1. LOCALITY IN DISTRIBUTED GRAPH ALGORITHMS Nathan Linial Presented by: Ron Ryvchin

2. The Main Results Tight bound on the complexity of 3-coloring of an n-cycle in a synchronous message passing model Lower bound proof Algorithm with the above complexity (by Cole and Vishkin)

3. Cycle coloring and the chromatic number

4. The Model Synchronous message passing No faults Message size unlimited 6 3 1 2 4 5

5. After t rounds, each processor knows the labeling of all nodes at distance t or less away After diameter(G) rounds every process knows the ID of every node and therefore can compute any function of G The Model

6. A general algorithm Let t be the number of rounds required to find the 3- coloring of an n-cycle The algorithm is: Each round, every processor will send all the information it knows to both of its neighbors After t rounds, each processor chooses the color of its node according to the information it has

7. A lower bound on 3 coloring of an n-cycle After t rounds, the data known to a processor P is an ordered list of 2t+1 labels, starting t places before it, through its own and on to the next t labels 14 3 2 5 5,1,4 1,4,3 4,3,2 3,2,5 2,5,1 t=1

8. A lower bound on 3 coloring of an n-cycle

11. 2 1 4 3

12. Given a graph G = (V, E), its line graph L(G) is the directed graph whose vertex set is E with (u, v) an edge if u,v is a directed path in G. Line Graphs 12 54 3 (1,2)(2,5)(5,4) (4,3)(1,3)(4,1)

13. L(G) coloring 12 24 2343 31 1 24 3 1 24 3

14. L(G) coloring xy

15. (1,2)(1,3)(1,4)(2,3)(2,4)(3,4)

16. (1,2)(1,3)(1,4)(2,3)(2,4)(3,4) (1) (3) (4) (2)

18. (1) (3) (4) (2)

20. lower bound – conclusion

21. Lower bound for 2 coloring an n-cycle, for n even

22. AN ALGORITHM FOR 3- COLORING OF AN N- CYCLE R. Cole and U. Vishkin, Deterministic coin tossing and accelerating cascades: micro and macro techniques for designing parallel algorithms

23. The algorithm has 3 phases In the first phase it will find a 6-coloring of the cycle In the second phase it will find a MIS of the cycle from the 6 coloring In the last phase it will find the 3-coloring of the cycle from the MIS 3 coloring algorithm

24. Phase #1: 6-coloring of an n-cycle XYZ

26. Phase #1: initial coloring 2 0 6 8 35 4 17 9 11 12 1415 16 18 19 1317 10

27. Phase #1: iteration 1 3 2 6 4 31 0 16 2 7 2 0 1 9 0 51 0 3

28. Phase #1: iteration 2 3 3 3 2 31 0 10 4 5 4 2 2 1 0 51 0 3

29. Phase #2: MIS Finding the MIS from the 6-coloring will take 6 iterations. Each node N will have an alive bit, which will initially be 1. When we add a node to the independent set it sends a signal to its neighbor, and at the beginning of the next iteration they will turn their alive bit off (to 0). At iteration #(i+1): If the color of N is i and alive[N] = 1: N will be added to the independent set, and will send a signal to its neighbors so they will turn their alive bit off.

30. Phase #2: MIS 3 3 3 2 31 0 10 4 5 4 2 2 1 0 51 0 3

31. Phase #3: Given an MIS I of an n-cycle, construct a 3-coloring 3 3 3 2 31 0 10 4 5 4 2 2 1 0 51 0 3

32. 3 3 3 2 31 0 10 4 5 4 2 2 1 0 51 0 3

33. 3 3 3 2 31 0 10 4 5 4 2 2 1 0 51 0 3 Color all remaining nodes blue

34. 3 3 3 2 31 0 10 4 5 4 2 2 1 0 51 0 3 Phase #3: Given an MIS I of an n-cycle, construct a 3-coloring Takes one round

35. The End