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52
Bounded VCdimension implies a fractional Helly theorem
, 2002
"... We prove that every set system of bounded VCdimension has a fractional Helly property. More precisely, if the dual shatter function of a set system F is bounded by o(m ), then F has fractional Helly number k. This means that for every ff ? 0 there exists a fi ? 0 such that if F 1 ; F 2 ; : : ..."
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We prove that every set system of bounded VCdimension has a fractional Helly property. More precisely, if the dual shatter function of a set system F is bounded by o(m ), then F has fractional Helly number k. This means that for every ff ? 0 there exists a fi ? 0 such that if F 1 ; F 2 ; : : : ; Fn 2 F are sets with i2I F i 6= ; for at least sets I ` f1; 2; : : : ; ng of size k, then there exists a point common to at least fin of the F i . This further implies a (p; k)theorem: for every F as above and every p k there exists T such that if G ` F is a finite subfamily where among every p sets, some k intersect, then G has a transversal of size T . The assumption about bounded dual shatter function applies, for example, to families of sets in R definable by a bounded number of polynomial inequalities of bounded degree; in this case, we obtain fractional Helly number d+1.
A Hierarchical Technique for Constructing Efficient Declustering Schemes for Range Queries
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A Counterexample to Beck’s Conjecture on the Discrepancy of Three Permutations
, 2011
"... Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. József Beck conjectured (c. 1982) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3 ..."
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Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. József Beck conjectured (c. 1982) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3 k, we exhibit three permutations whose corresponding set system has discrepancy Ω(log n). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck’s conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to ℓ permutations has discrepancy O(√ℓ) [SST01].
Bichromatic Discrepancy via Convex Partitions
"... Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept for measuring how mixed the elements of S = R ∪ B are. The discrepancy of a set X ⊂ S is X ∩ R  − X ∩ B. We say that a partition Π = {S1, S2,..., Sk} of S is convex if the convex hul ..."
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Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept for measuring how mixed the elements of S = R ∪ B are. The discrepancy of a set X ⊂ S is X ∩ R  − X ∩ B. We say that a partition Π = {S1, S2,..., Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition of S is the minimum discrepancy of the sets Si. The discrepancy of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the problem of computing the discrepancy of a bichromatic point set. We divide the study in general convex partitions for both general set of points and points in convex position, and also when the partition is given by a line. In this case we prove that this problem is 3SUMhard. 1
Consistent digital rays
 DISCRETE COMPUT. GEOM
, 2009
"... Given a fixed origin o in the ddimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment op between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axio ..."
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Given a fixed origin o in the ddimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment op between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log n) bound in the n × n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital starshaped regions centered at o which we use to design efficient algorithms for image processing problems.
Discrepancy of Point Sequences on Fractal Sets
"... We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 discrepancy (and consequently also the worstcase discrepancy) of an Npoint set with respect to halfspaces is at least of the ..."
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We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 discrepancy (and consequently also the worstcase discrepancy) of an Npoint set with respect to halfspaces is at least of the order N^(1/21/(2s)) , where s is the Hausdorff dimension of the fractal. We also show that for many fractals, this bound is tight for the L 2 discrepancy. Determining the correct order of magnitude of the worstcase discrepancy remains a challenging open problem.
THE DISCREPANCY OF A NEEDLE ON A CHECKERBOARD, II
, 811
"... Abstract. Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their blac ..."
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Abstract. Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding “finite ” problem (N × N checkerboard) we had proved that we can color it in such a way that the above quantity is at most C √ N log N, for any placement of the line segment. In this followup we show that it is possible to color the infinite checkerboard with two colors so that for any line segment I the excess of one color over another is bounded above by CɛI  1 2 +ɛ, for any ɛ> 0. We also prove lower bounds for the discrepancy of circular arcs. Finally, we make some observations regarding the L p discrepancies for segments and arcs, p < 2, for which our L 2based methods fail to give any reasonable estimates. Contents
Discrepancy after adding . . .
"... We show that the hereditary discrepancy of a hypergraph F on n pointsincreases by a factor of at most O(log n) when one adds a new edge to F. ..."
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We show that the hereditary discrepancy of a hypergraph F on n pointsincreases by a factor of at most O(log n) when one adds a new edge to F.
Quantum Lower Bounds by Entropy Numbers
, 2006
"... We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of informationbased complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional Lp ..."
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We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of informationbased complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional Lp spaces and of Sobolev embeddings. 1
Random numbers
"... have a peculiar power, even when they are only pseudoor quasirandom In the early 1990s Spassimir Paskov, then a graduate student at Columbia University, began analyzing an exotic financial instrument called a collateralized mortgage obligation, or CMO, issued by the investment bank Goldman Sachs. Th ..."
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have a peculiar power, even when they are only pseudoor quasirandom In the early 1990s Spassimir Paskov, then a graduate student at Columbia University, began analyzing an exotic financial instrument called a collateralized mortgage obligation, or CMO, issued by the investment bank Goldman Sachs. The aim was to estimate the current value of the CMO, based on the potential future cash flow from thousands of 30year mortgages. This task wasn’t just a matter of applying