid, and S i 4 i . The following lemma, INCIDENCES AND MANY FACES IN PSEUDOLINE ARRANGEMENTS 8 whose proof is omitted, is obtained by extending the results of Matousek [22] and Agarwal and Matou sek [3] Lemma 4.3 Let S and Gamma be as defined above, and let r be a parameter. Then there exists an elementary partition Xi = f(S 1 ; 4 1 ) S u ; 4 u )g of S so that m=r jS i j 2m=r and any (m=r) shallow arc of Gamma crosses O(log m) pseudo trapezoids of ....

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J. Matousek, Reporting points in halfspaces, Comput. Geom. Theory Appl., 2 (1992), 169--186.

....) 1=2 t) I Os. If we allow O(n log B n) space, the query time can be improved considerably, at least theoretically. In this abstract we only outline the basic approach. A line is called k shallow with respect to S, for k N , if at most k points of S lie below . Theorem 5. 2 (Matousek [32]) Let S be a set of n points in the plane, and let 1 r n=2 be a given parameter. Then there exists a balanced simplicial partition Pi of S so that any (N=r) shallow line crosses at most fi log 2 r triangles of Pi for some constant fi 1. Using this theorem we construct a partition tree ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....been studied extensively, especially in low dimension, where good solutions are known (see, for example [9] However, the combinatorial complexity of arrangements grows exponentially with the dimension, rendering the problem seemingly intractable. Indeed, following a long list of contributions [18, 12, 36, 28, 1, 29], currently the best algorithms can find a nearest neighbor in time poly(d; log n) but they need exponential (n Theta(d) storage. On the other hand, there is little evidence in the form of concrete lower bounds to support the curse of dimensionality conjecture [13] i.e. the belief that in ....

J. Matousek. Reporting points in halfspaces. In Proc. of 32nd FOCS, pp. 207--215, 1991.

....query halfspace is not shallow, for some = Omega Gamma n) then a simplex rangereporting data structure can be used to answer a query in time O(n 1 Gamma1=d k) O(k) For shallow hyperplanes, Matousek proves the following theorem, which is an analog of Theorem 4.2. Theorem 4. 3 (Matousek [179]) Let S be a set of n points in R d (d 4) and let 1 r n be a given parameter. Then there exists a family of pairs Pi = f(S 1 ; Delta 1 ) S m ; Delta m )g so that each S i S lies inside the simplex Delta i , n=r jS i j 2n=r, S i S j = for all i 6= j, and every ....

....can be improved slightly; see Table 3 for a summary of results. Problem d Size Query Time Source Reporting d = 2 n log n k [74] Emptiness d = 2 n log n [219] Reporting d = 3 n log n log n k [17] Emptiness d = 3 n log n [98] Reporting d 3 n log log n n 1 Gamma1=bd=2c polylog n k [179] Emptiness d 3 n n 1 Gamma1=bd=2c 2 O(log n) 179] Table 3. Asymptotic upper bounds for halfspace range searching in near linear space. Since the query time of a linear size simplex range searching data structure is only n 1=d factor better than the naive method, researchers have ....

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J. Matousek, Reporting points in halfspaces, Comput. Geom. Theory Appl., 2 (1992), 169--186.

....polyhedra satisfy this ideal condition. The result for terrains in [8] does not, since the best known algorithm for testing the intersection of two terrains has complexity O(n 4=3 ffl ) 8] In Section 5 we combine the approach of Chazelle et al. with recent results on halfspace range queries [20] obtaining a total running time O(n 4=3 ffl k 1=3 ffl n 1 ffl k log 2 n) This algorithm matches the best known intersection testing algorithm for k n, and is almost optimal for k n 3=2 (within a polylogarithmic factor) We refer to the algorithm in [8] as the CEGS Algorithm. For ....

....for inclusion in a 5 dimensional polytope in Plucker space. If we interpret p 0 (l; v) and p(l) as hyperplanes we can answer equivalent problems which are an half space range problem in 1 space and a half space emptiness problem in 5 space. Using a multi level data structure approach in [2] and [20] (see also [21] for an abstract treatment of multi level data structures) we obtain that such test can be computed in time O(n 1 ffl =s 1=2 ) using a data structure of size s with jB j s jB j 2 . The time used to build the data structure is O(s 1 ffl ) 5.3 The overall algorithm In ....

J. Matousek. Reporting points in halfspaces. In Proceedings of the 32th IEEE Symposium on Foundations of Computer Science, pages 207--215, 1991.

....of S. Using random sampling, Blumer et al. BEHW89] proved the following important result: any set system of VCdimension d has an # net of cardinality O( d # log 1 # ) Notice that this bound is independent of n, the number of points in the set system. More recently, Matousek and others [Mat92, MSW90] have given deterministic algorithms for finding # nets. These deterministic algorithms find frequent use in computational geometry as replacements for random sampling techniques. Indeed, Bronniman and Goodrich s algorithm can be seen as an application of # nets to derandomize an algorithm ....

J. Matousek. Reporting points in halfspaces. Computational Geom. Theory & Appl., 2:169--186, 1992.

.... in [2] that Chazelle et al. 5] reduce the problem of detecting if a line is above or below a set of lines to the problem to answering halfspace emptiness queries in Plucker space (5 dimensional projective space) This reduction, together with the method for solving halfspace emptiness queries in [13], gives us a new envelope structure that, for a set of M lines and for a parameter s in the range M s M 2 , is built in time O(s 1 ffl ) using O(s) storage. Given a query line l we can determine in time O(M 1 ffl =s 1=2 ) whether l is above all M lines or below all M lines. We call this ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169-- 186, 1992.

....O(k log 2 n) query time and O(n log n) preprocessing time and space. For a larger fixed dimension d 4, Clarkson and Shor [32] used shallow cuttings to achieve O(log n k) query time with O(n bd=2c ) preprocessing time and space, where is an arbitrarily small positive constant. Matousek [48] obtained a certain partition theorem that implies a data structure with O(n log n) preprocessing time, O(n log log n) space, and O(n 1 Gamma1=bd=2c log O(1) n k) query time. This method can be specialized to handle the d = 3 case; the query time appears to be O( log n) O(loglog n) k) ....

....then H Delta is just the union of the lists of planes of H below the v j s. One other important definition is the following: a simplex Delta is relevant if it intersects the region lev k (H) The sampling lemma below is stated implicitly in the proof of Matousek s shallow cutting lemma [48] (see Appendix A.2) and is needed in the analysis of our algorithm. Matousek s proof uses probabilistic arguments of Chazelle and Friedman [27] see Appendix A.1 for more details. Lemma 3.1 Let 1 r n and q = kr=n 1. Let f(n) be a regular function, i.e. a nondecreasing function satisfying ....

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J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....ratio) by switching to the greedy method when c becomes too large. The algorithm of Theorem 5.2 has been animated into a video [8] and Figure 2 shows snapshots illustrating different stages. But in three dimensions, we can actually do much better. In particular, we recall from Matousek et al. [39, 38] that the three dimensional halfspace set system admits (1=c) nets of size O(c) and as indicated in Section 3.2, their algorithm can be extended to the weighted case as well. The cost of computing a (1=c) net of size O(c) is O(nc) if c n ffi [38] and the cost of verifying that is it a ....

....particular, we recall from Matousek et al. 39, 38] that the three dimensional halfspace set system admits (1=c) nets of size O(c) and as indicated in Section 3.2, their algorithm can be extended to the weighted case as well. The cost of computing a (1=c) net of size O(c) is O(nc) if c n ffi [38], and the cost of verifying that is it a hitting set is O(n log n) as argued above. Therefore we obtain the following strengthening of our previous theorem: Theorem 5.2 Let Q P be two nested polyhedra in R 3 with a total of n facets, one of them being convex. It is possible to ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....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 ....

....kinds of intriguing questions immediately arise. If we insist on reporting the points, then the halfspace problem is considerably easier than the simplex version. For example, if we wish to achieve optimal query time, the former can be solved using only O(n bd=2c ) space, for any fixed ffl 0 [11, 21]. By contrast, simplex range reporting provably requires Omega Gamma n d Gammaffl ) storage (on a pointer machine) 8] In the counting version of the problem, however, no one has yet been able to take (significant) advantage of the fact that a halfspace is a restricted form of simplex to ....

Matousek, J. Reporting points in halfspaces, Proc. 32nd Ann. Symp. Found. Comput. Sci. (Oct.1991), 207--215.

....data structure in O(n log n) time, that allows halfspace emptiness queries to be answered in logarithmic time [3, 10, 14] In higher dimensions, a randomized algorithm due to Clarkson [12] answers halfspace emptiness queries in time O(log n) after O(n bd=2c ) 1 preprocessing time. Matousek [19] describes two halfspace emptiness data structures, one answering queries in time O(n 1;1=bd=2c polylog n) time after O(n log n) preprocessing time, and the other answering queries in time O(n 1;1=bd=2c 2 O(log n) 2 after O(n 1 ) preprocessing time. Combining Clarkson s and ....

....other answering queries in time O(n 1;1=bd=2c 2 O(log n) 2 after O(n 1 ) preprocessing time. Combining Clarkson s and Matousek s data structures, for a fixed parameter n s n bd=2c , one can answer queries in time O( n log n) s 1=bd=2c ) after O(s polylog n) preprocessing time [19,1,7]. For extensions and applications of halfspace range reporting, see [1,2,6,7,22,20] Given n points and m halfspaces, we can solvethe offline halfspace emptiness problem in time O i n log m (nm) bd=2c= bd=2c 1) polylog(n m) m log n j # using either Clarkson s data structure or ....

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J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....storage can be used to answer queries quickly. Dobkin and Lipton use O i n 2 d 1 j storage to allow O(2 d log n) search time. Clarkson [15] improves the storage requirement to O Gamma n (1 ffi)dd=2e Delta , paying d O(d) log n search time. Improvements by Yao and Yao [48] Matousek [36], and Agarwal and Matousek [1] still give exponential in d storage and search time. Finally, Meiser [37] gives the best result to date (in terms of search time) O(d 5 log n) search time using O Gamma n 2d ffi Delta storage. 3 2 For definiteness we say that a two sided error protocol ....

J. Matousek. Reporting points in halfspaces. In Proc. of 32nd FOCS, 1991.

....) into a halfspace (b) R d 1 . Let H i denote the intersection of the complements of these halfspaces (for i = 1 : n) Surely, q 2 L i if and only if (q) 2 H i . To check this condition, we preprocess H i in time O(n dd=2e ffi ) into a point location data structure D i , of Matousek [11], so that determining if q 2 H i is obtained in time O(log n) Hence determining if q 2 T ( is obtained in time O(n log n) We perform this query for each point of S (in time O(n d3d=2e 1 log n) Note that D i needs to be constructed only for a single L i , since the layers are just ....

J. Matousek, Reporting Points in Halfspaces, Comput. Geom. Theory Appl., 2 (1992), 169--186.

.... Xi of simplices which completely cover the cell of the arrangement of H containing the origin (and perhaps something more) such that no Delta 2 Xi is intersected by more than n hyperplanes of H . Existence of 0 shallow (1=r) cuttings of size O(r bd=2c ) for any H and r is established in [Mat92b] by the method of [CF90] and a deterministic algorithm is given for constructing them in O(nr bd=2c Gamma1 ) time, but only for r n 1 Gammaffi with an arbitrary positive constant ffi 0. For even dimension d, one can do better using the vertex accounting mechanism (see [AGR94] but in ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....was suggested by Clarkson [21] take a random sample R ae H of size O( 1=ffi) log(1=ffi) and canonically triangulate the arrangement of R. Derandomization of this method and refinements on the constants are discussed in several papers [16, 40, 48, 52] For a variant known as shallow cuttings [49], better bounds are still possible. Since the size of the cutting is not important when applying Lemma 2.1 (as long as it is bounded by a constant) the original method of Dyer and Megiddo or the randomized method of Clarkson is sufficient for our purposes. 5.1 Linear Programming with ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....bounds in higher dimensions, including preprocessing query time tradeoffs, are given by Theorem 4.1. Although the proof of these output sensitive bounds requires no new ideas (the main ingredients being Matousek s shallow cutting lemma and partition theorem for shallow hyperplanes [27]) they are worth noting due to the fundamental nature of range searching. 2 General Randomized Reductions In this section, we study the distance enumeration and selection problems in an abstract setting and describe general reductions that will be of use in some of the concrete algorithms of the ....

.... n point set P ae IR d in O (m) time and space such that the number k of points inside a query halfspace can be determined in time O 0 1 n m 1=bd=2c n bd=2c k dd=2e m 1=d 1 A : The previous query bounds are O (1 n=m 1=d ) 10, 26] and O (1 n=m 1=bd=2c k) [27]. The proof of this theorem is deferred to the appendix. In particular, it implies the following result: Corollary 4.2 An online sequence of n halfspace range counting queries on a given n point set in IR d require time O (n 2 Gamma2= bd=2c 1) n 2bd=2c k dd=2e ) 1= d 1) ....

[Article contains additional citation context not shown here]

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....O(k log 2 n) query time and O(n log n) preprocessing time and space. For a larger fixed dimension d 4, Clarkson and Shor [29] used shallow cuttings to achieve O(log n k) query time with O(n bd=2c ) preprocessing time and space, where is an arbitrarily small positive constant. Matousek [43] obtained a certain partition theorem that implies a data structure with O(n log n) preprocessing time, O(n log log n) space, and O(n 1 Gamma1=bd=2c polylog n k) query time. This method can be specialized to handle the d = 3 case; the query time appears to be O( log n) O(log log n) k) ....

....then H Delta is just the union of the lists of planes of H below the v j s. One other important definition is the following: a simplex Delta is relevant if it intersects the region lev k (H) The sampling lemma below is stated implicitly in the proof of Matousek s shallow cutting lemma [43] (see Appendix A.2) and is needed in the analysis of our algorithm. Matousek s proof uses probabilistic arguments of Chazelle and Friedman [24] see Appendix A.1 for more details. Lemma 3.1 Let 1 r n and q = kr=n 1. Let f(n) be a regular function, i.e. a nondecreasing function satisfying ....

[Article contains additional citation context not shown here]

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....data (from speech compression applications) to show that the general approach is significantly faster than the known practical approaches to the problem, including the k d tree. Matousek has conjectured that Omega Gamma n 1 Gamma1=bd=2c ) is a lower bound for the halfspace emptiness problem [14] (assuming linear space) The ball emptiness problem is a generalization of this problem (to spheres of infinite radius) and so any lower bound would also hold. However the ball emptiness problem can be reduced to a single nearest neighbor query (at the center of the ball) In light of this, if ....

J. Matousek. Reporting points in halfspaces. In Proc. 32nd Ann. Sympos. Foundations of Computer Science, pages 207--215, 1991.

....= Omega Gamma n) then a simplex rangereporting data structure can be used to answer a query in time O(n 1 Gamma1=d k) O(k) For shallow hyperplanes, Matousek proves the following theorem, which is an analog of 24 Pankaj K. Agarwal and Jeff Erickson Theorem 3. Theorem 4 (Matousek [199]) Let S be a set of n points in R d (d 4) and let 1 r n be a given parameter. Then there exists a family of pairs Pi = f(S 1 ; Delta 1 ) S m ; Delta m )g such that each S i S lies inside the simplex Delta i , n=r jS i j 2n=r, S i S j = for all i 6= j, and every ....

....the data structure can be improved slightly; see Table 3 for a summary of results. Problem d Size Query Time Source Reporting d = 2 n log n k [78] Emptiness n log n [242] Reporting d = 3 n log n log n k [16] Emptiness n log n [103] Reporting d 3 n log log n n 1 Gamma1=bd=2c polylog n k [199] Emptiness n n 1 Gamma1=bd=2c 2 O(log n) 199] Table 3. Asymptotic upper bounds for halfspace range searching in near linear space. Since the query time of a linear size simplex range searching data structure is only a n 1=d factor better than the naive method, researchers have developed ....

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J. Matousek, Reporting points in halfspaces, Comput. Geom. Theory Appl., 2 (1992), 169--186. 60 Pankaj K. Agarwal and Jeff Erickson

....bounds in higher dimensions, including preprocessing query time tradeoffs, are given by Theorem 4.1. Although the proof of these output sensitive bounds requires no new ideas (the main ingredients being Matousek s shallow cutting lemma and partition theorem for shallow hyperplanes [26]) they are worth noting due to the fundamental nature of range searching. 2 General Randomized Reductions Let U represent the object space and d : U Theta U IR represent the distance function, where d(p; q) d(q; p) for any p; q 2 U . Notation. Given sets P; Q ae U , define the multiset ....

.... an n point set P ae IR d in O (m) time and space such that the number k of points inside a query halfspace can be determined in time O 1 n m 1=bd=2c n bd=2c k dd=2e m 1=d : The previous query bounds are O (1 n=m 1=d ) 9, 25] and O (1 n=m 1=bd=2c k) [26]. Choosing the tradeoff parameter m appropriately, guessing the output size k, and applying Holder s inequality, we get: Corollary 4.2 An online sequence of n halfspace range counting queries on a given n point set in IR d require time O (n 2 Gamma2= bd=2c 1) n 2bd=2c k dd=2e ) ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....a standard duality transform, this problem can be reduced to the Range Searching 11 TABLE 4 Halfspace range searching. d S(n) Q(n) Source Notes d = 2 n log n k [41] Reporting d = 3 n log n log n k [14] Reporting d = 3 n log n [52] Emptiness d 3 n log log n n 1 Gamma1=bd=2c log c n [79] Reporting d 3 n n 1 Gamma1=d 2 O(log n) 79] Emptiness following problem: Given a set H of n hyperplanes, determine the number of hyperplanes of H lying above a query point. Since the same subset of hyperplanes lies above all points in a single cell of A(H) the arrangement of H , we ....

....to the Range Searching 11 TABLE 4 Halfspace range searching. d S(n) Q(n) Source Notes d = 2 n log n k [41] Reporting d = 3 n log n log n k [14] Reporting d = 3 n log n [52] Emptiness d 3 n log log n n 1 Gamma1=bd=2c log c n [79] Reporting d 3 n n 1 Gamma1=d 2 O(log n) [79] Emptiness following problem: Given a set H of n hyperplanes, determine the number of hyperplanes of H lying above a query point. Since the same subset of hyperplanes lies above all points in a single cell of A(H) the arrangement of H , we can answer a halfspace range counting query by locating ....

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J. Matousek, Reporting points in halfspaces, Comput. Geom. Theory Appl., 2 (1992), 169--186.

.... jeffe We derive a lower bound of Omega Gamma n log m n 2=3 m 2=3 m log n) for the following halfspace emptiness problem: Given a set of n points and m hyperplanes in IR 5 , is every point above every hyperplane This matches the best known upper bound to within polylogarithmic factors [3], and improves the previous best lower bound of Omega Gamma n log m m log n) 1] Our lower bounds apply to polyhedral partitioning algorithms, a restriction of the class of partitioning algorithms introduced to study the complexity of Hopcroft s problem [2] Informally, a polyhedral ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....than the known practical approaches to the problem, including the k d tree. We also show the following purely theoretical result on exact nearest neighbor queries (improving Yao and Yao s bound [20] It follows from an application of recent results on halfspace range reporting due to Matousek [13]. Theorem 1.2. Given a set of n points in E d , one can preprocess it for exact nearest neighbor queries in O(n log n) time and O(n log log n) space, such that queries can be answered in O(n 1 Gamma1=b(d 1) 2c (log n) O(1) expected time. Matousek has conjectured that Omega Gamma n ....

....it for exact nearest neighbor queries in O(n log n) time and O(n log log n) space, such that queries can be answered in O(n 1 Gamma1=b(d 1) 2c (log n) O(1) expected time. Matousek has conjectured that Omega Gamma n 1 Gamma1=bd=2c ) is a lower bound for the halfspace emptiness problem [13] (assuming linear space) The ball emptiness problem is a generalization of this problem (to spheres of infinite radius) and so any lower bound would also hold. However the ball emptiness problem can be reduced to a single nearest neighbor query (at the center of the ball) In light of this, if ....

[Article contains additional citation context not shown here]

J. Matousek. Reporting points in halfspaces. In Proc. 32nd Ann. Sympos. Foundations of Computer Science, pages 207--215, 1991.

.... logarithmic query time [10, 25] whereas, the same query time can be achieved Space Preprocessing Query Time Source d = 2; 3 O(n) O(n log n) O(log n) 13, 3, 16] d 4 O(n bd=2c = log bd=2c n) O(n bd=2c = log bd=2c n) O(log n) 27] O(n) O(n 1 ) O(n 1 1=bd=2c 2 O(log n) [22] O(n) O(n log n) O(n 1 1=bd=2c polylog n) 27] n s n bd=2c O(spolylog n) O( npolylog n) s 1=bd=2c ) 27] Table 2. Best known upper bounds for halfspace emptiness queries. with o(n bd=2c ) space if we only want to know whether the halfspace is empty [27] Table 2 lists the resource ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl. 2(3):169--186, 1992.

....space, the problem was first considered by Dobkin and Lipton [11] They showed an exponential in d search algorithm using (roughly) a doubleexponential in d (summing up time and space) data structure. This was improved and extended in subsequent work of Clarkson [5] Yao and Yao [34] Matousek [27], Agarwal and Matousek [1] and others, all requiring query time exponential in d. Recently, Meiser [28] obtained a polynomial in d search algorithm using an exponential in d size data structure. For approximate nearest neighbor search, Arya et al. 3] gave an exponential in d time search ....

J. Matousek. Reporting points in halfspaces. In Proc. of 32nd FOCS, 1991.

....(although a small one) over the lower bound in Corollary 4.3. 24 Jeff Erickson Space Preprocessing Query Time Source d = 2; 3 O(n) O(n log n) O(log n) 20, 3, 25] d 4 O(n bd=2c = log bd=2c n) O(n bd=2c = log bd=2c n) O(log n) 42] O(n) O(n 1 ) O(n 1 1=bd=2c 2 O(log n) [37] O(n) O(n log n) O(n 1 1=bd=2c polylog n) 42] n s n bd=2c O(spolylog n) O( npolylog n) s 1=bd=2c ) 42] Table 2. Best known upper bounds for halfspace emptiness queries. 8 Halfspace Emptiness Queries The space and time bounds for the best hyperplane (or simplex) emptiness query data ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl. 2(3):169--186, 1992.

....ratio) by switching to the greedy method when c becomes too large. The algorithm of Theorem 5.2 has been animated into a video [8] and Figure 2 shows snapshots illustrating different stages. But in three dimensions, we can actually do much better. In particular, we recall from Matousek et al. [39, 38] that the three dimensional halfspace set system admits nets of size O(1= and as indicated in Section 3.2, their algorithm can be extended to the weighted case as well. The cost of the computation of a (1=c) net of size O(c) is O(nc) if c n ff [38] and the cost of verifying is O(n log n) ....

....we recall from Matousek et al. 39, 38] that the three dimensional halfspace set system admits nets of size O(1= and as indicated in Section 3.2, their algorithm can be extended to the weighted case as well. The cost of the computation of a (1=c) net of size O(c) is O(nc) if c n ff [38], and the cost of verifying is O(n log n) as argued above. Therefore we obtain the following strengthening of our previous theorem: Theorem 5.2 Let Q P be two nested polyhedra in IR 3 with a total of n facets, one of them being convex. It is possible to deterministically find a separator of ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....algorithms for the above problems for various types of the set L. In fact, this approach works for any set L for which we can determine whether it lies entirely on one side of a query hyper plane. Such a test has been called a strong violation oracle [GLS88] or half space emptiness query [Mat92] When L is the unit ball centered at the origin, the corresponding optimization problems will be called MaxScale B and MinCover B, and when L is a polytope containing the origin, L will be replaced by P . The dimensions encountered will be d = 6 for objects in three dimensions, and d = 3 ....

....on one side of a hyper plane can be done in time O(dn) with a trivial algorithm. This is repeated O(c log(n=c) times over the entire algorithm. For variable d, this incurs a total cost of O(ncd log(n=c) However, for fixed d, using sophisticated data structures for half plane range queries [Mat92] the time complexity can be significantly improved by trading off preprocessing time for query time (see [Cla93] The resulting time complexity is O i n 1 ffi (nc) 1= 1 fl= 1 ffi) j ; where ffi is an arbitrarily small positive number and fl = 1=bd=2c. Finally, let us describe the ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....data structure in O(n log n) time, that allows halfspace emptiness queries to be answered in logarithmic time [3, 10, 14] In higher dimensions, a randomized algorithm due to Clarkson [12] answers halfspace emptiness queries in time O(log n) after O(n bd=2c ) 1 preprocessing time. Matousek [19] describes two halfspace emptiness data structures, one answering queries in time O(n 1 Gamma1=bd=2c polylog n) time after O(n log n) preprocessing time, and the other answering queries in time O(n 1 Gamma1=bd=2c 2 O(log n) 2 after O(n 1 ) preprocessing time. Combining Clarkson s ....

....answering queries in time O(n 1 Gamma1=bd=2c 2 O(log n) 2 after O(n 1 ) preprocessing time. Combining Clarkson s and Matousek s data structures, for a fixed parameter n s n bd=2c , one can answer queries in time O( n log n) s 1=bd=2c ) after O(s polylog n) preprocessing time [19, 1, 7]. For extensions and applications of halfspace range reporting, see [1, 2, 6, 7, 22, 20] Given n points and m halfspaces, we can solve the offline halfspace emptiness problem in time O i n log m (nm) bd=2c= bd=2c 1) polylog(n m) m log n j ; using either Clarkson s data structure or ....

[Article contains additional citation context not shown here]

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....was suggested by Clarkson [17] take a random sample R ae H of size O( 1=ffi) log(1=ffi) and canonically triangulate the arrangement of R. Derandomization of this method and refinements on the constants are discussed in several papers [12, 33, 38, 42] For a variant known as shallow cuttings [39], better bounds are still possible. Since the size of the cutting is not important when applying Lemma 2.1 (as long as it is a constant) the original construction of Dyer and Megiddo is sufficient for our purposes. Feasible case. As a first example, we consider the linear programming problem ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169--186, 1992.

....is completed by observing, as in the proof of Theorem 2.3, that E[N ] is at least half the total number of k th level vertices. 2 Instead of the above probabilistic argument, an alternative proof can be obtained from results on geometric cuttings, specifically, the shallow cuttings of Matousek [23]: Let H be a collection of n hyperplanes in IR d in general position. A (1=r) cutting of the first k levels is a collection of simplices f i g covering all points of levels k, such that jH i j n=r for each i, where H i is the set of all hyperplanes of H that intersect the interior of i ....

..... Hence, the total number of k th level vertices is O( n=k) Delta k 8=3 ) O(nk 5=3 ) However, we prefer the earlier proof, as the proof of the shallow cutting lemma itself requires actually more involved probabilistic techniques, including an argument similar to one of the previous proofs [23]. 2 Remark: As before, any improvement in the worst case k insensitive upper bound would imply an improvement in our k sensitive bound. Extensions to d dimensional arrangements of hyperplanes are also immediate: if we have an O(n d Gammac d ) bound on the complexity of a single level, then ....

J. Matousek, Reporting points in halfspaces, Comput. Geom. Theory Appl. 2 (1992), 169--186.

.... additive terms of the form O(polylog n) or O(n ) All the data structures we describe can be constructed in time O(s 1 ) and can be modified to support insertions and deletions in time O(s 1 =n) at the expense of at most an O(n ) factor in both the preprocessing and query times [1, 5, 43]. We also Raising Roofs, Crashing Cycles, and Playing Pool 5 use the standard technique of composing several geometric range searching data structures into a single multi level data structure. For further details on space time tradeoffs, dynamization, multilevel data structures, and other ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl. 2(3):169--186, 1992.

....algorithms for the above problems for various types of the set L. In fact, this approach works for any set L for which we can determine whether it lies entirely on one side of a query hyper plane. Such a test has been called a strong violation oracle [GLS88] or half space emptiness query [Mat92] When L is a ball centered at the origin, the corresponding optimization problems will be called MaxScale B and MinCover B, and when L is a polytope containing the origin, L will be replaced by P . The dimensions encountered will be d = 6k for k objects in three dimensions, and 3k for k ....

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

....storage can be used to answer queries quickly. Dobkin and Lipton use O i n 2 d 1 j storage to allow O(2 d log n) search time. Clarkson [14] improves the storage requirement to O Gamma n (1 ffi)dd=2e Delta , paying d O(d) log n search time. Improvements by Yao and Yao [46] Matousek [34], and Agarwal and Matousek [1] still give exponential in d storage and search time. Finally, Meiser [35] gives the best result to date (in terms of search time) O(d 5 log n) search time using O Gamma n 2d ffi Delta storage. 3 In the approximate nearest neighbor search (approximate ....

J. Matousek. Reporting points in halfspaces. In Proc. of 32nd FOCS, 1991.

.... by Meiser [24] to O(d 5 log n) with pre processing O(n d ffl ) Recently, Kleinberg [17] has developed a scheme for approximate nearest neighbor problem that achieves query time O(d 2 log n) with preprocessing n O(d) There have been a number of other approaches and extensions (e.g. [31, 23, 25, 1, 2]) The best approaches from these studies are still impractical for the values of d encountered in the retrieval applications above. Overview of Paper. In Section 3 we give a construction for locality preserving hash functions in two dimensions; this is extended to higher dimensions in Section 4. ....

J. Matousek, "Reporting points in halfspaces," Proc. 32nd FOCS, 1991.

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J. Matousek. Reporting points in halfspaces. In Proc. 32. IEEE Symposium on Foundations of Computer Science, pages 207--215, 1991. 14

....O(n 1 Gamma1=bd=2c (log n) O(1) If m is a parameter, n m n bd=2c , and if space and preprocessing time O(m 1 ffi ) for some fixed ffi 0 are allowed, a query time O( n m 1=bd=2c log 4d Gamma1 n) can be achieved. The proof combines recent results on halfspace range reporting ([Mat91b]) with a multidimensional version of Megiddo s parametric search technique (a similar technique has been applied previously e.g. in [CSY87] CM89] NPT90] With a small modification of this data structure and when the query time for n queries and the preprocessing time are balanced ....

....quickly determine whether fl contains some point of P . We will call this the halfspace emptiness problem (there are actually two versions of this problem, since we may consider an open halfspace or a closed one) The best known results for this problem in dimension d 4 were recently attained in [Mat91b]. Here we outline a relatively simple solution. To this end, we will need several definitions. Let P be an n point set in E d . A simplicial partition of P is a collection Pi = f(P 1 ; s 1 ) Pm ; s m )g; where the P i s are nonempty sets (called the classes of Pi) forming a ....

[Article contains additional citation context not shown here]

J. Matousek. Reporting points in halfspaces. In Proc. 32. IEEE Conference on Foundations of Computer Science, 1991.

.... one can store S in a data structure of size O(n #d 2# # ) so that a half space query can be answered in time O(log n k) 12] Recently Matousek showed that a half space query can be answered in time O(n 1 1 #d 2# log O(1) n k) using O(n log n) preprocessing time and O(n log log n) space [24]. Combining these two approaches, for a 1 Throughout this paper we assume d to be some fixed positive integer and # to be an arbitrarily small constant. The constants in the time complexity of the algorithms depend on d and #. given parameter m, n # m # n #d 2# , one can answer a half space ....

....to the points in S. Answering an empty half space query for S reduces to determining whether a query point p lies above all hyperplanes of H; we refer to the dual problem as the upper envelope problem for H. We describe our data structure in this dual formulation. We need the following results of [24]. The (# k) level of H is the set of points p # R d such that at most k hyperplanes of H lie (strictly) above p. For a parameter r # n, we define a (1 r) cutting of (# k) level of H to be a set # of pairwise disjoint simplices such that # covers the (# k) level of H and that each simplex ....

[Article contains additional citation context not shown here]

J. Matousek. Reporting points in halfspaces. Proc. 32nd IEEE Symp. Found. Computer Science (1991) 207--215.

No context found.

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

No context found.

J. Matousek, "Reporting points in halfspaces," Proc. 32nd IEEE FOCS, 1991.

No context found.

J. Matousek. Reporting points in halfspaces. In Proc. of 32nd FOCS, 1991.

No context found.

J. Matousek. Reporting points in halfspaces. In Proc. of 32nd FOCS, 1991.

No context found.

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

No context found.

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

No context found.

J. Matousek, Reporting Points in Halfspaces, Comput. Geom. Theory Appl., 2 (1992), 169--186.

No context found.

J. Matousek, Reporting Points in Halfspaces, Comput. Geom. Theory Appls. 2 (1992), 169--186.

No context found.

J. Matousek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2(3):169--186, 1992.

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