Tree-Like Structure in Graphs and Embedability to Trees A
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TREE-LIKE STRUCTURE IN GRAPHS AND EMBEDABILITY TO TREES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Muad Mustafa Abu-Ata May 2014 Dissertation written by Muad Mustafa Abu-Ata B.S., Yarmouk University, 2000 M.Sc., Yarmouk University, 2003 Ph.D., Kent State University, 2014 Approved by Dr. Feodor F. Dragan , Chair, Doctoral Dissertation Committee Dr. Ruoming Jin , Members, Doctoral Dissertation Committee Dr. Ye Zhao Dr. Artem Zvavitch Accepted by Dr. Javed Khan , Chair, Department of Computer Science Dr. James L. Blank , Dean, College of Arts and Sciences ii TABLE OF CONTENTS LIST OF FIGURES . vii LIST OF TABLES . ix Acknowledgements . xi Dedication . xii 1 Introduction . 1 1.1 Research contribution . 5 1.2 Publication notes . 7 1.3 Preliminaries and Notations . 7 1.3.1 Tree-decomposition . 11 1.4 Related work . 13 1.4.1 Low distortion embedding . 13 1.4.2 Embedding into a metric of a (weighted) tree. 14 1.4.3 Tree spanners . 16 1.4.4 Sparse spanners . 17 1.4.5 Collective tree spanners . 19 1.4.6 Spanners with bounded tree-width. 20 iii 2 Metric tree-like structures in real-life networks: an empirical study . 21 2.1 Introduction . 21 2.2 Datasets . 24 2.3 Layering Partition, its Cluster-Diameter and Cluster-Radius . 28 2.4 Hyperbolicity . 33 2.5 Tree-Distortion . 39 2.6 Tree-Breadth, Tree-Length and Tree-Stretch . 46 2.7 Use of Metric Tree-Likeness . 53 2.7.1 Approximate distance queries . 53 2.7.2 Approximating optimal routes . 56 2.7.3 Approximating diameter and radius . 58 2.8 Conclusion . 61 3 Collective Additive Tree Spanners and the Tree-Breadth of a Graph with Con- sequences . 65 3.1 Introduction . 65 3.2 Collective Additive Tree Spanners and the Tree-Breadth of a Graph . 68 3.3 Hierarchical decomposition of a graph with bounded tree-breadth . 69 3.4 Construction of collective additive tree spanners . 72 3.5 Additive spanners for graphs admitting (multiplicative) tree t-spanners . 80 4 Collective Additive Tree Spanners of Graphs with Bounded k-Tree-Breadth, k ≥ 2 81 4.1 Introduction . 81 iv 4.2 Balanced separators for graphs with bounded k-tree-breadth . 82 4.3 Decomposition of a graph with bounded k-tree-breadth . 85 4.4 Construction of a hierarchical tree . 87 4.5 Construction of collective additive tree spanners . 89 4.6 Additive Spanners for Graphs Admitting (Multiplicative) t-Spanners of Bounded Tree-width. 93 4.6.1 k-Tree-breadth of a graph admitting a t-spanner of bounded tree- width . 93 4.6.2 Consequences . 95 5 Embedding of Weighted Graphs into Trees: Theoretical Grounds and Empirical Analysis on Real Datasets . 97 5.1 Layering partition for weighted graphs . 98 5.2 Properties of layering partition for weighted graphs . 99 5.3 Construction of tree embedding . 102 5.4 Experiment . 107 5.4.1 Datasets . 107 5.4.2 Layering partition results . 113 5.4.3 Non-contractive embedding results . 113 5.4.4 Edge subdivision (h ≤ w)...................... 115 5.4.5 Contractive embedding: weighting clusters with their own diameters118 5.4.6 Embedding with recursive partitioning of clusters . 118 6 Conclusion and Future Work . 123 v BIBLIOGRAPHY . 127 vi LIST OF FIGURES 1 A graph and its tree-decomposition of width 3, of length 3, and of breadth 2......................................... 12 2 Layering partition and associated constructs. 29 3 Illustration to the proof of Proposition 3. 38 0 4 Embedding into trees H; H` and H`...................... 42 5 Illustration to the proof of Proposition 9. 48 6 Distortion distribution for embedding of a graph dataset into its canonic tree H...................................... 55 7 Four tree-likeness measurements scaled. 64 8 Tree-likeness measurements: pairwise comparison. 64 9 A graph G with a disk-separator Dr(v; G) and the corresponding graphs + + G1 ;:::;G4 obtained from G. c1; : : : ; c4 are meta vertices representing the disk Dr(v; G) in the corresponding graphs. 70 10 a) A graph G and its balanced disk-separator D1(13;G). b) A hierarchical 0 0 tree H(G) of G. We have G = G(# Y ), Y = D1(13;G). Meta vertices are shown circled, disk centers are shown in bold. c) The graph G(# Y 1) with 1 1 1 its balanced disk-separator D1(23;G(# Y )) = Y . G(# Y ) is a minor of G(# Y 0). d) The graph G(# Y 2), a minor of G(# Y 1) and of G(# Y 0). Y 2 = V (G(# Y 2)) is a leaf of H(G)...................... 73 vii 11 Illustration to the proof of Lemma 4: \unfolding" meta vertices. 75 12 Illustration to the proof of Lemma 7. 77 3 13 A graph G with a balanced Dr -separator and the corresponding graphs + + + G1 ;:::;G4 obtained from G. Each Gi has three meta vertices represent- ing the three disks. 86 14 Illustration to the proof of Lemma 14. A tree-decomposition for G is obtained from a tree-decomposition of H................... 96 15 A layering partition of a weighted graph G.................. 100 16 Illustration of proof of Lemma 17. 105 17 Cluster-width versus average distortion, maximum distortion and number of dummy vertices for the Celegans dataset. 116 18 Cluster-width versus average distortion, maximum distortion and number of dummy vertices for the CornellKing dataset. 117 viii LIST OF TABLES 1 Known results on approximate embedding problems for multiplicative dis- tortion; λ is used to denote the optimal distortion and n to denote the number of points in the input metric. The table contains only the results that hold for the multiplicative definition of the distortion; there is a rich body of work that applies to other definitions of distortion, notably the additive or average distortion, see [17] for an overview. 15 2 Graph datasets and their parameters: number of vertices, number of edges, diameter, radius. 25 3 Layering partitions of the datasets and their parameters. ∆s(G) is the largest diameter of a cluster in LP(G; s), where s is a randomly selected start vertex. For all datasets, the average diameter of a cluster is between 0 and 1. For most datasets, more than 95% of clusters are cliques. 31 4 Frequency of diameters of clusters in layering partition LP(G; s) (three datasets). 32 5 δ-hyperbolicity of the graph datasets. 35 6 Relative frequency of δ-hyperbolicity of quadruplets in our graph datasets that have less than 10K vertices. 36 7 Distortion results of embedding datasets into a canonic tree H....... 44 ix 8 Distortion results of non-contractive embedding of datasets into trees H` 0 and H`..................................... 45 9 Lower and upper bounds on the tree-breadth of our graph datasets. 50 10 Estimation of diameters and radii. 59 11 Summary of tree-likeness measurements. 62 12 Real datasets parameters: n: the number of vertices, m: the number of edges, the largest edge weight, the smallest edge weight and the diameter of the graph. 108 13 Layering partitions of the datasets and their parameters. h is the cluster- width of LP(s; h) and set equal to the longest edge weight. s is a randomly selected start vertex. 114 14 Distortion results for non-contractive embedding of the datasets into tree H. Cluster-width is equal to the largest edge weight (h = w). 115 15 Distortion results for non-contractive embedding of the datasets into tree H. Cluster-width is less than or equal the largest edge weight (h ≤ w). 117 16 Distortion results for embedding of the datasets into tree H0. Edges inside each cluster C are weighted equal to diam(C)=2. 119 17 Percentage of vertex pairs with distortion up to a given value by embedding datasets into tree H0 with own diameter weighting. 120 18 Distortion results for embedding with P-centers partitioning for datasets into tree H0. P-centers has negligible improvement of distortion for other datasets of table 12 . 122 x Acknowledgements I would like to express my deepest gratitude and thank my research advisor, Dr. Feodor F. Dragan, for mentoring me during my PhD study and research. I have learned a lot from him. Without his persistent help, patience and guidance, this dissertation would not have materialized. I cannot thank him enough for his sincere and overwhelming help and support. Also, I would like to thank my dissertation committee, Dr. Ruoming Jin, Dr. Ye Zhao and Dr. Artem Zvavitch, for their participation, comments and feedback. Finally, I would like to thank the faculty and staff of the Department of Computer Science at Kent State University for their help and support. xi This dissertation is dedicated to the memory of my mother, Hajar Ibdah, her endless love, care and support have sustained me throughout my life. Her passion, strength and faith are the greatest lessons in my life. xii CHAPTER 1 Introduction The problem of embedding a graph metric into a \nice" and \simpler" metric space with low distortion has been a subject of extensive research, motivated from several applications in various domains and for its intrinsic mathematical interest. \Nice" met- ric spaces are those with well-studied structural properties, allowing to design efficient approximation algorithms, such as Euclidean or `1 space, lines, weighted trees and dis- tributions over them. A very incomplete list of applications includes approximation algorithms for graph and network problems, such as sparsest cut [14, 126], minimum bandwidth [34, 89], low-diameter decompositions [126], buy-at-bulk network design [16], distance and routing labeling schemes [77, 79, 102, 164], and optimal group Steiner trees [48, 93], and online algorithms for metrical task systems and file migration problems [24,26].