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

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)

Cycle coloring and the chromatic number

The Model Synchronous message passing No faults Message size unlimited

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

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

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 ,1,4 1,4,3 4,3,2 3,2,5 2,5,1 t=1

A lower bound on 3 coloring of an n-cycle

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 (1,2)(2,5)(5,4) (4,3)(1,3)(4,1)

L(G) coloring

L(G) coloring xy

(1,2)(1,3)(1,4)(2,3)(2,4)(3,4)

(1,2)(1,3)(1,4)(2,3)(2,4)(3,4) (1) (3) (4) (2)

(1) (3) (4) (2)

lower bound – conclusion

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

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

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

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

Phase #1: initial coloring

Phase #1: iteration

Phase #1: iteration

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.

Phase #2: MIS

Phase #3: Given an MIS I of an n-cycle, construct a 3-coloring

Color all remaining nodes blue

Phase #3: Given an MIS I of an n-cycle, construct a 3-coloring Takes one round

The End