Results 11  20
of
99
On Distinct Sums and Distinct Distances
, 2001
"... The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bo ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bound on the number of distinct sums. As an application we improve the SolymosiToth bound on an old Erd}os problem: the number of distinct distances n points determine in the plane. Our bound also nds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.
On the Number of Congruent Simplices in a Point Set
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2002
"... We derive improved bounds on the number of kdimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex, for k < or = d  1. Let f(sup d)(sub K)(n) be the maximum number of ksimplices spanned by a set of n points in R(sup d) that are congruent to a given ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
We derive improved bounds on the number of kdimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex, for k < or = d  1. Let f(sup d)(sub K)(n) be the maximum number of ksimplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex. We prove that f(sup 3)(sub 2) (n) = O(n sup 5/3) times 2 sup O(alpha (sup 2)(n)), f(sup 4)(sub 2)(n) = O(n (sup 2) + epsilon), f(sup 5)(sub 2) (n)) = theta (n (sup 7/3), and f(sup 4)(sub 3)(n) = O(n (sup 9/4) plus epsilon). We also derive a recurrence to bound f(sup d)(sub k) (n) for arbitrary values of k and d, and use it to derive the bound f(sup d)(sub k(n) = O(n(sup d/2)) for d < or = 7 and k < or = d  2. Following Erdo's and Purdy, we conjecture that this bound holds for larger values of d as well, and for k < or = d  2.
Reconstructing Sets From Interpoint Distances
 of Algorithms Combin
, 2002
"... Which point sets realize a given distance multiset? Interesting cases include the "turnpike problem" where the points lie on a line, the "beltway problem" where the points lie on a loop, and multidimensional versions. We are interested both in the algorithmic problem of determining such point sets f ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Which point sets realize a given distance multiset? Interesting cases include the "turnpike problem" where the points lie on a line, the "beltway problem" where the points lie on a loop, and multidimensional versions. We are interested both in the algorithmic problem of determining such point sets for a given collection of distances and the combinatorial problem of finding bounds on the maximum number of different solutions. These problems have applications in genetics and crystallography.
Distinct distances in three and higher dimensions
 Combin. Probab. Comput
, 2003
"... Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in threedimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to th ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in threedimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to the remaining elements of P. The same result holds for points on the threedimensional sphere. As a consequence, we obtain analogous results in higher dimensions. 1
Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)
, 2010
"... We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.
Distance sets of welldistributed planar sets, preprint
, 2002
"... Abstract. Let X be a 2dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX may be if X contains a welldistributed set whose distance set ∆ satisfies the estimate  ∆ ∩ [0, N]  ≤ CN 3/2−ǫ. We also give a necessary and s ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. Let X be a 2dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX may be if X contains a welldistributed set whose distance set ∆ satisfies the estimate  ∆ ∩ [0, N]  ≤ CN 3/2−ǫ. We also give a necessary and sufficient condition for the existence of a welldistributed set with  ∆ ∩ [0, N]  ≤ CN.
Geometric Representations of Graphs
 IN PAUL ERDÖS, PROC. CONF
, 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
Isosceles Triangles Determined By a Planar Point Set
"... It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5 ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5e 1 , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi{C. Toth and Tardos.
Some scalable parallel algorithms for geometric problems
 Journal of Parallel and Distributed Computing
, 1999
"... This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary interconnection network. It introduces efficient scalable parallel algorithms for a number of geometric problems including the rectangle finding problem, the maximal equally spaced collinear points problem, and the point set pattern matching problem. All of the algorithms presented are scalable in that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scalability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.
Combinatorial and experimental methods for approximate point pattern matching
 Algorithmica
, 2003
"... Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.