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60
Incremental Clustering and Dynamic Information Retrieval
, 1997
"... Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retri ..."
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Cited by 151 (4 self)
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Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application, and which should also be useful in other applications. The goal is to efficiently maintain clusters of small diameter as new points are inserted. We analyze several natural greedy algorithms and demonstrate that they perform poorly. We propose new deterministic and randomized incremental clustering algorithms which have a provably good performance. We complement our positive results with lower bounds on the performance of incremental algorithms. Finally, we consider the dual clustering problem where the clusters are of fixed diameter, and the goal is to minimize the number of clusters. 1 Introduction We consider the following problem: as a sequence of points from a metric...
On The Complexity Of Computing Mixed Volumes
 SIAM J. Comput
, 1998
"... . This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several #Phardness results that focus on the di#erence of com ..."
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Cited by 29 (1 self)
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. This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several #Phardness results that focus on the di#erence of computing mixed volumes versus computing the volume of polytopes. We show that computing the volume of zonotopes is #Phard (while each corresponding mixed volume can be computed easily) but also give examples showing that computing mixed volumes is hard even when computing the volume is easy. On the positive side, we derive a randomized algorithm for computing the mixed volumes V ( m 1 z } { K 1 , . . . , K 1 , m 2 z } { K 2 , . . . , K 2 , . . . , ms z } { Ks , . . . , Ks ) of wellpresented convex bodies K 1 , . . . , Ks , where m 1 , . . . , ms # N 0 and m 1 # n  #(n) with #(n) = o( log n log log n ). The algorithm is an interpolation method based on polynomialtime ra...
On Transfinite Barycentric Coordinates
, 2006
"... A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The GordonWixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and WachspressWarren coordinates ar ..."
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Cited by 18 (0 self)
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A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The GordonWixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and WachspressWarren coordinates are studied as particular cases of that general construction. Motivated by finite element/volume applications, we study capabilities of transfinite barycentric interpolation schemes to approximate harmonic and quasiharmonic functions. Finally we establish and analyze links between transfinite barycentric coordinates and certain inverse problems of di#erential and convex geometry.
On 64% Majority Rule
 Econometrica
, 1988
"... Many electoral rules (such as those governing the U.S. Constitution) require a supermajority vote to change the status quo. It is well known that without some restriction on preferences, supermajority rules have paradoxical properties. For example, electoral cycles are possible with anything other ..."
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Cited by 16 (0 self)
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Many electoral rules (such as those governing the U.S. Constitution) require a supermajority vote to change the status quo. It is well known that without some restriction on preferences, supermajority rules have paradoxical properties. For example, electoral cycles are possible with anything other than 100%majority rule. Can these problems still arise if there is sufficient similarity of attitudes among the voting population? We introduce a definition of social consensus which involves two restrictions on domain: one on individual preferences, the other on the distribution of preferences. Individuals vote for the proposal closest (in Eucidean distance) to their most preferred point. The density of voters ' ideal points is concave over its support in Rn. Under these conditions, there exists an unbeatable proposal according to 64%majority rule. In addition, no electoral cycles are possible. For ndimensional decision problems, the precise majority size necessary to avoid cycles is 1 (n/(n + 1)) " which rises monotonically to 1 (l/e), just below 64%. Our approach is based on the SimpsonKramer minmax rule. We compare this rule with Condorcet's original proposal for an electoral system immune to his paradox of voting. We conclude by considering the properties of a voting constitution based on 64%majority rule.
Redundant PicardFuchs systems for Abelian integrals
 J. of Differential Equations
"... Abstract. We derive an explicit system of Picard–Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two time ..."
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Cited by 9 (2 self)
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Abstract. We derive an explicit system of Picard–Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two times greater than the standard Picard–Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were able to majorize the number of all (real and complex) zeros. In the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and nonhomogeneity of uni and bivariate complex polynomials are studied. Contents
Transboundary Extremal Length
 J. d'Analyse Math
, 1993
"... . We introduce two basic notions, `transboundary extremal length' and `fat sets', and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably man ..."
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Cited by 8 (3 self)
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. We introduce two basic notions, `transboundary extremal length' and `fat sets', and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably many boundary components is conformally equivalent to a circle domain. This theorem is further generalized in two directions. We show that the same statement is true for a wide class of domains with uncountably many boundary components, in particular for domains bounded by quasicircles and points. Moreover, these domains admit more general uniformizations. For example, every circle domain is conformally equivalent to a domain whose complementary components are heartshapes and points. Introduction The concept of extremal length (see, e.g. [21]) is a very useful tool in the study of conformal and quasiconformal mappings. Basically, one can say that its usefulness stems from two basic pro...
Approximating the volume of convex bodies
 Discrete Comput. Geom
, 1993
"... Abstract. It is a well known fact that for every polynomial time algorithm which gives an upper bound V (K) and a lower bound V (K) for the volume of a convex set K ⊂ E d, the ratio V (K)/V (K) is at least (cd / log d) d. Here we describe an algorithm which gives for ǫ> 0 in polynomial time an upper ..."
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Cited by 8 (1 self)
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Abstract. It is a well known fact that for every polynomial time algorithm which gives an upper bound V (K) and a lower bound V (K) for the volume of a convex set K ⊂ E d, the ratio V (K)/V (K) is at least (cd / log d) d. Here we describe an algorithm which gives for ǫ> 0 in polynomial time an upper and lower bound with the property V (K)/V (K) ≤ d!(1 + ǫ) d. 1.
Algebraic methods for computing smallest enclosing and circumscribing cylinders of simplices
 Appl. Algebra Eng. Commun. Comput
"... Abstract. We provide an algebraic framework to compute smallest enclosing and smallest circumscribing cylinders of simplices in Euclidean space E n. Explicitly, the computation of a smallest enclosing cylinder in E 3 is reduced to the computation of a smallest circumscribing cylinder. We improve exi ..."
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Cited by 7 (4 self)
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Abstract. We provide an algebraic framework to compute smallest enclosing and smallest circumscribing cylinders of simplices in Euclidean space E n. Explicitly, the computation of a smallest enclosing cylinder in E 3 is reduced to the computation of a smallest circumscribing cylinder. We improve existing polynomial formulations to compute the locally extreme circumscribing cylinders in E 3 and exhibit subclasses of simplices where the algebraic degrees can be further reduced. Moreover, we generalize these efficient formulations to the ndimensional case and provide bounds on the number of local extrema. Using elementary invariant theory, we prove structural results on the direction vectors of any locally extreme circumscribing cylinder for regular simplices. 1.
Decay of meanvalues of multiplicative functions
 Canad. J. Math
"... Given a multiplicative function f with f(n)  ≤ 1 for all n, we are concerned with obtaining explicit upper bounds on the meanvalue 1 ..."
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Cited by 7 (4 self)
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Given a multiplicative function f with f(n)  ≤ 1 for all n, we are concerned with obtaining explicit upper bounds on the meanvalue 1
Smoothed Analysis of Condition Numbers
"... The running time of many iterative numerical algorithms is dominated by the condition number of the input, a quantity measuring the sensitivity of the solution with regard to small perturbations of the input. Examples are iterative methods of linear algebra, interiorpoint methods of linear and conv ..."
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Cited by 6 (5 self)
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The running time of many iterative numerical algorithms is dominated by the condition number of the input, a quantity measuring the sensitivity of the solution with regard to small perturbations of the input. Examples are iterative methods of linear algebra, interiorpoint methods of linear and convex optimization, as well as homotopy methods for solving systems of polynomial equations. Thus a probabilistic analysis of these algorithms can be reduced to the analysis of the distribution of the condition number for a random input. This approach was elaborated for averagecase complexity by many researchers. The goal of this survey is to explain how averagecase analysis can be naturally refined in the sense of smoothed analysis. The latter concept, introduced by Spielman and Teng in 2001, aims at showing that for all real inputs (even illposed ones), and all slight random perturbations of that input, it is unlikely that the running time will be large. A recent general result of Bürgisser, Cucker and Lotz (2008) gives smoothed analysis estimates for a variety of applications. Its proof boils down to local bounds on the volume of tubes around a real algebraic hypersurface in a sphere. This is achieved by bounding the integrals of absolute curvature of smooth hypersurfaces in terms of their degree via the principal kinematic formula of integral geometry and Bézout’s theorem.