DOCSLIB.ORG

Flips and Spanners

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Flips and Spanners FLIPSANDSPANNERS alexander jozef hubertus verdonschot A thesis submitted to the Faculty of Graduate and Post Doctoral Affairs in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science Carleton University Ottawa, Ontario, Canada arXiv:1509.02563v1 [cs.CG] 8 Sep 2015 © 2015 Alexander Jozef Hubertus Verdonschot ii ABSTRACT In this thesis, we study two different graph problems. The first problem revolves around geometric spanners. Here, we have a set of points in the plane and we want to connect them with straight line segments, such that there is a path between each pair of points and these paths do not require large detours. If we achieve this, the resulting graph is called a spanner. We focus our attention on two graphs (the Θ-graph and Yao-graph) that are constructed by connect- ing each point with its nearest neighbour in a number of cones. Al- though this construction is very straight-forward, it has proven chal- lenging to fully determine the properties of the resulting graphs. We show that if the construction uses 5 cones, the resulting graphs are still spanners. This was the only number of cones for which this ques- tion remained unanswered. We also present a routing strategy (a way to decide where to go next, based only on our current location, its direct neighbourhood, and our destination) on the half-Θ6-graph, a variant of the graph with 6 cones. We show that our routing strat- egy avoids large detours: it finds a path whose length is at most a constant factor from the straight-line distance between the endpoints. Moreover, we show that this routing strategy is optimal. In the second part, we turn our attention to flips in triangulations. A flip is a simple operation that transforms one triangulation into an- other. It turns out that with enough flips, we can transform any trian- gulation into any other. But how many flips is enough? We present an improved upper bound of 5.2n - 33.6 on the maximum flip distance between any pair of triangulations with n vertices. Along the way, we prove matching lower bounds on each step in the current algorithm, including a tight bound of (3n - 9)=5 flips needed to make a trian- b c gulation 4-connected. In addition, we prove tight Θ(n log n) bounds on the number of flips required in several settings where the edges have unique labels. iii iv ACKNOWLEDGMENTS I would not have been able to write this thesis without the help and support of many people. First of all, I would like to thank my supervisors – Prosenjit Bose, Pat Morin, and Vida Dujmovi´c.They are all I could have wished for in my supervisors and more. Combining a wealth of knowledge with a burning curiosity and a penchant for finding fascinating, yet ap- proachable, open problems, they made these past five years into a journey of exploration and excitement. I also want to thank the other members and students of the Com- putational Geometry lab for making it such a nice place to work (and occasionally not work). I am especially grateful to fellow PhD stu- dents André, Carsten, Dana, and Luis, for being great friends and collaborators. In fact, I am very grateful to all the researchers and students who I got to work with during my PhD studies. Working to- gether was always a pleasure, and they taught me more than classes ever could. Finally, I would like to thank my family and friends for their sup- port during these long, and at times stressful, years. I am especially grateful to my mother for encouraging me to take the leap of faith that is an international PhD, and to my partner, Gehana, for her un- failing love and support. Thank you! v vi CONTENTS abstract iii acknowledgments v contents vii 1 summary of the thesis1 1.1 Geometric spanners 1 1.2 Flips in triangulations 2 i geometric spanners5 2 an introduction to yao- and Θ-graphs7 2.1 Geometric spanners 7 2.2 Preliminaries 9 2.3 Yao-graphs 10 2.4 Θ-graphs 14 2.5 History 16 3 the Θ5 -graph is a spanner 21 3.1 Introduction 21 3.2 Connectivity 23 3.3 Spanning ratio 25 3.4 Lower bound 35 3.5 Conclusions 38 4 competitive routing in the half-Θ6 -graph 41 4.1 Introduction 41 4.2 Preliminaries 43 4.3 Spanning ratio of the half-Θ6-graph 45 4.4 Remarks on the spanning ratio 48 4.5 Routing in the half-Θ6-graph 49 4.5.1 Positive routing 51 4.5.2 Negative routing 57 4.6 A stateful algorithm 60 4.7 Bounding the maximum degree 61 4.7.1 Routing in G12 63 4.7.2 Routing in G9 67 4.8 Conclusions 68 ii flips in triangulations 73 5 a history of flips in combinatorial triangula- tions 75 5.1 Introduction 75 5.2 Wagner’s bound 77 5.3 Komuro’s bound 79 5.4 Mori et al.’s bound 81 5.5 Lower bounds 84 vii viii contents 6 making triangulations 4-connected using flips 87 6.1 Introduction 87 6.2 Upper bound 88 6.3 Lower bound 102 6.4 Conclusions and open problems 104 6.5 Lemmas and proofs 105 7 edge-labelled flips 109 7.1 Introduction 109 7.2 Convex polygons 111 7.2.1 Upper bound 111 7.2.2 Lower bound 114 7.2.3 Simultaneous flips 117 7.3 Combinatorial triangulations 120 7.3.1 Upper bound 120 7.3.2 Lower bound 123 7.3.3 Simultaneous flips 124 7.4 Pseudo-triangulations 126 7.4.1 Pointed pseudo-triangulations 128 7.4.2 General pseudo-triangulations 134 7.5 Conclusions and open problems 139 SUMMARYOFTHETHESIS 1 This thesis is comprised of two main parts. The first part, found in Chapters 2 through 4, deals with geometric spanners. Chapters 5 through 7 contain the second part, which focuses on flips in triangula- tions. A brief introduction and summary of each part is given below. The first chapter of each part provides a more detailed introduction. The common theme in the two parts is that both deal with graphs. A graph consists of a set of vertices, some of which are connected by edges. In this thesis, all graphs will be simple, which means that there is at most one edge connecting each pair of vertices, and edges cannot connect a vertex to itself. 1.1 geometric spanners Spanners can be informally described as graphs in which one never needs to make a large detour. That is, the shortest path between two vertices is proportional to their actual distance. Road networks are a good example; nearby cities are typically connected by a direct road, so that the total distance travelled is not much more than the distance ‘as the crow flies’. Spanners have been studied in many different con- texts, but we will focus on geometric spanners, where the vertices are points in the plane, and the length of an edge is the Euclidean dis- tance between its endpoints. The spanning ratio is the maximum ratio between the shortest path in the graph and the straight-line distance between any pair of vertices. Chapter 2 gives an in-depth introduction to geometric spanners in general, and simple cone-based spanners in particular. The Θ-graph is one such cone-based spanner. To construct it, we partition the plane around each vertex into a number of equiangular cones and add an edge to the ‘closest’ vertex in each cone, where the closest vertex is defined as the vertex whose projection on the bisector of the cone is closest. It has been shown that for any desired spanning ratio t, there is a number of cones k such that the Θ-graph with k cones (typically written as Θk) is guaranteed to have spanning ratio t. However, it was not known exactly for which values of k the span- ning ratio of Θk is bounded by a constant. It was known that Θ3 and below are not constant spanners, while Θ6 and up are. Recently, Θ4 was shown to be a constant spanner as well, leaving the question unanswered only for Θ5. In Chapter 3, we prove that Θ5 is, indeed, a constant spanner. With the earlier results, this implies that Θk is a spanner for all k > 4. This result was first published in the proceed- 1 2 summaryofthethesis ings of the 39th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2013)[8] and later appeared in Computa- tional Geometry: Theory and Applications [9]. Of course, knowing that there exists a short path to where you want to go is not the end of the story: you also have to know how to find it. This is called routing, or competitive routing if the spanning ratio of the resulting path is bounded by a constant. If you know the entire graph, routing is nothing more than computing a path, but most settings consider the more restricted scenario where you know your destination, but you can only see your current location and its neighbours. This is referred to as local routing. In Chapter 4, we present a local, competitive routing strategy for the half-Θ6-graph, which is closely related to Θ6. Our strategy achieves a routing ratio of 5=p3 = 2.886 : : : , which seems slightly disappointing compared to the spanning ratio of 2. This makes it all the more surprising that we managed to show that our algorithm is, in fact, optimal: no other routing strategy can achieve a better routing ratio, under the same restrictions.
Load more

Recommended publications
  • Weighted Additive Spanners
    Weighted Additive Spanners Reyan Ahmed1, Greg Bodwin2, Faryad Darabi Sahneh1, Stephen Kobourov1, and Richard Spence1 1 Department of Computer Science, University of Arizona 2 Department of Computer Science, Georgia Institute of Technology Abstract. A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic +2 and +4 unweighted spanners (both all-pairs and pairwise) to +2W and +4W weighted spanners, where W is the maximum edge weight. For a technical reason, the +6 unweighted spanner becomes a +8W weighted spanner; closing this error gap is an interesting remaining open problem. Keywords: Additive spanner · Pairwise spanner · Shortest-path neigh- borhood 1 Introduction An f(·)-spanner of an undirected graph G = (V; E) with jV j = n nodes and jEj = m edges is a subgraph H which preserves pairwise distances in G up to some error prescribed by f; that is, distH (s; t) ≤ f(distG(s; t)) for all nodes s; t 2 V: Spanners were introduced by Peleg and Sch¨affer[25] in the setting with multi- plicative error of type f(d) = cd for some positive constant c.
    [Show full text]
  • Light Spanners for High Dimensional Norms Via Stochastic Decompositions
    1 Light Spanners for High Dimensional Norms via 2 Stochastic Decompositions 1 3 Arnold Filtser 4 Ben-Gurion University of the Negev ,Beer-Sheva, Israel 5 [email protected] 2 6 Ofer Neiman 7 Ben-Gurion University of the Negev, Beer-Sheva, Israel 8 [email protected] 9 Abstract 10 Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling 11 metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, 12 where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that 1+1/t2 13 any n-point Euclidean metric has an O(t)-spanner with O˜(n ) edges, little is known. 14 In this paper we study several aspects of spanners in high dimensional normed spaces. First, 15 we build spanners for finite subsets of `p with 1 < p ≤ 2. Second, our construction yields a 16 spanner which is both sparse and also light, i.e., its total weight is not much larger than that of 17 the minimum spanning tree. In particular, we show that any n-point subset of `p for 1 < p ≤ 2 1+O˜(1/tp) O˜(1/tp) 18 has an O(t)-spanner with n edges and lightness n . 19 In fact, our results are more general, and they apply to any metric space admitting a certain 20 low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)- 1/t 21 spanner with lightness O(n ). We exhibit the following tradeoff: metrics with decomposability 1/t 22 parameter ν = ν(t) admit an O(t)-spanner with lightness O˜(ν ).
    [Show full text]
  • On the Edge Crossings of the Greedy Spanner
    On the Edge Crossings of the Greedy Spanner David Eppstein ! Department of Computer Science, University of California, Irvine Hadi Khodabandeh ! Department of Computer Science, University of California, Irvine Abstract The greedy t-spanner of a set of points in the plane is an undirected graph constructed by considering pairs of points in order by distance, and connecting a pair by an edge when there does not already exist a path connecting that pair with length at most t times the Euclidean distance. We prove that, for any t > 1, these graphs have at most a linear number of crossings, and more strongly that the intersection graph of edges in a greedy t-spanner has bounded degeneracy. As a consequence, we prove a separator theorem for greedy spanners: any k-vertex subgraph of a greedy spanner√ can be partitioned into sub-subgraphs of size a constant fraction smaller, by the removal of O( k) vertices. A recursive separator hierarchy for these graphs can be constructed from their planarizations in linear time, or in near-linear time if the planarization is unknown. 2012 ACM Subject Classification Theory of computation → Sparsification and spanners; Theory of computation → Computational geometry; Theory of computation → Design and analysis of algorithms Keywords and phrases Geometric Spanners, Greedy Spanners, Separators, Crossing Graph, Sparsity Funding This work was supported in part by the US National Science Foundation under grant CCF-1616248. 1 Introduction Geometric spanners are geometric graphs whose distances approximate distances in complete graphs, while having fewer edges than complete graphs. Given a set of points V on the Euclidean plane (or in any other metric space), a t-spanner on V can be defined as a graph S having V as its set of vertices V and satisfying the following inequality for every pair of points (P, Q): dS(P, Q) ≤ t · d(P, Q) (1) where dS(P, Q) is the length of the shortest path between P and Q using the edges in S, and d(P, Q) is the Euclidean distance of P and Q.
    [Show full text]
  • Experimental Study of Kinetic Geometric T-Spanner Algorithms
    Experimental Study of Kinetic Geometric t-Spanner Algorithms Master thesis by Jonas Suhr Christensen, 20032491 [email protected] Supervised by Gerth Stølting Brodal Department of Computer Science Aarhus University Denmark 31 August, 2010 Abstract In this thesis we describe, implements and experiments with two algorithms that constructs and maintains geometric t-spanners. The algorithms are the Kinetic Spanner in Rd by Abam et al. and the Deformable Spanner by Gao et al. The implementations has only been done in R2. The experiments are focused on two different aspects of the algorithms namely the qualities of the spanners constructed by the algorithms and the kinetic qualities of the algorithms. The results of the experiments are compared with other known non-kinetic spanner algorithms and a kinetic Delaunay triangulation. The results shows that the qualities of the spanners in general is worse than the spanners constructed by the non-kinetic algorithms. However on input with specific characteristics properties of the constructed spanners are on level with spanners of the non-kinetic algorithms. The experiments on the kinetic qualities can be summed up to that the kinetic spanner in Rd is a lot more expensive to maintain than the kinetic Delaunay triangulation. The deformable spanner is not a part of experiments on the kinetic properties and the experiments do not consider the responsiveness of the kinetic spanner in Rd. 1 Acknowledgements I would like to thank Mohammad Ali Abam for guidance on implementing and understanding the kinetic algorithms. Furthermore I would like to thank Gerth Stølthing Brodal for advises doing the process of writing the thesis.
    [Show full text]
  • Sparse Hop Spanners for Unit Disk Graphs
    Sparse Hop Spanners for Unit Disk Graphs Adrian Dumitrescu Department of Computer Science, University of Wisconsin–Milwaukee, WI, USA [email protected] Anirban Ghosh School of Computing, University of North Florida, Jacksonville, FL, USA [email protected] Csaba D. Tóth Department of Mathematics, California State University Northridge, Los Angeles, CA, USA Department of Computer Science, Tufts University, Medford, MA, USA [email protected] Abstract A unit disk graph G on a given set of points P in the plane is a geometric graph where an edge exists between two points p, q ∈ P if and only if |pq| ≤ 1. A subgraph G0 of G is a k-hop spanner if and only if for every edge pq ∈ G, the topological shortest path between p, q in G0 has at most k edges. We obtain the following results for unit disk graphs. I Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n. II Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges. III Every n-vertex unit disk graph has a 2-hop spanner with O(n log n) edges. This is the first nontrivial construction of 2-hop spanners. IV For every sufficiently large n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4.
    [Show full text]
  • Graph Spanners: a Tutorial Review
    Graph Spanners: A Tutorial Review Reyan Ahmeda, Greg Bodwinc, Faryad Darabi Sahneha, Keaton Hammb, Mohammad Javad Latifi Jebellib, Stephen Kobourova, Richard Spencea aDepartment of Computer Science, University of Arizona bDepartment of Mathematics, University of Arizona cDepartment of Computer Science, Georgia Institute of Technology Abstract This survey provides a guiding reference to researchers seeking an overview of the large body of literature about graph spanners. It surveys the current liter- ature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners. Contents 1 Introduction 3 1.1 Organization and Layout . .3 1.2 Notation and Preliminaries.....................4 2 Graph Spanners 5 2.1 Classical Results . .6 2.2 Sparsity and Lightness . .7 2.3 Relation to Minimum Spanning Trees . .9 2.4 Erd}os'Girth Conjecture . .9 2.5 Open Problems . 11 3 NP-hardness and Hardness of Approximation 11 arXiv:1909.03152v2 [cs.DM] 14 Mar 2020 3.1 NP-hardness of the basic t{spanner problem . 11 3.2 Label Cover and hardness of approximation of the basic t{spanner problem . 12 3.3 NP{Hardness for Planar Graphs . 13 3.4 NP{Hardness of Additive Spanners . 13 3.5 Open Problems . 13 This work is supported in part by NSF grants CCF-1740858, CCF-0721503 and DMS- 1839274. 4 The Greedy Algorithm for Multiplicative Spanners 13 4.1 Kruskal's Algorithm for Computing Minimum Spanning Trees . 14 4.2 The Greedy Algorithm for Multiplicative Spanners .
    [Show full text]
  • Master's Thesis
    Eindhoven University of Technology MASTER Geometric spanner networks Hoedemakers, C.G.J.J. Award date: 2015 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Department of Mathematics and Computer Science Algorithms and Visualization Research Group Geometric Spanner Networks Master Thesis Chris Hoedemakers Thesis committee: prof.dr. Mark de Berg prof.dr. Prosenjit Bose (supervisor) dr.ir. Huub Van De Wetering final version Eindhoven, August 2015 Abstract The topic of this thesis is geometric spanner networks. Geometric spanners are networks defined by a set of points P and a set of straight line segments E, called edges, that connect the points in P . The aim of spanners is to balance the total number of edges in the network against the largest detour one has to make when taking the shortest path of edges from an arbitrary point in P to any other.
    [Show full text]
  • On Euclidean Steiner (1 + Ε)-Spanners
    On Euclidean Steiner (1 + ε)-Spanners Sujoy Bhore ! Université Libre de Bruxelles, Brussels, Belgium Csaba D. Tóth ! California State University Northridge, Los Angeles, CA, USA Tufts University, Medford, MA, USA Abstract Lightness and sparsity are two natural parameters for Euclidean (1 + ε)-spanners. Classical results show that, when the dimension d ∈ N and ε > 0 are constant, every set S of n points in d-space admits an (1 + ε)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ε > 0 for constant d ∈ N have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1 + ε)-spanner. They gave upper bounds of O˜(ε−(d+1)/2) for the minimum lightness in dimensions d ≥ 3, and O˜(ε−(d−1))/2) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed √ Steiner (1 + ε)-spanners of lightness O(ε−1 log ∆) in the plane, where ∆ ∈ Ω( n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair ofpoints. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1 + ε)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ε−d/2) for the lightness and Ω(ε−(d−1)/2) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2.
    [Show full text]
  • Eficient and Simple Algorithms for Fault-Tolerant Spanners
    Eficient and Simple Algorithms for Fault-Tolerant Spanners Michael Dinitz∗ Caleb Robelle [email protected] [email protected] Johns Hopkins University University of Maryland, Baltimore County Baltimore, Maryland, USA Baltimore, Maryland, USA ABSTRACT that approximately preserve distances (as opposed to other notions It was recently shown that a version of the greedy algorithm gives of graph sparsiication which approximately preserve cuts [6], the a construction of fault-tolerant spanners that is size-optimal, at spectrum [5, 26], or other graph properties). When considering least for vertex faults. However, the algorithm to construct this spanners through the lens of sparsiication, perhaps the most im- spanner is not polynomial-time, and the best-known polynomial portant goal in the study of graph spanners is understanding the time algorithm is signiicantly suboptimal. Designing a polynomial- tradeof between the stretch and the sparsity. The main result in time algorithm to construct (near-)optimal fault-tolerant spanners this area, which is tight assuming the łErdős girth conjecturež [16], was given as an explicit open problem in the two most recent pa- was given by Althöfer et al.: pers on fault-tolerant spanners ([Bodwin, Dinitz, Parter, Vassilevka Theorem 1.1 ([1]). For every positive integer �, every weighted Williams SODA ’18] and [Bodwin, Patel PODC ’19]). We give a graph � = (�, �) has a (2� − 1)-spanner with at most � (�1+1/� ) surprisingly simple algorithm which runs in polynomial time and edges. constructs fault-tolerant spanners that are extremely close to op- timal (of by only a linear factor in the stretch) by modifying the This notion of graph spanners was irst introduced by Peleg greedy algorithm to run in polynomial time.
    [Show full text]
  • The Greedy Spanner Is Existentially Optimal∗
    The Greedy Spanner is Existentially Optimal∗ Arnold Filtser y Shay Solomon z Abstract The greedy spanner is arguably the simplest and most well-studied spanner construction. Experi- mental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families, in terms of both the size and weight. Roughly speaking, the greedy spanner is said to be existentially optimal (or near-optimal) for a graph family if the worst performance of the greedy spanner over all graphs in is just as good (or nearly as good)G as the worst performance of an optimal spanner over all graphs in G. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphsG (due to Chechik and Wulff-Nilsen [SODA’16]) and doubling metrics (due to Gottlieb [FOCS’15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near- optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area. Further, we observe that approximate-greedy spanners are existentially near-optimal as well. Con- sequently, we provide an O(n log n)-time construction of (1 + )-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb’s construction, whose runtime is O(n log2 n) and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al.
    [Show full text]
  • Games of Geometry
    Ruler of the plane - Games of geometry Citation for published version (APA): Beekhuis, S., Buchin, K., Castermans, T., Hurks, T., & Sonke, W. (2017). Ruler of the plane - Games of geometry. In B. Aronov, & M. J. Katz (Eds.), 33rd International Symposium on Computational Geometry (SoCG) (pp. 631-635). (Leibniz International Proceedings in Informatics (LIPIcs); Vol. 77). Schloss Dagstuhl - Leibniz- Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2017.63 Document license: CC BY DOI: 10.4230/LIPIcs.SoCG.2017.63 Document status and date: Published: 01/06/2017 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
    [Show full text]