D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... We finally note that our approach could also be applied to other, non geometric optimization problems, provided that the search for the optimal value of the parameter can be guided by an appropriate range searching mechanism, for which the underlying range space has finite VC dimension (see [14, 22] for details) However, we will not elaborate on this possibility, and leave it as another open problem to explore. Acknowledgement We wish to thank Noga Alon for several helpful discussions that provided valuable information concerning expanders and their applications, and Pankaj Agarwal for ....

D. Haussler and E. Welzl, Epsilon nets and simplex range queries, Discrete Comput. Geom. 2 (1987), 127--151.

....30, 2000 Abstract The path VC dimension of a graph G is the size of the largest set U of vertices of G such that each subset of U is the intersection of U with a subpath of G. The VC dimension for graphs was introduced by Kranakis, et al. KKR 97] building on an idea of Haussler and Welzl [HW87] We show that computing the path VC dimension of a graph is # 3 complete. This adds a rare natural 3 complete problem to the repertoire. 1 Introduction A set system is said to shatter a set A if for each subset S of A there is a set C such that S = A#C. If shatters a set of ....

....has Vapnik Cervonenkis dimension (VC dimension) at least k. The VC dimension was introduced by Vapnik and Cervonenkis in their study of uniform convergence of relative frequencies. It has become a successful tool in areas such as computaional learning theory, and computational geometry [AB92, HW87] Roughly speaking, it is used to measure the complexity of set systems. For example, the VC dimension of a concept class is finite precisely if the class is learnable in the PAC learning model, and the VC dimension of the class can be used to determine the necessary sample size. It is not ....

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David Haussler and Emo Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2:127--151, 1987.

....s as (t; s) t s k( tu svk : 1) Note that (t; s) is a concave function on a because it is a quotient of a linear function and a convex function. The lemma is an easy consequence of the concavity property. 2 Using Lemma 4. 1 and the standard random sampling technique [9], we construct a two level data structure to decide whether (A; B) The rst level constructs a complete bipartite decomposition for the pairs (a; b) 2 A B for which (a ; b) The second level processes each bipartite clique A i B i in the decomposition, checking whether (a ; b) ....

....2 A j (e ) 2 g, let B B be the set of edges b for which the surface (b) crosses , and let B B be the set of edges b for which the surface (b) lies completely below . Set n = jA j and m = jB j. Obviously, n = n and jB j m. By the theory of random sampling [9], m m=r with high probability, for all . By construction, a ; b) for any pair e 2 A and b 2 B . We use the second level data structure, sketched below, to determine whether (A ; B ) If m or n is less than a prespeci ed constant, then we use a naive procedure ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.

....though some of the results also extend to other MAX metrics. We are not aware of any previous provable results for SUM metrics. Techniques: Our algorithms are inspired by and closely related to the algorithms for Set Cover with bounded VC dimension in [5] which are based on the use of nets [12]. The analysis of the algorithms uses an argument presented by Clarkson in [8] and is also similar to those in [20, 32] 5 Approximation Algorithms In this section, we describe our algorithms, which are all based on the same basic approach. We first focus on the case of ; 698 , and ....

....given quality metric. Definition 5.1 Given a weight function , we say that a partitioning is 6 # good if every tile satisfies 06 ) 8 # 6 ) 8 . We remark that 8 good partitionings correspond to the nets used in [5] and originally introduced in [12], which have found many applications in approximation algorithms and computational geometry. 5.2 Approximating the Number of Tiles for MAX Metrics We now present approximation results for the partitioning problem where the cumulative metric is a MAX metric, i.e. the heft of a partition is ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2, 127--151, 1987.

....a random subset # of size ##, for a sufficiently large constant parameter #, where each subset is chosen with equal probability. Since # is a partition of # ## # #. By the theory of # nets, an appropriate choice of # guarantees that, with high probability, ## ##### ### #,forany# # [19]. This proves part (i) As for (ii) observe that if # ##,fora vertex or edge # in ####,then# is also a vertex or an edge, respectively, in the arrangement of the intersection curves ##. Since this arrangement has ### # vertices and edges, the bound in part (ii) follows. A ....

....## does not lie in any surface of #, therefore by the theory of # nets and with an appropriate choice of #, # ## ##### ### # with high probability, for all such # s. Similarly, one can argue that # ## ##### ### # for each cell # ## , as such a cell lies in exactly one surface of #. See [13, 19] for details. This completes the proof of the lemma. Bounding incidences. Let # be a subset of # satisfying the conditions of Lemma 6.5. We compute # as defined above. Then #### ### ### # ## ##### # # # Since each point in ## lies on every curve in ## and two curves in # intersect in ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127--151.

....E f(r) for any r = 1; 2; n, where the expectation is over a random choice of an r element set R H . Then the expected running time of the algorithm is 2 An net argument. We are going to estimate the function f(r) E . General results of Haussler and Welzl [HW87] imply that for a suitable constant c = c(d) a random r element sample R H has the following property with probability at least 1 Gamma 1=r : Any line segment s that does not intersect any hyperplane of R intersects at most c r log r hyperplanes of H . This is usually expressed by saying ....

....sample R H has the following property with probability at least 1 Gamma 1=r : Any line segment s that does not intersect any hyperplane of R intersects at most c r log r hyperplanes of H . This is usually expressed by saying that R is an net with respect to segments, with = c log r=r [HW87] Let NET(R) be a predicate expressing this property, that is, NET(R) is true if and only if R has this property. We can write, using conditional expectations = E Delta Pr[ NET(R) j j NOT NET(R) Delta Pr[NOT NET(R) Since has never more than O(r ) ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

....Such a routine exists, but it is rather complicated. This example illustrates the power of randomized algorithms: they are often much simpler than their deterministic counterparts, which makes them easier to implement and often faster in practice. Since the pioneering work of Haussler and Welzl [81], Clarkson and Shot [55] and Mulmuley [104] randomized algorithm have become a major design paradigm in computational geome try. They have become so popular that there is now a textbook on computational geometry that focusses on randomized algorithms [108] In this section we will not consider ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Cornput. Geom., 2:127- 151, 1987.

....CutRandomInc depends heavily on the order in which the lines are inserted into the arrangement. Indeed, the set of active trapezoids maintained by CutRandomInc falls outside the classical frameworks of Clarkson and Shor [CS89] lazy randomized incremental construction [dBDS95] and epsilon nets [HW87] See Figures 8.4, 8.5, for situations that illustrate the difference between these frameworks and ours. In order to analyze our algorithms, new techniques need to be developed. In the following, we denote by R a selection of S of length r n, i.e. an ordered sequence of r distinct elements ....

....of the line l 1 , then the trapezoid Delta is not created, because the ray emanating downward from l 2 l 3 intersects it. This implies that there is no locality in the determination of which trapezoids arise in the execution of CutRandomInc, so the standard techniques of [CS89, dBDS95, HW87] can not be applied directly in analyzing CutRandomInc. b c a b c a b c Delta (i) ii) iii) Figure 8.5: Even if merging is used by CutRandomInc, an active trapezoid Delta 2 C i might disappear if we skip an insertion of a line which is completely unrelated to Delta. The ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

....are added. Our algorithm is a variant of the Paterson Yao algorithm. We construct the BSP for S in such a way that there is a close relationship between the BSP and the planar arrangement of lines supporting the edges of the xy projections of the triangles in S. We use results from net theory [17] and on arrangements of lines [15] to bound the expected number of vertices in the convex subdivision of R induced by the BSP and the expected running time of the algorithm. Finally, we present a deterministic algorithm (Section 5) for constructing a BSP for a set S of n triangles in R . If ....

....total time spent in processing f is O(jS Delta j log jS Delta j) O(k f log k f ) Let Z be the set of all active faces of A(L ) that are intersected by i . The total time spent in processing i is O(k f log k f ) Using the Zone Theorem [9, 15] and the theory of random sampling [13, 17], we can show that the expected value of k f is O(n log n) which implies the following theorem: Theorem 4.2 Let S be a set of n non intersecting triangles . We can compute a BSP for S of expected size O(n in expected time O(n Remark: Using a similar argument, we can also prove that ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127-- 151.

....vertices in S, but no vertex of X S. Question: Is V C path (G) # k Reference: Schaefer [Sch00] Comments: Special case of the GRAPH VC DIMENSION problem defined for types of subgraphs of a given graph. Introduced by Kranakis, et al. KKR 97] building on an idea of Haussler and Welzl [HW87] The problem is also # p 3 complete for cyles instead of paths [Sch00] All other cases investigated so far turn out to be in P (stars, neighborhoods) or NP complete (trees, connected sets) KKR 97] Also see VC DIMENSION. 7 2.2 Sets and Partitions [SP1] VC DIMENSION Given: A Boolean ....

David Haussler and Emo Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2:127--151, 1987.

....that the minimum is achieved at a unique vertex de ned by d constraints (non degeneracy) 1=r) nets and (1=s) semi cuttings. For p 2 R d , let op be the ray connecting p to the origin o, and let H jop H consist of those hyperplanes that intersect the open segment op. N H is a (1=r) net [9] for H with respect to rays if jH jop j jHj=r implies N jop 6= A random sample N H of size S net (r) C net (dr) log(dr) where C net is an absolute constant, is a (1=r) net with large probability [9] and it can be computed deterministically in time O(d) 2d ( dr) log(dr) d n [2] ....

....H consist of those hyperplanes that intersect the open segment op. N H is a (1=r) net [9] for H with respect to rays if jH jop j jHj=r implies N jop 6= A random sample N H of size S net (r) C net (dr) log(dr) where C net is an absolute constant, is a (1=r) net with large probability [9] and it can be computed deterministically in time O(d) 2d ( dr) log(dr) d n [2] and in time O(n log r) for r = O(n 1=d ) 12] A (1=s) semi cutting is a triangulation T of T H such that X 2T jH j j c C sc n s c s bd=2c ; 1 For a collection of sets X , we ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2 (1987), 127-151.

....F to subsets X 0 of X it is necessary and sufficient that for every Y and ffl 0 there is an ffl net of size at most f(ffl) and this is equivalent to the VCdimension of F being finite. When talking about a family of hypergraphs the VC dimension should be uniformly bounded. Haussler and Welzl [14] proved that f(ffl) O(d(1=ffl) log(1=ffl) where d is the VC dimension, and Koml os, Pach and Woeginger [19] gave examples showing this cannot be further improved. Ding, Seymour and Winkler [8] characterized when is bounded by a function of for a hypergraph and all of its restrictions. ....

....plus FH(2; ff; fi) for some ff 1 and fi 0 are sufficient or not. Fractional Helly and weak ffl nets. In the second main part of the proof of the (p; q) theorem for convex sets, the existence of weak ffl nets for convex sets is used. This important notion was introduced by Haussler and Welzl [14] and further studied in several papers, such as [1] 7] As far as we know, at least three different proofs of existence of weak ffl nets for convex sets are known. Two are given in Alon et al. [1] a direct geometric argument, leading to a weak ffl net of size O( 1=ffl) Gamma2 d Gamma1 ) for ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2(1987), 127--151.

.... extremal graph theoretic arguments (see [15] that the number of its edges, and hence the number of mixed edge touching regular vertices on #U is O(mn 1 2 n) We next choose an integer parameter r, to be fixed below, and construct a (1 r) cutting on the edges of the polygons in C 2 (see [4, 11] for details) This yields a tiling of the plane by O(r 2 ) pairwise openly disjoint vertical trapezoids, each crossed by at most n r edges. For each trapezoid # , consider the set # # of all the arcs in # that cross # , clipped to within # ; some of these arcs may intersect # in two connected ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom. 2 (1987), 127--151.

....Street, London WC2A 2AE z School of Mathematical Studies, University of North London, 2 16 Eden Grove, London N7 8EA 1 1 Introduction In this paper we investigate a parameter de ned for any graph: the VapnikChervonenkis dimension. The VC dimension of a graph was de ned by Haussler and Welzl [7] and is an interesting special case of the more general and well established notion of the Vapnik Chervonenkis dimension of a set system, rst introduced in [11] The Vapnik Chervonenkis dimension has proved useful in a number of areas of mathematics and computer science; in probability theory ....

.... of the more general and well established notion of the Vapnik Chervonenkis dimension of a set system, rst introduced in [11] The Vapnik Chervonenkis dimension has proved useful in a number of areas of mathematics and computer science; in probability theory [11, 10, 8] in computational geometry [7] and in the theory of machine learning [4, 2] for example. We start by presenting the necessary de nitions and making a few preliminary observations. Our main aim is to determine, for each positive integer d, the exact edge probabililty threshold function for a random graph G(n; p) to have VC ....

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David Haussler and Emo Welzl, Epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2: 127-151, 1987.

....q; RECTANGLEQUERY(1, ABSTRIPE, CDSTRIPE, RZB) end end. 4 The Adaptive Quadtree There are several strategies of partioning which may be applied in the algorithm of the previous section. Examples are the polygon trees [10] the conjugation trees or ham sandwich tree [4] 3] the ffl nets [6], and the partition trees [9] All of them were developed to minimize the aymptotic worst case complexity in time and space. For example, the ham sandwich tree requires O(n) space, O(n log n) preprocessing time, and O(n fi ) query time for n points, fi = log 0:695. The query times of the later ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. In Proceedings 2th ACM Symp. on Computational Geometry, 1986.

....most ff p r triangles of Pi. 7 Figure 5. A balanced simplicial partition of size 7. We use this theorem to build a range searching data structure for S called a partition tree T . Partition trees are one of the most commonly used internal memory data structure for geometric range searching [3, 25, 31, 45]. Each node v in a partition tree T is associated with a subset S v S of points and a triangle Delta v . For the root u of T , we have S u = S and Delta u = R 2 . Let N v = jS v j and n v = dN v =Be. Given a node v, we construct the subtree rooted at v as follows. If N v B, then v is a leaf ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

....by de Berg [2] Furthermore, the worst case free space complexity of a (bounded reach) robot in a guardable scene is considerably smaller than the worst case free space complexity in unrestricted scenes [5] 2. 2 Relation with # nets In a geometric setting, one can define # nets as follows [8, 9]. A subset N of a given set O of n objects in R d is called an # net with respect to a family R of ranges, if any empty range, i.e. any range not intersecting an object from N , intersects at most #n objects from O. The actual definition of # nets is more general than the definition we ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

....cardinality at least n=2. Let such two sets be called representative for h. To complete the proof of the theorem we only must show that S intersects each representative set for every hyperplane h in IR d . And this follows from the result on the randomized construction of nets (with = 2 ) [13]. Indeed, from the result due to Haussler and Welzl [13] it follows that the set S is with probability at least 2 3 an ( 2) net of each of R and B for the range space (IR d ; H) Therefore, by the definition of nets (see, e.g. 13] S intersects every representative of each hyperplane h ....

....representative for h. To complete the proof of the theorem we only must show that S intersects each representative set for every hyperplane h in IR d . And this follows from the result on the randomized construction of nets (with = 2 ) 13] Indeed, from the result due to Haussler and Welzl [13] it follows that the set S is with probability at least 2 3 an ( 2) net of each of R and B for the range space (IR d ; H) Therefore, by the definition of nets (see, e.g. 13] S intersects every representative of each hyperplane h in IR d . One can also easily improve the query ....

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2:127--151, 1987.

....For example, Fowler et al. 78] proved that it is NP Hard to decide whether a given set of n points can be covered by k unit squares. The greedy algorithm can be used for computing an O(log n) approximation. However, one can do slightly better if the VC dimension of the set system is nite; see [95] for the de nition of VC dimension. Clarkson [44] modi ed the ITERATIVE lp algorithm for computing a convex polytope of small complexity that lies between two nested convex polytopes, by reducing it to a geometric set cover problem. Later Br onnimann and Goodrich [33] showed that Clarkson s ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127-151.

....of already existing columns. E) The Approximation Lemma is not an isolated discovery. In several fields results of a similar flavour have been obtained, for instance: i) the VC dimension [VC 71] and its applications in computational learning theory [LLW 91] AB 92] and computational geometry [HW 87] ii) Monte Carlo approximations [KL 83] KLM 89] LV 91] iii) uniformity and irregularities (discrepancies) of set systems and matrices [BF 81] LSV 86] iv) stochastic information theory [Ahl 78] Ahl 79] On the other hand there are some statistics papers by Wald and others [DWW 51] ....

Haussler, D. and Welzl, E. 1987. Epsilon--nets and simplex range queries. Discrete and Computational Geometry 2: 127--151.

....an appendix that includes some proofs omitted in the main sections. 2 2 Sampling in Con guration and Range Spaces Two useful abstractions in the theory of random sampling in computational geometry are con guration spaces of bounded degree [25, 54, 48, 4] and range spaces of bounded VCexponent [67, 36, 57, 49]. 2.1 Con guration Spaces A con guration space 1 is a 5 tuple (X; C; T ; D; K) where (i) X is a set of objects; ii) C is a set of cells; iii) T : 2 X 2 C is a mapping that indicates for each subset R X of objects the set T (R) of cells determined by R; iv) D : C 2 X is a mapping ....

.... smallest e such that for any Y X, jR jY j = O(jY j e ) Two useful sampling concepts are approximations and nets: A X is a (1=r) approximation for (X; R) if for every R 2 R, jA Rj jAj jRj jXj 1 r : 3 This concept is simpler and more convenient than the original VC dimension [36, 67]. 4 N X is an net for (X; R) if for every R 2 R with jRj jXj, N R 6= A random sample of appropriate size is a (1=r) approximation [67] or an (1=r) net [36] Lemma 2.4 Let (X; R) be a range space with bounded VC exponent. An s sample A X with s = O(r 2 log(r= is a ....

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2 (1987), 127-151.

....2 n) expected time. Our algorithm is a variant of the Paterson Yao algorithm. We construct the BSP for S in such a way that there is a close relationship between the BSP and the planar arrangement of lines supporting the edges of the xy projections of the triangles in S. We use results on nets [20] and on arrangements of lines [17] to bound the expected number of vertices in the convex subdivision of R 3 induced by the BSP and the expected running time of the algorithm. Finally, we present a deterministic algorithm in Section 5 for constructing a BSP for a set S of n triangles in R 3 . ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127-151. 27

....Kalai and Matousek [2] gave an affirmative answer to the question of Kavraki et al. when d = 2. Specifially, they showed that g( 1 ffl ) C 1 ffl log 1 ffl suffices for some (rather large) constant C. Their proof uses the concepts of VC dimensionality (cf. 9] and ffl nets (cf. [1, 4]) The essential result is that the set system formed by the visibility polygons for points in any compact simply connected region in 2 space has bounded VC dimension. Following the same approach as Kalai and Matousek, Valtr [7] provided both a significantly tighter constant C as well as an ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2:127-151, 1987.

....jeffeg June 16, 1997 1 Introduction About ten years ago, the field of range searching, especially simplex range searching, was wide open. At that time, neither efficient algorithms nor nontrivial lower bounds were known for most range searching problems. A series of papers by Haussler and Welzl [149], Clarkson [85, 86] and Clarkson and Shor [89] not only marked the beginning of a new chapter in geometric searching, but also revitalized computational geometry as a whole. Led by these and a number of subsequent papers, tremendous progress has been made in geometric range searching, both in ....

....partition of R d by d hyperplanes is not always possible for d 5; the problem is still open for d = 4. However, weaker partitioning schemes were proposed in [90, 259] After the initial improvements and extensions on Willard s partition tree, a major breakthrough was made by Haussler and Welzl [149]. They formulated range searching in an abstract setting and, using elegant probabilistic methods, gave a randomized algorithm to construct a linear size partition tree with O(n ff ) query time, where ff = 1 Gamma 1 d(d Gamma1) 1 for any 0. The constant of proportionality hidden in ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127--151.

....sets) if it intersects each convex set K with jK X j n. It is shown that there are point sets X ae R d for which every weak 1 50 net has at least const Delta e p d=2 points. Weak nets with respect to convex sets, as defined in the abstract, were introduced by Haussler and Welzl [6] and later applied in results in discrete geometry, most notably in the spectacular proof of the Hadwiger Debrunner (p; q) conjecture by Alon and Kleitman [2] For a finite X ae R d , let f(X; denote the smallest size of a weak net for X, 0 1, and let f(d; supff(X; X ae R ....

....No. 158 99 and 159 99 and by ETH Zurich. 1 One can also consider weak nets for set systems F other than convex sets in R d , such as the family of all balls, or all ellipsoids, and so on (then N should intersect all F 2 F such that jF Xj n) According to results of Haussler and Welzl [6], for the two just mentioned examples of F , weak nets exist of size bounded by a polynomial in d (for fixed 0) This result is fairly general and, roughly speaking, applies whenever the sets of F can be defined by a formula of length polynomial in d. Proof of Theorem 1. Instead of a ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

....is the subgraph of G d induced by the vertices V d H for some oriented halfspace H . Since the Vapnik Chervonenkis Dimension of d space with respect to the range of halfspaces is d 1, there can be no more than P d 1 i=0 jP j i jP j d 1 sets of the form P H [3]. Choosing P = V d , this upper bound of 2 d(d 1) illustrates the gain achieved by considering MSS instead of the double exponentially many sliceable subsets. Furthermore, a vertex subset P V d can easily be tested for being of the form P = V d H . Indeed, the existence of an ....

D. Haussler and E. Welzl: "Epsilon-nets and simplex range queries", pp.127-151 in Discrete Comput. Geom. 2 (1987).

....2 n) expected time. Our algorithm is a variant of the PatersonYao algorithm. We construct the BSP for S in such a way that there is a close relationship between the BSP and the planar arrangement of lines supporting the edges of the xy projections of the triangles in S. We use results on nets [54] and on arrangements of lines [40] to bound the expected number of vertices in the convex subdivision of R 3 induced by the BSP and the expected running time of the algorithm. In Section 6.2, we consider the problem of whether we can use a suitable measure of geometric complexity to construct a ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127--151.

....of the DobkinKirkpatrick algorithm, each edge in E intersects O(log r) tetrahedra of the triangulation of a cell of Vor F (R) and they can be computed in O(log r) time. Hence, X 4 m4 = O(n 2 r log 2 r) 5.1) and the sets E4 can be computed in O(n 2 r log 2 r) time. By the net theory [22], n4 n=r for all 4 with high probability, so t 4 = O(n 2 =r 2 ) 5.2) for all 4 with high probability. As shown in [2] S4 , for all 4 2 Delta, can be computed in overall time O(nr log 2 r) Since Vor F (S4 ) can be computed and triangulated in O(n 2 4 ) time, the total time spent in ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127--151.

....CDSTRIPE, RZB) end f for q in Q . g end. f main algorithm g 4 The adaptive quadtree There are several strategies of partitioning which may be applied in the algorithm of the previous section. Examples are the polygon trees [11] the conjugation trees[4] or ham sandwich tree[3] the ffl nets [6], and the partition trees [10] All of them were developed to minimize the asymptotic worst case complexity in time and space. For example, the ham sandwich tree requires O(n) space, O(n log n) preprocessing time, and O(n fi ) query time for n points, fi = log 0:695. The query times of the later ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. In Proceedings 2th ACM Symp. on Computational Geometry, 1986. 25

....geometry. randomization did not become popular in computational geometry until the late 1980s. In the mid 1980s Clarkson was developing his random sampling technique, which he extended to a surprisingly general framework in his 1988 paper [46, 54] Around the same time Haussler and Welzl [100] introduced the idea of nets and VC dimensions. These two techniques brought randomization to the forefront of computational geometry and revolutionized the field. Numerous randomized divideand conquer and incremental algorithms, dynamic data structures, and analysis techniques (e.g. backward ....

....Fowler et al. 82] proved that it is NP Hard to decide whether a given set of n points can be covered by k unit squares. The greedy algorithm can be used for computing an O(log n) approximation. However, one can do slightly better if the VC dimension of the set system Sigma is finite; see [100] for the definition of VC dimension. Clarkson [48] modified the ITERATIVE lp algorithm for computing a convex polytope of small complexity that lies between two nested convex polytopes, by reducing it to a geometric set cover problem. Later Bronnimann and Goodrich [35] showed that Clarkson s ....

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127--151.

....P y2Y w(y) for any subset Y of X . Definition 1. Given a weight function w, we say that a p Theta p partitioning is ff good if every tile r i;j of H satisfies w(R i;j ) ff Delta w(X) We remark that our ff good partitionings correspond to the ffl nets used in [4] and originally introduced in [12], which have found many applications in computational geometry. 14 6.3 Upper Bounds for MAX Metrics We now present approximation results for the p Theta p partitioning problem where the cumulative metric is the MAX metric, i.e. the heft of a partition is the largest heft of any of the tiles. ....

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2:127--151, 1987.

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Haussler, D., E. Welzl, "Epsilon-nets and simplex range queries", Discrete Computational Geometry, 2, 1987, pp. 127-151.

....d is called a weak net for (convex ranges of) S if, for any subset T of n points of S, the convex hull of T intersects W . If the points of W could be chosen from among those of S, then W would have been called a strong net (or just an net) of S. Such nets, introduced by Haussler and Welzl [11], are defined for general range spaces, where a range space is a pair (S; R) where S is a set and R is a set of subsets of S, called ranges. A set N S is an net if every range of S that contains at least n points intersects N . Haussler and Welzl have shown that if the range space has finite ....

....N . Haussler and Welzl have shown that if the range space has finite VC dimension then there always exists an net of size O( 1 log 1 ) in particular, this size is independent of the size of S. Moreover, a random subset of S of this size will be an net with high probability. See [11] for more details. In the setup that we are concerned with, the ranges are all intersections of S with convex sets; we will refer to these ranges as convex ranges. Unfortunately, in this setup, the resulting range space has infinite VC dimension, which, in particular, is manifested by the fact ....

Haussler, D., Welzl, E. Epsilon nets and simplex range queries, Discrete Comput. Geom. 2 (1987), 127--151.

....of V onto I, i.e. V j I = f(v i 1 ; v i k ) v 1 ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The Vapnik Chervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] this terminology comes from [HW87]) We will denote this number by d. Hence d = maxfk : 9 I = i 1 ; i k ) 1 i j n; with V j I = f0; 1g k g: This quantity plays a important role in certain areas of statistics, in particular in the theory of empirical processes ....

..... i k ) 1 i j n; with V j I = f0; 1g k g: This quantity plays a important role in certain areas of statistics, in particular in the theory of empirical processes [Dud78,Vap82,GZ84,Dud84,Pol84,Tal87a,Tal87b,Tal88,Pol90] It has also been used recently in the fields of computational geometry [HW87] [Wel88] MSW90] EGS88] CF88] CW89] and machine learning [BEHW89,HP88,RHW89,FC90,VW91] Let jV j denote the cardinality of V . The following result is well known, and was independently discovered by several people, including Sauer [Sau72] and Vapnik and Chervonenkis (see [Ass83] for a review, ....

David Haussler and Emo Welzl. Epsilon nets and simplex range queries. Disc. Comp. Geometry, 2:127--151, 1987.

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Computational Geometry, 2:127--151, 1987.

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.

No context found.

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.

No context found.

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Computational Geometry, 2: 127--151, 1987.

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D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), pp. 127-151.

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

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D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127--151.

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D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.

No context found.

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127--151, 1987.

No context found.

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2, 127--151, 1987.

No context found.

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2(1987), 127-151.

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D. Haussler, E. Welzl, Epsilon-nets and simplex range queries, Discrete & Computational Geometry, 2 (1987), 127--151.

No context found.

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987.

No context found.

D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Computational Geometry, 2:127--151, 1987.

No context found.

D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), 127-151.

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