Results 1  10
of
57
Lower bounds for algebraic computation trees
 IN STOC
, 1983
"... A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying ..."
Abstract

Cited by 231 (0 self)
 Add to MetaCart
A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying the method to decision trees we extend all the apparently known lower bounds for linear decision trees to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20]. We also show how this new method can be used to establish lower bounds on the complexity of constructions with ruler and compass in plane Euclidean geometry.
Bounding the VapnikChervonenkis dimension of concept classes parameterized by real numbers
 Machine Learning
, 1995
"... Abstract. The VapnikChervonenkis (VC) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the VC dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are au ..."
Abstract

Cited by 92 (1 self)
 Add to MetaCart
Abstract. The VapnikChervonenkis (VC) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the VC dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are automatic for discrete concept classes, but hitherto little has been known about what general conditions guarantee polynomial bounds on VC dimension for classes in which concepts and examples are represented by tuples of real numbers. In this paper, we show that for two general kinds of concept class the VC dimension is polynomially bounded in the number of real numbers used to define a problem instance. One is classes where the criterion for membership of an instance in a concept can be expressed as a formula (in the firstorder theory of the reals) with fixed quantification depth and exponentiallybounded length, whose atomic predicates are polynomial inequalities of exponentiallybounded degree. The other is classes where containment of an instance in a concept is testable in polynomial time, assuming we may compute standard arithmetic operations on reals exactly in constant time. Our results show that in the continuous case, as in the discrete, the real barrier to efficient learning in the Occam sense is complexitytheoretic and not informationtheoretic. We present examples to show how these results apply to concept classes defined by geometrical figures and neural nets, and derive polynomial bounds on the VC dimension for these classes. Keywords: Concept learning, information theory, VapnikChervonenkis dimension, Milnor’s theorem 1.
Linear decision trees: volume estimates and topological bounds
 PROC. 24TH ACM SYMP. ON THEORY OF COMPUTING
, 1992
"... We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a line ..."
Abstract

Cited by 51 (5 self)
 Add to MetaCart
(Show Context)
We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a linear decision tree for this problem is Θ(nlog(n/k)). The upper bound is easy; the lower bound can be established for k = O(n 1/4−ε) by a volume argument; for the whole range, however, our proof is more complicated and it involves the use of some topology as well as the theory of Möbius functions.
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
Abstract

Cited by 36 (7 self)
 Add to MetaCart
We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
Abstract

Cited by 28 (4 self)
 Add to MetaCart
We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
Better Lower Bounds on Detecting Affine and Spherical Degeneracies
, 1995
"... We show that in the worst case,\Omega (n d ) sidedness queries are required to determine whether a set of n points in IR d is affinely degenerate, i.e., whether it contains d +1points on a common hyperplane. This matches known upper bounds. Wegive a straightforward adversary argument, b ..."
Abstract

Cited by 26 (8 self)
 Add to MetaCart
(Show Context)
We show that in the worst case,\Omega (n d ) sidedness queries are required to determine whether a set of n points in IR d is affinely degenerate, i.e., whether it contains d +1points on a common hyperplane. This matches known upper bounds. Wegive a straightforward adversary argument, based on the explicit construction of a point set containing\Omega (n d ) "collapsible" simplices, anyoneof which can be made degenerate without changing the orientation of anyother simplex. As an immediate corollary,wehavean\Omega (n d )lower bound on the number of sidedness queries required to determine the order type of a set of n points in IR d . Using similar techniques, wealsoshowthat\Omega (n d+1 ) insphere queries are required to decide the existence of spherical degeneracies in a set of n points in IR d . 1 Introduction A fundamental problem in computational geometry is determining whether a given set of points is in "general position." A simple example of ...