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Evi Papaioannou PhD Thesis

Independent Sets and Graph Coloring with Applications to the Frequency Allocation Problem in Wireless Networks. Evi Papaioannou PhD Thesis. Department of Computer Engineering and Informatics University of Patras. Subject. Wireless networks

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Evi Papaioannou PhD Thesis

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  1. Independent Sets and Graph Coloring with Applications to the Frequency Allocation Problem in Wireless Networks Evi Papaioannou PhD Thesis Department of Computer Engineering and Informatics University of Patras

  2. Subject • Wireless networks • Frequency allocation problem in cellular wireless netowrks • Call control problem in wireless networks with cellular, planar, arbitrary topology • Networks of autonomous transmitters • Maximum independent set problem • Minimum coloring problem

  3. Methodology • On-line problems • Users/transmitters appear gradually and the sequence can stop arbitrarily • On-line algorithms • Algorithms cannot change their choices • Performance evaluation • Competitive analysis • Metric = value of competitive ratio

  4. Cellular wireless networks • The geographical area is divided in regions (cells) • Each cell is the calling area of a base station • Base stations are interconnected via a high speed network

  5. Communication • Communication between user and base station is always required • Frequency Division Multiplexing(FDM) Technology : many users within the same cell can simultaneously communicate with their base station using different frequencies [Hale 80]

  6. Interference graph • Irregular networks

  7. Interference graph • Cellular networks • Reuse distance (k): the min distance between two cells where the same frequency can be used

  8. Frequency allocation Input • A cellular network and users that wish to communicate with their base station Output • Frequency allocation to all users, so that: • Users in the same or adjacent cells are assigned distinct frequencies • The number of frequencies used in minimized

  9. Graph coloring Imagine: • Frequencies  colors • Users that wish to communicate with their base station nodes of the interference graph of the wireless network Then: • Frequency allocation problem problem of multicoloring the nodes of the interference graph • The interference graph is constructed gradually • Nodes are added gradually as calls appear

  10. Call control Input • A cellular network supporting w frequencies and users that wish to communicate with their base station Output • Frequency allocation to some of the users, so that: • Users in the same or adjacent cells are assigned distinct frequencies • At most w frequencies are used • The number of the users served is maximized

  11. Independent sets Imagine: • Frequencies colors • Users that wish to communicate with their base station nodes of the interference graph of the wireless network Then: • Call control problem Maximum independent set problem in the interference graph • The interference graph is constructed gradually • Nodes are added gradually as calls appear

  12. Frequency allocation Cost: Number of frequencies used Competitive ratio: Call Control Benefit: Number of users served Competitive ratio: Competitive analysis

  13. Frequency allocation

  14. Previous results • Off-line algorithms • 4/3-προσέγγιση [NS97, MR97, JKNS98] • Even if the sequence of calls is know a priori, the frequency allocation problem cannot be solved optimally in polynomial time [MR97] • Simple 3/2- and 17/12-approximation algorithms [JKNS98] • On-line algorithms • Fixed Allocation algorithm: competitive ratio 3 [JKNS98] • No deterministic algorithm can have a competitive ratio smaller than 2 [JKNS98]

  15. The Greedy algorithm • Frequencies: positive integers 1, 2, 3, ... • When a call appears, it is assigned the smaller available frequency, so that • There is no interference between calls in the same or adjacent cells (according to reuse distance of the network) • The greedy algorithm is at most 2.5-and at least 2.429-competitive, against off-line adversaries • New [ΝΤ04] lower bound = 2.5  Tight analysis

  16. Proof – Upper bound

  17. Proof – Upper bound D

  18. ...α1 ...α2 ...α6 ...α3 ...α5 ...α4 Proof – Upper bound D ...α0

  19. ...α1 ...α2 ...α6 ...α3 ...α5 ...α4 Proof – Upper bound D ...α0 a0 2.5D

  20. Proof – Lower bound

  21. Call control

  22. Previous results • Greedy algorithm, networks of maximum degree Δ that support one frequency [PPS97] Greedy algorithm Optimal algorithm Benefit = 1 Benefit = Δ

  23. Previous results • «Classify and Randomly Select» paradigm for networks with chromatic number χ that support one frequency [ΑΑFLR96, PPS97]

  24. Previous results • «Classify and Randomly Select» paradigm for networks with chromatic number χ that support one frequency [ΑΑFLR96, PPS97] Chromatic number = 4 4 times worse Networks of max degree Δ Chromatic number  Δ+1  Δ+1 times worse

  25. Previous results • Lower bounds for arbitrary networks [BFL96] • Simple way of transforming an algorithm designed for networks that support one frequency to an algorithm for networks that support arbitrarily many frequencies [AAFLR01] • Upper bounds for networks with planar and arbitrary interference graphs using the «Classify and Randomly Select» paradigm [PPS02]

  26. The Greedy algorithm • The greedy algorithm in networks that support one frequency achieves a competitive ratio equal to the size of the maximum independent set of every node of the interference graph

  27. Benefit = 1 Benefit = 3 Deterministic algorithms • The greedy algorithm in cellular networks that support one frequency • Optimal in the class of deterministic on-line algorithms • Competitive ratio: 3

  28. Randomized algorithms • Based on the «Classify and Randomly Select» paradigm • Competitive ratio = number of colors used for the coloring of the interference graph • Competitive ratio for cellular networks: 3

  29. Idea Accept the call with probability p

  30. Idea Marking Technique Accept the call with probability p (1-p)t0: w.h.p. one of the calls is accepted

  31. Marking Technique Idea Accept the call with probability p (1-p)t0: w.h.p. one of the calls is accepted

  32. Algorithm p-Random • Initially all cells are unmarked • For each new call c in a cell v • If vis marked, reject c • If there is an accepted call in cell vor in its adjacent cells, reject c • Otherwise: • With probability p, accept c • With probability 1-p, reject cand mark cell v

  33. Upper bounds • Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

  34. Upper bounds • Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

  35. Upper bounds • Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

  36. Upper bounds • Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

  37. Upper bounds • Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

  38. Upper bounds • Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p • Best upper bound: 2.651

  39. Upper bounds • Algorithmp-Random achieves better competitive ratio than all deterministic algorithms in all networks that support one frequency • 27/28Δ • Extend the analysis for sparse networks of degree 3 or 4 • Disadvantages • No improvement fro networks that support arbitrarily many frequencies • Uses randomness proportional to the size of sequence of calls

  40. CRS-based algorithms Objective • Randomized algorithms • Arbitrarily many frequencies • Whatever reuse distance • Few randomness • Weak random sources • Constant number of random bits Given • «Classify and Randomly Select» paradigm • Simple • Use randomness only once at the beginning • Behaves «well» independently of the number of supported frequencies

  41. 0 1 0 1 0 1 0 1 0 1 2 3 2 3 2 3 2 3 2 1 0 1 0 1 0 1 0 1 0 3 2 3 2 3 2 3 2 3 0 1 0 1 0 1 0 1 0 1 2 3 2 3 2 3 2 3 2 1 0 1 0 1 0 1 0 1 0 Algorithm CRS-A • Color the interference graph with 4 colors 0,1,2,3 • Select one of the colors, ignore calls in cells colored with the selected color and execute the greedy algorithm for all other calls

  42. 0 1 0 1 0 1 0 1 0 1 2 3 3 2 3 3 2 3 3 2 3 3 2 1 0 1 0 1 0 1 0 1 0 3 3 2 3 3 2 3 3 2 3 3 2 3 3 0 1 0 1 0 1 0 1 0 1 2 3 3 2 3 3 2 3 3 2 3 3 2 1 0 1 0 1 0 1 0 1 0 AlgorithmCRS-A • Color the interference graph with 4 colors 0,1,2,3 • Select one of the colors, ignore calls in cells colored with the selected color and execute the greedy algorithm for all other calls

  43. AlgorithmCRS-A: analysis • The greedy algorithm will accept at least half of the optimal calls • Work on average on the 3/4 of the total calls • Competitive ratio = 8/3

  44. CRS-based algorithms • Network that supports w frequencies • CRS-based algorithms: • Color the interference graph • Define v color classes from the colors used • Select equiprobably one out of v color classes • Execute the greedy algorithm only for cells colored with colors from the selected color class • If: • Each color belongs to at least λ different color classes, and • Each connected component of the subgraph of G containing nodes colored with colors of the same color class is a clique then, the CRS-based algorithm is v/λ-competitive against oblivious adversaries

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