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.

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Haussler, D., E. Welzl, "Epsilon-nets and simplex range queries", Discrete Computational Geometry, 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 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.

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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 (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.

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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 and 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), 127-151.

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D. Haussler, E. Welzl, Epsilon-nets and simplex range queries, Discrete & Computational Geometry, 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.

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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 (1987), 127-151.

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