1 On Channel Assignment Of Graphs Author : Hsin-Ju Wu Adviser : Yung-Ling Lai Speaker : Shr-Jia Hung

2 Outline Motivation and Definition Off-line Labeling On-line Labeling Conclusion and Future work

3 Outline Motivation and Definition Off-line Labeling On-line Labeling Conclusion and Future work

4 Motivation Channel assignment problem  the number of finite frequencies Use a graph to model it.

5 Motivation Vertices transmitters Edges Distances Adjacent very close Distance 2 Close A CB

6 Definition k-L(p,q) labeling f for a given graph G=(V,E), is a function f : V→{ 0,1,…,k } such that | f(x)-f(y) | p if d(x,y)= 1 and | f(x)-f(y) | q if d(x,y)= 2. The L(p,q) labeling number of graph G is then defined as:

7 Outline Motivation and Definition Off-line Labeling On-line Labeling Conclusion and Future work

8 Off-line Labeling L(d,1) labeling on

9 Outline Motivation and Definition Off-line Labeling On-line Labeling Conclusion and Future work

10 Online base application

11 Online base application

12 Online base application

13 Online base application

14 Online base application

15 Online base application

16 Definition of online labeling Given a graph G. The vertices are given one-by-one arbitrarily. Only the adjacency relation between the given vertices are known. Satisfy the condition of L(2,1)-labeling. Give a label right away which is not changeable later.

17 Online L(2,1) labeling of path Path algorithm Call Function Get_available_Number(N 1,N 2 )

18 Get_available_Number(N1,N2) N 1 =0N 1 =1N 1 =2 N 2 =0 N 2 =1N 2 =0N 2 =1N 2 =2

19 N 1 =0 and N 2 =0 x

20 N 1 =1 and N 2 =0 L1L1 x

21 N 1 =1 and N 2 =1 L1L1 xvjvj L1L1 x

22 N 1 =2 and N 2 =0,1,2 L 2 xL1L1 L2L2 xL1L1 L2L2 xL1L1

23 Time Complexity Path algorithm 3.1 Call Function O(n) Get_available_Number(N 1,N 2 )

24 Time Complexity Path algorithm O(n) Get_available_Number(N 1,N 2 )

25 Time Complexity Path algorithm 3.1 Call Function O(n 2 ) Get_available_Number(N 1,N 2 )

26 Online L(2,1) labeling of path Pattern A

27 Online L(2,1) labeling of path

28 Online L(2,1) labeling of cycle Pattern B

29 Online L(2,1) labeling of cycle

30 Star S

31 Online L(2,1) labeling of star 1 0 n 2 n-11

32 Online L(2,1) labeling of star n

33 Online---Other graph bound Double star: Full binary tree:

34 Outline Motivation and Definition Off-line Labeling On-line Labeling Conclusion and Future work

35 Conclusion Off-line L(d,1) labeling - On-line L(2,1) labeling - Path - Cycle - Star - Double star - Full binary tree

36 Future Work On-line L(2,1) labeling  K 2 xP n, K 2 xC n  Wheel  Complete bipartite graph  relation with max degree  relation with radius, diamter  relation with density(size/order)