Frequency Allocation in Wireless Communication Networks

Transcription

1 Frequency Allocation in Wireless Communication Networks Joseph Wun-Tat Chan King s College London Joint work with: Francis Y. L. Chin University of Hong Kong Yong Zhang University of Hong Kong Deshi Ye Zhejiang University Hong Zhu Fundan University

2 Outline Introduction Frequency Allocation Problem Offline problem Online problem Variations of the Problem Future Directions

3 Wireless Communication Networks Mobile phone A cell Frequency h 2 Frequency h 1 Frequency f f h 1 and f h 2 and f h 3 Frequency f Base station Frequency h 3 To avoid interference, two communications in the same cell or adjacent cells do not use the same frequency

4 Frequency Allocation Problem Problem Formulation Given the hexagon graph and the calls in each cell Frequencies are represented by {1, 2, 3, } Two calls from the same or adjacent cells can t be assigned the same frequency The goal is to assign frequencies to all calls using the minimum span of frequencies (i.e., the difference between the largest and smallest frequencies)

5 Offline Problem The calls are given in advance Multicoloring of weighted hexagon graphs NP-complete [McDiarmid and Reed 2000] 4/3-approximation algorithm [McDiarmid McDiarmid and Reed 2000, Narayanan and Shende 2001]

6 Online Problem The calls are given one at a time Assign a frequency to a call before the next call is presented The assigned frequency to a call cannot be changed Calls will not terminate

7 Fixed Allocation Color the cells with three colors, R, G,, and B Partition the frequencies into 3 disjoint subsets S R ={1, 4, 7, } S G ={2, 5, 8, } S B ={3, 6, 9, } For the cells with color R For the cells with color G For the cells with color B Frequency allocation rule: Use frequencies according to the color of the cells, starting from the lowest frequency 3 calls No adjacent cells are of same color {3, 6, 9} 9 The span of frequencies used is at most 3 times that of the optimal allocation

8 Greedy Allocation Frequency allocation rule: For each call in a cell, assign the lowest frequency which has not been assigned by the cells nor the neighboring cells {1,4} { } {2} {3} cell X {4,6} {1,4,8} {2,5} E.g., if a call request appears in cell X, frequency 7 will be assigned to the call The span of frequencies used is at most 17/7( 2.43) times that of the optimal allocation [Chan, Chin, Ye, Zhang, Zhu 2006] The ratio is tight [Caragiannis, Kaklamanis, Papaioannou 2000]

9 Hybrid Allocation Color the cells with three colors, R, G, and B Partition the frequencies into 4 disjoint subsets S S ={1, 5, 9, } S R ={2, 6, 10, } S G ={3, 7, 11, } S B ={4, 8, 12, } Frequency allocation rule Shared by all cells For the cells with color R For the cells with color G For the cells with color B For a call request in a cell with color x, assign the lowest frequency in S S S x that has not been used by the cell nor the neighboring cells

10 Hybrid Allocation The span of frequencies used is at most 2 times that of the optimal allocation Why? Suppose cell X uses the highest frequency 4k-3 4k-2 4k-1 4k cell X cell Y Cell Y has the highest frequency in S S among the neighbouring cells of cell X

11 Lower Bound 2 is the best that we can do for any online algorithm Adversary sequence: One call for each of the cells labelled a One call for each of the cells labelled b One call for each of the cells labelled c One call for each of the cells labelled d a c b a d d d a :1 :1 b :2 :4 :3 a c :3 :5 :6 :2 :1 :1

12 Asymptotic Performance To break the barrier of 2 Focus on the performance when there are large amount of calls When the span of frequencies used in the optimal allocation is very large ( ( ), hybrid algorithm can achieve a ratio close to With some modification in partitioning the available frequencies The ratio of the number of frequencies in S S and in each of S R, S G, S B : 1 The lower bound of the asymptotic competitive ratio is 1.5

13 Other Variations Dynamic frequency allocation Any call may terminate at any time Coverage of radio stations represented by a graph other than a hexagon graph Interference constraints The shortest distance that the same frequency can be reused (without creating interference) is a parameter r A simple assumption is r=2 (cells apart, inclusive) General r?

14 Dynamic Problem Generalized version of the online problem A call may terminate at any time The goal is to minimize the span of frequencies used over all time Performance: Fixed Allocation: 3-competitive3 Greedy Allocation: 3-competitive 3 Zhang, Zhu 2006] competitive [Chan, Chin, Ye, No online algorithm is known for better than 3-3 competitive

15 Performance of Greedy lower bound = 3 The base case: 1 k calls k calls 1 call k calls The center cell gets the highest frequency 3k The general case: 3k+2 3k+1 3k+4 3k+3 3k+1 3k+2 3k+1 3k+1

16 Coverage by Linear Networks The geographical coverage area is divided into cells aligned in a line [Chan, Chin, Ye, Zhang, Zhu 2006] Online: Upper bound=lower bound= 1.5 Asymptotic upper bound= Asymptotic lower bound=4/ Dynamic: Upper bound=lower bound=5/ Asymptotic upper bound=5/ Asymptotic lower bound=14/

17 Concept: Algorithm for Dynamic Case To reuse (borrow) frequencies from nearby cells which are from distance 2 that does not create interference Apply greedy if no frequency to borrow X

18 k-colorable graphs Other Coverage Graphs Online case: Hybrid algorithm is (k+1)/2- competitive E.g., planar graphs are 4-colorable 4 5/2-competitive Unit disk graphs Its vertices can be put in equal-size circles in a plane in such a way that two vertices are joined by an edge if and only if the corresponding circles interested

19 Frequency: {1, 2, 3, } Reuse distance The same frequency cannot be used for two calls within the same cell (distance=0) or adjacent cells (distance=1) Current setting reuse distance=2 In general, reuse distance=r Offline r=3 2-approximation [Kchikech, Togni 2005] r 4 3-approximation [Kchikech, Togni 2005] Online (and Dynamic) By a fixed allocation scheme r=3 3-competitive [Jordan, Schwabe 1996] r=4 4-competitive [Jordan, Schwabe 1996]

20 Future Directions Better than 3-competitive 3 for dynamic frequency allocation in hexagon graphs For general reuse distance, any online algorithm better than fixed allocation? Online or dynamic allocation on unit disk graphs Multicoloring on general graphs

21 Q & A