scholarly journals Output sensitive algorithms for covering many points

2015 ◽  
Vol Vol. 17 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Hossein Ghasemalizadeh ◽  
Mohammadreza Razzazi
Keyword(s):  
Positive Integer ◽  
Time Algorithm ◽  
Covering Problems ◽  
Discrete Algorithms ◽  
International Audience ◽  
Previous Algorithm ◽  
Set Of Points

Discrete Algorithms International audience In this paper we devise some output sensitive algorithms for a problem where a set of points and a positive integer, m, are given and the goal is to cover a maximal number of these points with m disks. We introduce a parameter, ρ, as the maximum number of points that one disk can cover and we analyse the algorithms based on this parameter. At first, we solve the problem for m=1 in O(nρ) time, which improves the previous O(n2) time algorithm for this problem. Then we solve the problem for m=2 in O(nρ + 3 log ρ) time, which improves the previous O(n3 log n) algorithm for this problem. Our algorithms outperform the previous algorithms because ρ is much smaller than n in many cases. Finally, we extend the algorithm for any value of m and solve the problem in O(mnρ + (mρ)2m - 1 log mρ) time. The previous algorithm for this problem runs in O(n2m - 1 log n) time and our algorithm usually runs faster than the previous algorithm because mρ is smaller than n in many cases. We obtain output sensitive algorithms by confining the areas that we should search for the result. The techniques used in this paper may be applicable in other covering problems to obtain faster algorithms.

scholarly journals A randomized algorithm for finding a maximum clique in the visibility graph of a simple polygon

2015 ◽  
Vol Vol. 17 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Sergio Cabello ◽  
Maria Saumell
Keyword(s):  
Hamiltonian Cycle ◽  
Randomized Algorithm ◽  
Maximum Clique ◽  
Simple Polygon ◽  
Maximum Size ◽  
Visibility Graph ◽  
Probability Of Error ◽  
Discrete Algorithms ◽  
International Audience ◽  
Previous Algorithm

Discrete Algorithms International audience We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.

scholarly journals Isomorphism of graph classes related to the circular-ones property

2013 ◽  
Vol Vol. 15 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Andrew R. Curtis ◽  
Min Chih Lin ◽  
Ross M. Mcconnell ◽  
Yahav Nussbaum ◽  
Francisco Juan Soulignac ◽  
...  
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Circular Arc ◽  
Graph Classes ◽  
Discrete Algorithms ◽  
International Audience ◽  
Related Graph ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Discrete Algorithms International audience We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.

scholarly journals A linear time algorithm for finding an Euler walk in a strongly connected 3-uniform hypergraph

2012 ◽  
Vol Vol. 14 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Zbigniew Lonc ◽  
Pawel Naroski
Keyword(s):  
Linear Time ◽  
Natural Extension ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Uniform Hypergraph ◽  
Graph Theoretic ◽  
Uniform Hypergraphs ◽  
Discrete Algorithms ◽  
Strongly Connected ◽  
International Audience

Discrete Algorithms International audience By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

LABELING POINTS WITH RECTANGLES OF VARIOUS SHAPES

2002 ◽  
Vol 12 (06) ◽  
pp. 511-528 ◽  
Author(s):  
ATSUSHI KOIKE ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI ◽  
TAKESHI TOKUYAMA ◽  
SHUHEI WATANABE
Keyword(s):  
Time Algorithm ◽  
Scaling Factor ◽  
Left Part ◽  
Set Of Points

We deal with a map-abeling problem, named LOFL (Left-part Ordered Flexible Labeling), to label a set of points in a plane in the presence of polygonal obstacles. The label for each point is selected from a set of rectangles with various shapes satisfying the left-part ordered property, and is placed near to the point after scaled by a scaling factor σ which is common to all points. In this paper, we give an optimal O((n + m) log (n + m)) algorithm to decide the feasibility of LOFL for a fixed scaling factor σ, and an O((n + m) log 2 (n + m)) time algorithm to find the largest feasible scaling factor σ, where n is the number of points and m is the total number of edges of the polygonal obstacles.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

scholarly journals Tameness of Complex Dimension in a Real Analytic Set

2013 ◽  
Vol 65 (4) ◽  
pp. 721-739
Author(s):  
Janusz Adamus ◽  
Serge Randriambololona ◽  
Rasul Shafikov
Keyword(s):  
Positive Integer ◽  
Complex Manifold ◽  
Analytic Set ◽  
Complex Dimension ◽  
Real Analytic ◽  
Set Of Points

AbstractGiven a real analytic set X in a complex manifold and a positive integer d, denote by Ad the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that Ad is a closed semianalytic subset of X.

AN APPROXIMATE MORPHING BETWEEN POLYLINES

2005 ◽  
Vol 15 (02) ◽  
pp. 193-208 ◽  
Author(s):  
SERGEY BEREG
Keyword(s):  
Shortest Path ◽  
Optimization Problem ◽  
Linear Time ◽  
Medial Axis ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Geodesic Path ◽  
Minimum Width ◽  
Approximation Factor ◽  
Previous Algorithm

We consider the problem of continuously transforming or morphing one simple polyline into another polyline so that every point p of the initial polyline moves to a point q of the final polyline using the geodesic shortest path from p to q. The width of a morphing is defined as the longest geodesic path between corresponding points of the polylines. The optimization problem is to compute a morphing that minimizes the width. We present a linear-time algorithm for finding a morphing with width guaranteed to be at most two times the minimum width of a morphing. This improves the previous algorithm10 by a factor of log n. We develop a linear-time algorithm for computing a medial axis separator. We also show that the approximation factor is less than two for κ-straight polylines.

scholarly journals Explicit Correlation Amplifiers for Finding Outlier Correlations in Deterministic Subquadratic Time

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3306-3337
Author(s):  
Matti Karppa ◽  
Petteri Kaski ◽  
Jukka Kohonen ◽  
Padraig Ó Catháin
Keyword(s):  
Positive Integer ◽  
Binary Data ◽  
Randomized Algorithm ◽  
Time Algorithm ◽  
High Dimensionality ◽  
Parameter Range ◽  
Technical Tool ◽  
Similarity Join ◽  
Explicit Correlation ◽  
Main Technical Tool

Abstract We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders by Reingold et al. (Ann Math 155(1):157–187, 2002). We say that a function $$f:\{-1,1\}^d\rightarrow \{-1,1\}^D$$ f : { - 1 , 1 } d → { - 1 , 1 } D is a correlation amplifier with threshold $$0\le \tau \le 1$$ 0 ≤ τ ≤ 1 , error $$\gamma \ge 1$$ γ ≥ 1 , and strength p an even positive integer if for all pairs of vectors $$x,y\in \{-1,1\}^d$$ x , y ∈ { - 1 , 1 } d it holds that (i) $$|\langle x,y\rangle |<\tau d$$ | ⟨ x , y ⟩ | < τ d implies $$|\langle f(x),f(y)\rangle |\le (\tau \gamma )^pD$$ | ⟨ f ( x ) , f ( y ) ⟩ | ≤ ( τ γ ) p D ; and (ii) $$|\langle x,y\rangle |\ge \tau d$$ | ⟨ x , y ⟩ | ≥ τ d implies $$\left (\frac{\langle x,y\rangle }{\gamma d}\right )^pD \le \langle f(x),f(y)\rangle \le \left (\frac{\gamma \langle x,y\rangle }{d}\right )^pD$$ ⟨ x , y ⟩ γ d p D ≤ ⟨ f ( x ) , f ( y ) ⟩ ≤ γ ⟨ x , y ⟩ d p D .

Sign in / Sign up

Export Citation Format

Share Document