Addressing and Distances for Cellular Networks with Holes
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ADDRESSING AND DISTANCES FOR CELLULAR NETWORKS WITH HOLES A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Science by Robert Allan Harbart August 2009 Thesis written by Robert Allan Harbart B.S., Kent State University, 2004 M.S., Kent State University, 2009 Approved by Dr. Feodor F. Dragan, Advisor Dr. Robert Walker,Chair, Department of Computer Science Dr. John Stalvey, Dean, College of Arts and Sciences ii TABLE OF CONTENTS LIST OF FIGURES . v LIST OF TABLES . xii Acknowledgements . xiii Dedication . xiv 1 Introduction . 1 2 Previous Work and Our Problem . 6 3 Solution For Routing And Location Update With A K-disc Hole . 12 3.1 Creating a Cut . 14 3.2 Lemma 1 . 17 3.3 Lemma 2 . 25 3.4 Create the Four Region / Four Cut graph . 29 3.5 Theorem 1 . 34 4 Extension To Any Single Convex Hole . 39 4.1 General forms of convex holes . 41 4.2 Expansion Method . 44 4.3 Creating convex shapes with the Expansion Method . 51 4.4 Applying Theorem 1 to a convex shape hole . 54 iii 5 Possible Extension To Graphs With A Few Convex Holes . 57 5.1 Case 1 - Cuts do not intersect another hole . 58 5.2 Case 2 - Cuts do intersect the second hole . 62 5.3 Case 3 - Chaining holes . 70 5.3.1 A two hole graph . 71 5.3.2 A three hole graph . 76 5.3.3 A four hole graph . 80 6 Conclusion . 86 7 References . 89 iv LIST OF FIGURES 1 A view of a cellular network as a triangular system. 2 2 The hexagonal system view of the network. 3 3 The dual graph (trigraph) system view of the network. 3 4 The process of paging outward. 7 5 A triangular system, producing three trees with isometric embedding. 9 6 A network graph decomposed into four sub-graphs, each without a hole. 10 7 Hexagonal structure. 12 8 Triangular structure. 12 9 Example of a 2-disc hole. 13 10 General convex hole. 13 11 Cuts start with choosing an arbitrary edge of the hole. 14 12 Cuts contain all nodes in a perpendicular line until reaching a terminal node. 15 13 Example of a graph with a cut. 15 14 An opposite cut will be created parallel to the first cut. 16 15 The cuts must be equal distance apart in either direction. 16 16 Both cuts and the hole create two partitions in the graph. 17 17 Two nodes found in the same partition. 17 18 Path found without crossing any cuts. 18 19 Paths found when using cuts. 18 v 20 This path can be shortened without crossing a cut. 19 21 This path can also be shortened without crossing a cut. 19 22 Two nodes found in alternate partitions. 19 23 Paths found when using minimal cut crossings. 20 24 Paths found using multiple cut crossings. 20 25 Paths can be shortened by reducing the number of cut crossings. 21 26 Paths can also be shortened by reducing the number of cut crossings. 21 27 Returning to the previously designed partition graph. 22 28 The second pair of cuts will be perpendicular to the first pair of cuts. 22 29 All cuts will be an equal distance apart from those adjacent to them. 23 30 When the (A) edge is chosen on one side the (B) edge must be chosen on the alternate side of the hole. 23 31 A second two partition graph by cut3, cut4, and the hole. 24 32 Combining both two partition graphs results in a Four Cut / Four Region graph. 24 33 The Four Cut / Four Region graph. 24 34 Two nodes are found within the same region. 25 35 Two nodes are found in adjacent regions. 25 36 Two nodes are found in opposite regions of the graph. 26 37 Using a Lemma 1(B) composite to solve Case C. 26 38 Application of Lemma 1(B) on the cut1 and cut2 partition graph. 27 39 Application of Lemma 1(B) on the cut3 and cut4 partition graph. 27 40 Recalling the general graph with hole. 28 vi 41 Decomposition of the original graph. 28 42 Two Cut / Two Partition graph. 29 43 Sub-graphs G1 and G2............................. 29 44 Two sub-graphs lack some node pairs. 30 45 Using a third cut returns a path for the node pair. 30 46 Selecting non-perpendicular cuts. 31 47 Both cuts sever the shortest path. 31 48 Shortest path crossing two cuts. 31 49 This is the collection of the four sub-graphs. 31 50 Sub-graph composite of Four Cut / Four Region graph. 32 51 A shortest path placed into the Four Cut / Four Region graph. 32 52 The shortest path does not cross cut1 or cut3. 33 53 Completely labeled Four Cut / Four Region graph. 33 54 Two nodes are found within the same region. 34 55 Two nodes are found within adjacent regions. 34 56 Two nodes are found within opposite regions. 35 57 Shortest path must cross one horizontal cut. 35 58 Shortest path must cross one vertical cut. 35 59 One node in a cut, the other within an opposite region. 36 60 One node in a cut, the other within an adjacent region. 36 61 Two nodes within the same cut. 37 62 Two nodes within adjacent cuts. 37 63 Two nodes within opposite cuts. 38 vii 64 Differences between convex and isometric shapes. 39 65 An isometric shaped hole . 40 66 Isometric shaped hole with two cuts. 41 67 Isometric shaped hole with four cuts. 41 68 A shortest path crossing all four cuts. 41 69 K-disc structure hole . 42 70 Rectangle structure hole . 42 71 Square structure hole . 43 72 Pyramid structure hole . 43 73 Trapezoid structure hole . 43 74 Understanding the orientation of expansion hexagons. 44 75 Basic convex expansion. 45 76 Convex expansion with left terminal hexagon. 45 77 Convex expansion with right terminal hexagon. 45 78 Convex expansion with left and right terminal hexagons. 46 79 Single line graph . 46 80 The results of all expansions on a single line graph. 46 81 Increasing line graph . 47 82 The results of all expansions on an increasing line graph. 47 83 Decreasing line graph . 48 84 The results of all expansions on a decreasing line graph. 48 85 Constant line graph . 49 86 The results of all expansions on a decreasing line graph. 49 viii 87 Expansion legend. 51 88 K-disc structure. 51 89 Rectangle structure. 51 90 Square structure. 52 91 Pyramid structure. 52 92 Trapezoid structure. 52 93 Random convex structure 1. 53 94 Random convex structure 2. 53 95 Random convex structure 3. 53 96 Two cut graph for a rectangle. 54 97 Four cut graph for a rectangle. 54 98 Two cut graph for a square. ..