FREQUENCY ASSIGNMENT IN RADIO NETWORKS A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Science By Uday Kiran Viyyure May 2008 Thesis written by Uday Kiran Viyyure M.S., Kent State University, USA, 2008 B.S., Madras University, India 2001 Approved by Dr. Feodor F. Dragan , Advisor Dr. Robert A. Walker , Chair, Department of Computer Science Dr. Jerry Feezel , Dean, College of Arts and Sciences ii TABLE OF CONTENTS LIST OF FIGURES ......................................................................................................... iv LIST OF TABLES ........................................................................................................... vi ACKNOWLEDGEMENTS ........................................................................................... vii CHAPTER 1 INTRODUCTION..................................................................................... 1 CHAPTER 2 PRELIMINARIES .................................................................................... 5 2.1 Distance K-Chromatic Number Problem .................................................................. 7 2.2 The L(h,k) Graph Coloring Problems. ...................................................................... 8 CHAPTER 3 COLORING METHODS ....................................................................... 10 3.1 The L(1) Graph Coloring ........................................................................................ 10 3.2 The L(1,1) Graph Coloring ..................................................................................... 12 3.3 The L(1,1,1) Graph Coloring .................................................................................. 16 3.4 The L(2,1) Graph Coloring ..................................................................................... 19 3.5 The L(2,1,1) Graph Coloring .................................................................................. 22 3.6 The L(3,1,1) Graph Coloring .................................................................................. 28 CHAPTER 4 CONCLUSIONS AND FUTURE RESEARCH ................................... 36 APPENDIX A FREQUENCY ASSIGNMENT TOOL ............................................... 37 REFERENCES................................................................................................................ 40 iii LIST OF FIGURES Figure1. Cells showing co-channel and adjacent channel interference…………………..3 Figure 2. Hexagonal cell structure of a cellular network…………………………………4 Figure 3. Cellular graph network corresponding to a cellular network of Figure 2……..5 Figure 4. Clique for L(1) coloring. ................................................................................... 11 Figure 5. L(1) coloring for a 5×5 cellular network........................................................... 11 Figure 6. L(1) coloring for a 7×7 cellular network.......................................................... 12 Figure 7. L(1) coloring for a 10×10 cellular network...................................................... 12 Figure 8. Clique for the L(1,1) coloring of a cellular graph network.. ............................ 13 Figure 9. Color Spectrum for the L(1,1) coloring............................................................ 14 Figure 10. L(1,1) coloring of a cellular graph network. ................................................... 14 Figure 11. L(1,1) coloring of a 5×5 cellular network. ...................................................... 15 Figure 12. L(1,1) coloring of a 7×7 cellular network. ...................................................... 15 Figure 13. L(1,1) coloring of a 10×10 cellular network. .................................................. 15 Figure 14. Clique for L(1,1,1) coloring in a cellular graph network. ............................... 16 Figure 15. Honey comb cellular graph with L(1,1,1) graph coloring technique. ............. 17 Figure 16. Color spectrum used for L(1,1,1) coloring..................................................... 18 Figure 17. A 5×5 cellular network with L(1,1,1) coloring. .............................................. 18 Figure 18. A 7×7 cellular network with L(1,1,1) coloring. .............................................. 18 Figure 19. A 10×10 cellular network with L(1,1,1) coloring. .......................................... 19 iv Figure 20. Color spectrum used in L(2,1) Coloring.......................................................... 20 Figure 21. A 5×5 cellular network with L(2,1) coloring. ................................................. 20 Figure 22. A 7×7 cellular network with L(2,1) coloring. ................................................. 21 Figure 23. A 10×10 cellular network with L(2,1) coloring. ............................................. 21 Figure 24. Clique of a cellular graph with the integers of the colors associated. ............. 22 Figure 25. Color spectrum used for L(2,1,1) graph coloring............................................ 23 Figure 26. Cliques of a cellular graph with L(2,1,1) coloring. ......................................... 23 Figure 27. A 5×5 cellular network with L(2,1,1) coloring. .............................................. 24 Figure 28. A 7×7 cellular network with L(2,1,1) coloring. .............................................. 24 Figure 29. A 10×10 cellular network with L(2,1,1) coloring. .......................................... 25 Figure 30. Coloring spectrum for L(3,1,1) coloring. ........................................................ 29 Figure 31. A 5×5 cellular network with L(3,1,1) coloring. .............................................. 29 Figure 32. A 7×7 cellular network with L(3,1,1) coloring. .............................................. 29 Figure 33. A 10×10 cellular network with L(3,1,1) coloring. .......................................... 30 Figure 34. Tool interface with number of rows and columns entered............................. 34 Figure 35. Tool generating graph without coloring.......................................................... 37 Figure 36. Tool coloring the graph with L(3,1,1) coloring method.................................. 37 Figure 37. Tool coloring the graph with L(3,1,1) coloring method.................................. 38 v LIST OF TABLES Table 1. Relationship between Manhattan distance and Minimum distance................... 26 Table 2. Relationship between consecutive vertices....................................................... 26 Table 3. Manhattan distance between consecutive colors for L(3,1,1) coloring. ............ 31 Table 4. Manhattan distance between colors (a,b), where |a-b| = 2................................. 32 vi Acknowledgement I would like to take this opportunity to express my sincere gratitude to my advisor Dr. Feodor F. Dragan for kindly providing guidance throughout my research work. I thank him for his continuous encouragement and making this thesis possible. It was a great pleasure to work on my thesis under his supervision. vii CHAPTER 1 Introduction Technological advances and the tremendous growth of wireless network and wireless terminals have caused the wireless communications and mobile computing to grow rapidly. The present trends in the telecommunications industry to provide information access from any where to any place in the world and shifting of present applications to a multimedia environment has caused a rapid growth in the mobile users community. This enormous growth in telecommunication filed has put a lot of constraints on the availability of the radio frequency spectrum. Due this the frequency reuse and channel assignment problems and algorithms have been studied extensively in the last two decades by radio and electrical engineers, operations researchers, graph theorists and computer scientists. Different people approached this problem in different ways (1) Classic methods like graph coloring and integer programming. (2) Heuristic methods like neural networks genetic algorithms. (3) Local search such as simulated annealing and Tabu search. (4) Constrained programming. 1 2 In cellular networks a geographical area is divided into smaller service areas called cells and each of these cell’s have a base station and all the wireless terminals or the users in those cells communicate with their corresponding cell area base stations. For these communication links to be established the available frequency spectrum should be used and reused very efficiently. The efficient reuse in the spectrum helps to reduce the cost of service by reducing the number of base stations and also accommodating more number of users per base stations. A channel assignment problem or the frequency assignment problem is nothing but the task of assigning frequency from a radio spectrum to a set of transmitters and receivers satisfying certain conditions. In order to divide a radio spectrum in an optimal way many techniques have been given in literature, such as frequency division, time division and code division. In Frequency division the spectrum is divided in to disjoint frequency bands. In time division the same channel is used by different base stations at different times and thus attaining channel allocation with respect to time. In channel division the channel allocation is achieved by using different modulation code. We can also achieve channel assignment by a combination of above techniques. Here in this paper we discuss with respect to frequency division, so we view the channel assignment as nothing but frequency assignment