EMBEDDING THE DOUBLE CIRCLE IN A SQUARE GRID OF MINIMUM SIZE

2014 ◽  
Vol 24 (03) ◽  
pp. 247-258 ◽  
Author(s):  
S. BEREG ◽  
R. FABILA-MONROY ◽  
D. FLORES-PEÑALOZA ◽  
M. A. LOPEZ ◽  
P. PÉREZ-LANTERO
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Grid Size ◽  
Lattice Points ◽  
Minimum Size ◽  
Small Integer ◽  
Square Grid ◽  
Similar Construction ◽  
Asymptotic Size ◽  
Point Set

In 1926, Jarník investigated the drawing of a curve that visits a large number of lattice points relative to its curvature. To this end, he constructed a convex n-gon with vertices on a “small” integer grid [0, c.n3/2]2, where c > 0 is a constant, and proved that this grid size is optimal up to a constant factor. We consider a similar construction for the double circle of 2n points and prove that it can be embedded in a grid of the same asymptotic size. Moreover, we give an O(n)-time algorithm to generate the corresponding point set.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

scholarly journals The complex moment problem: determinacy and extendibility

2019 ◽  
Vol 124 (2) ◽  
pp. 263-288 ◽  
Author(s):  
Dariusz Cichoń ◽  
Jan Stochel ◽  
Franciszek Hugon Szafraniec
Keyword(s):  
Partial Solution ◽  
Positive Definite ◽  
Lattice Points ◽  
Moment Sequence ◽  
Representing Measure ◽  
Straight Line Passing ◽  
Straight Line ◽  
Plane Algebraic Curve ◽  
Line Passing ◽  
Point Set

Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb{C} $ as well as complex moment sequences which are constant on a family of parallel “Diophantine lines”. All this is supported by a bunch of illustrative examples.

scholarly journals Sign-rank Can Increase under Intersection

2021 ◽  
Vol 13 (4) ◽  
pp. 1-17
Author(s):  
Mark Bun ◽  
Nikhil S. Mande ◽  
Justin Thaler
Keyword(s):  
Communication Complexity ◽  
Function Class ◽  
Time Algorithm ◽  
Constant Factor ◽  
Pac Learning ◽  
New Techniques ◽  
Know How ◽  
Communication Problem ◽  
Analytic Complexity ◽  
Class Of Functions

The communication class UPP cc is a communication analog of the Turing Machine complexity class PP . It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f , let f ∧ f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP cc ( f ) = O (log n ), and UPP cc ( f ∧ f ) = Θ (log 2 n ). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP cc , the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n Omega Ω(log n ) . This matches an upper bound of (Klivans, O’Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.

NEAREST NEIGHBOR PROBLEMS

1992 ◽  
Vol 02 (04) ◽  
pp. 383-416 ◽  
Author(s):  
GORDON WILFONG
Keyword(s):  
Nearest Neighbor ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Minimum Cardinality ◽  
Nearest Neighbor Rule ◽  
Np Complete ◽  
Cover Problem ◽  
Consistent Subset ◽  
Special Case

Suppose E is a set of labeled points (examples) in some metric space. A subset C of E is said to be a consistent subset ofE if it has the property that for any example e∈E, the label of the closest example in C to e is the same as the label of e. We consider the problem of computing a minimum cardinality consistent subset. Consistent subsets have applications in pattern classification schemes that are based on the nearest neighbor rule. The idea is to replace the training set of examples with as small a consistent subset as possible so as to improve the efficiency of the system while not significantly affecting its accuracy. The problem of finding a minimum size consistent subset of a set of examples is shown to be NP-complete. A special case is described and is shown to be equivalent to an optimal disc cover problem. A polynomial time algorithm for this optimal disc cover problem is then given.

ALGORITHMS FOR POINT SET MATCHING WITH k-DIFFERENCES

2006 ◽  
Vol 17 (04) ◽  
pp. 903-917
Author(s):  
TATSUYA AKUTSU
Keyword(s):  
Euclidean Space ◽  
Related Problem ◽  
Time Algorithm ◽  
Two Dimensions ◽  
Efficient Algorithms ◽  
Common Point ◽  
Dimensional Euclidean Space ◽  
Maximum Cardinality ◽  
Point Set ◽  
Special Case

The largest common point set problem (LCP) is, given two point set P and Q in d-dimensional Euclidean space, to find a subset of P with the maximum cardinality that is congruent to some subset of Q. We consider a special case of LCP in which the size of the largest common point set is at least (|P| + |Q| - k)/2. We develop efficient algorithms for this special case of LCP and a related problem. In particular, we present an O(k3n1.34 + kn2 log n) time algorithm for LCP in two-dimensions, which is much better for small k than an existing O(n3.2 log n) time algorithm, where n = max {|P|,|Q|}.

Cell Growth Problems

1967 ◽  
Vol 19 ◽  
pp. 851-863 ◽  
Author(s):  
David A. Klarner
Keyword(s):  
Cell Growth ◽  
Equivalence Class ◽  
Equivalence Classes ◽  
Lattice Points ◽  
The Other ◽  
Square Lattice ◽  
Cut Set ◽  
Point Set ◽  
Infinite Set ◽  
Interior Points

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn, which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn, or (ii) in Tn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

scholarly journals Computing the Partition Function for Perfect Matchings in a Hypergraph

2011 ◽  
Vol 20 (6) ◽  
pp. 815-835 ◽  
Author(s):  
ALEXANDER BARVINOK ◽  
ALEX SAMORODNITSKY
Keyword(s):  
Partition Function ◽  
Polynomial Time ◽  
Pairwise Disjoint ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Perfect Matchings ◽  
Constant Factor ◽  
Higher Dimensional ◽  
Element Set

Given non-negative weightswSon thek-subsetsSof akm-element setV, we consider the sum of the productswS1⋅⋅⋅wSmover all partitionsV=S1∪ ⋅⋅⋅ ∪Sminto pairwise disjointk-subsetsSi. When the weightswSare positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial inmfactor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman–Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

2012 ◽  
Vol 22 (01) ◽  
pp. 27-44 ◽  
Author(s):  
HEE-KAP AHN ◽  
CHRISTIAN KNAUER ◽  
MARC SCHERFENBERG ◽  
LENA SCHLIPF ◽  
ANTOINE VIGNERON
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Fréchet Distance ◽  
Planar Case ◽  
Running Time ◽  
Frechet Distance ◽  
Axis Parallel ◽  
Polygonal Curves ◽  
Orthogonal Case ◽  
Improved Algorithm

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

GEOMETRIC OPTIMIZATION PROBLEMS OVER SLIDING WINDOWS

2006 ◽  
Vol 16 (02n03) ◽  
pp. 145-157 ◽  
Author(s):  
TIMOTHY M. CHAN ◽  
BASHIR S. SADJAD
Keyword(s):  
Optimization Problems ◽  
Simple Algorithm ◽  
Sliding Window ◽  
Constant Factor ◽  
Geometric Optimization ◽  
One Dimension ◽  
Sliding Windows ◽  
One Dimensional ◽  
Previous Solution ◽  
Point Set

We study the problem of maintaining a (1 + ∊)-factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store [Formula: see text] points at any time, where the parameter R denotes the "spread" of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.

FITTING FLATS TO POINTS WITH OUTLIERS

2011 ◽  
Vol 21 (05) ◽  
pp. 559-569
Author(s):  
GUILHERME D. DA FONSECA
Keyword(s):  
Lower Bound ◽  
Computer Science ◽  
Upper Bound ◽  
Arbitrary Constant ◽  
Fundamental Problem ◽  
Constant Factor ◽  
Infinite Cylinder ◽  
Running Time ◽  
Point Set ◽  
Set Of Points

Determining the best shape to fit a set of points is a fundamental problem in many areas of computer science. We present an algorithm to approximate the k-flat that best fits a set of n points with n - m outliers. This problem generalizes the smallest m-enclosing ball, infinite cylinder, and slab. Our algorithm gives an arbitrary constant factor approximation in O(nk+2/m) time, regardless of the dimension of the point set. While our upper bound nearly matches the lower bound, the algorithm may not be feasible for large values of k. Fortunately, for some practical sets of inliers, we reduce the running time to O(nk+2/mk+1), which is linear when m = Ω(n).

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