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Cabello S.,IMFM | Cabello S.,University of Ljubljana | Cheong O.,KAIST | Knauer C.,University of Bayreuth | Schlipf L.,Free University of Berlin
Computational Geometry: Theory and Applications | Year: 2015

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O(n3). We also give (1-ε)-approximation algorithms that take time O(ε-1/2log n+ε-3/2). © 2015 Elsevier B.V.

Kozak J.,University of Ljubljana | Krajnc M.,University of Ljubljana | Vitrih V.,IMFM | Vitrih V.,University of Primorska
Computer Aided Geometric Design | Year: 2016

In this paper a new approach for a construction of polynomial surfaces with rational field of unit normals (PN surfaces) is presented. It is based on bivariate polynomials with quaternion coefficients. Relations between these coefficients are derived that allow one to construct PN surfaces of general odd and even degrees. For low degree PN surfaces the theoretical results are supplemented with algorithms and illustrated with numerical examples. © 2016 Elsevier B.V.

Zerovnik J.,University of Ljubljana | Kuzma B.,IMFM | Kuzma B.,University of Primorska
Proceedings of the 11th International Symposium on Operational Research in Slovenia, SOR 2011 | Year: 2011

Densities of ball packing with two different sizes of balls is studied. For several ratios between the radii estimates for densities on unbounded regions are given.

Cabello S.,IMFM | Cabello S.,University of Ljubljana | Jejcic M.,University of Ljubljana
Computational Geometry: Theory and Applications | Year: 2015

Let G be a unit disk graph in the plane defined by n disks whose positions are known. For the case when G is unweighted, we give a simple algorithm to compute a shortest path tree from a given source in O(nlog n) time. For the case when G is weighted, we show that a shortest path tree from a given source can be computed in O(n1+ε) time, improving the previous best time bound of O(n4/3+ε). © 2014 Published by Elsevier B.V.

Dolinar G.,University of Ljubljana | Guterman A.E.,Moscow State University | Kuzma B.,University of Primorska | Orel M.,IMFM
European Journal of Combinatorics | Year: 2011

Let F be a finite field of characteristic different from 2. We show that no bijective map transforms the permanent into the determinant when the cardinality of F is sufficiently large. We also give an example of a non-bijective map when F is arbitrary and an example of a bijective map when F is infinite which do transform the permanent into the determinant. The technique developed allows us to estimate the probability of the permanent and the determinant of matrices over finite fields having a given value. Our results are also true over finite rings without zero divisors. © 2010 Elsevier Ltd.

Kutnar K.,University of Primorska | Marusic D.,University of Primorska | Marusic D.,University of Ljubljana | Sparl P.,IMFM | And 2 more authors.
European Journal of Combinatorics | Year: 2013

A vertex-transitive graph X is said to be half-arc-transitive if its automorphism group acts transitively on the set of edges of X but does not act transitively on the set of arcs of X. A classification of half-arc-transitive graphs on 4p vertices, where p is a prime, is given. Apart from an obvious infinite family of metacirculants, which exist for p ≡ 1 ( mod 4) and have been known before, there is an additional somewhat unique family of half-arc-transitive graphs of order 4p and valency 12; the latter exists only when p ≡ 1( mod 6) is of the form 22k + 2k + 1, k > 1. © 2013 Elsevier Ltd.

Sparl P.,University of Maribor | Zerovnik J.,IMFM | Zerovnik J.,University of Ljubljana | Witkowski R.,Adam Mickiewicz University
Algorithmica | Year: 2012

In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by frequencies assigned to them, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where every vertex knows its position in the graph. We present a 1-local 7/5-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 17/12-competitive algorithm. © 2011 The Author(s).

Cabello S.,IMFM | Cabello S.,University of Ljubljana | Colin De Verdiere E.,Ecole Normale Superieure de Paris | Lazarus F.,CNRS GIPSA Laboratory
Computational Geometry: Theory and Applications | Year: 2012

Let G be an unweighted graph of complexity n embedded in a surface of genus g, orientable or not. We describe improved algorithms to compute a shortest non-contractible and a shortest non-separating cycle in G. If k is an integer, we can compute such a non-trivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edge-width or face-width of a graph is bounded from above by a constant. This also implies an output-sensitive algorithm to compute a shortest non-trivial cycle that runs in O(gn k0) time, where k0 is the length of the cycle. We also give an approximation algorithm for the shortest non-trivial cycle. If a parameter 0<ε<1 is given, we compute in O(gn/ε) time a non-trivial cycle whose length is at most 1+ε times the length of the shortest non-trivial cycle. © 2011 Elsevier B.V.

Changat M.,Kerala University | Lekha D.S.,Kerala University | Peterin I.,University of Maribor | Subhamathi A.R.,Nss College Rajakumari | And 2 more authors.
Discrete Optimization | Year: 2015

Abstract We introduce a game which is played by two players on a connected graph G. The players I and II alternatively choose vertices of the graph until all vertices are taken. The set of vertices chosen by player I is denoted by ΠI, and by II is denoted by ΠII. Let d(x,π)=Σyâπd(x,y) and let M(π)=min{d(x,π)â£xâπ} be the median value of a profile π⊆V(G). The objective of player I is to maximize M(πII)-M(πI) and the objective of player II is to minimize M(πII)-M(πI). The winner of the game is the player with the smaller median value of her profile. We give a necessary condition for a tree so that player I (who begins the game) has a winning strategy for the game. We prove also that for hypercubes and some other symmetric graphs the player II has a strategy to draw the game. Complete bipartite graphs are considered as well. © 2015 Elsevier B.V.

Vodopivec A.,IMFM | Kaatz F.H.,University of Advancing Technology | Kaatz F.H.,Chandler Gilbert Community College | Kaatz F.H.,Mesa Community College | Mohar B.,Simon Fraser University
Journal of Mathematical Chemistry | Year: 2010

The topographical Wiener index is calculated for two-dimensional graphs describing porous arrays, including bee honeycomb. For tiling in the plane, we model hexagonal, triangular, and square arrays and compare with topological formulas for the Wiener index derived from the distance matrix. The normalized Wiener indices of C 4, T 13, and O(4), for hexagonal, triangular, and square arrays are 0.993, 0.995, and 0.985, respectively, indicating that the arrays have smaller bond lengths near the center of the array, since these contribute more to the Wiener index. The normalized Perron root (the first eigenvalue, λ 1), calculated from distance/distance matrices describes an order parameter, Φ = λ 1/n, where Φ = 1 for a linear graph and n is the order of the matrix. This parameter correlates with the convexity of the tessellations. The distributions of the normalized distances for nearest neighbor coordinates are determined from the porous arrays. The distributions range from normal to skewed to multimodal depending on the array. These results introduce some new calculations for 2D graphs of porous arrays. © Springer Science+Business Media, LLC 2009.

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