1. Distributed Coloring Discrete Mathematics and Algorithms Seminar Melih Onus November 10 2005

2. Outline Vertex Coloring Model Luby’s Algorithm Coloring Constant Degree Oriented Graphs Coloring Oriented Graphs Conclusion & Open Problems

3. Vertex Coloring Given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color Proper Coloring Improper Coloring

4. Model G(V, E), V represents the set of processors and E represents communication links Communication links are bidirectional Processors are synchronized Each node knows n,  and its neighbors The edges in E have an orientation (edge {u, v} is oriented either as u  v or v  u)

5. How can we use orientation? If nodes u and v choose the same color during any round of algorithm, in the existing algorithms both nodes remain uncolored With orientation, u can be colored provided that there is no edge w  v and node w also chooses the same color uvuv With existing algorithms both remain uncolored Using orientation, u gets colored red

6. Luby’s algorithm In each round –Each uncolored node chooses a color uniformly random –If there is no conflict, node is colored with that color Distributed  +1-coloring algorithm Works in O(log n) rounds w.h.p.

7. a v bc Luby’s algorithm (Example) Round 1 uv a bc Round 2 uv a bc Round 3 uv a bc Round 4 u

8. Coloring Constant Degree Oriented Graphs Special case: Constant degree graphs Algorithm Color-Random: In each round –Each uncolored node v chooses an available color c v uniformly at random –If no neighbor node u with higher priority ( u  v) chooses the same color c v, node v is colored with c v u  v : u has higher priority

9. Algorithm Color-Random (Example) Round 1 uv a bc Round 2 uv a bc Round 3 uv a bc

10. Algorithm Color-Random For constant degree graph with n nodes provided with -acyclic orientation, our algorithm obtains a  +1 coloring in O( ) rounds, w.h.p.. An orientation of the edges of a graph is said to be m-acyclic if and only if the orientation does not have cycles of length at most m.

11. p p p p p p Analysis(Part I) Lemma: After O((logn) 1/2 ) rounds, every path of length (logn) 1/2 has at least one colored node, w.h.p.. Proof: (logn) 1/2 Each node has constant number of neighbors, so the probability that a node is not colored at a round is at most p (constant). The probability that none of the nodes at the path is colored at a round is at most p (logn)1/2. pp (logn)1/2

12. Analysis(Part I) p p p p p p (logn) 1/2 nodes p (logn)1/2 c(logn) 1/2 rounds p (logn)1/2 p clogn =1/n -clogp p (logn)1/2

13. Analysis(Part II) After O((logn) 1/2 ) rounds, every connected component of uncolored nodes have diameter at most (logn) 1/2, w.h.p.. Orientation is (logn) 1/2 -acyclic, so there can be no cycles on connected component of uncolored nodes. This provides a topological ordering.

14. Analysis(Part II) label(u)= 0 if no entering edge v  u 1+max v:v  u label v otherwise 0 0 1 1 2 2 3 4 Maximum label is (logn) 1/2. All nodes will be colored after (logn) 1/2 rounds.

15. Lowerbound For every Las Vegas algorithm A, there is infinite family of oriented graphs G such that A has complexity of at least  ((logn) 1/2 ), on expectation, to compute a proper vertex coloring. A Las Vegas algorithm is a randomized algorithm that always produces a correct result, with the only variation being its runtime.

16. Coloring Oriented Graphs General Case: Arbitrary degree graphs Algorithm Color-Wait For each round –If u is uncolored and does not have any uncolored neighbor w such that w  u then node u is colored with the lowest available color

17. uuv cb Algorithm Color-Wait (Example) Round 1 v a cb Round 2 a v cb Round 3 u a no node with entering edge for node u no node with entering edge for node b no uncolored node with entering edge for node c no uncolored node with entering edge for node v

18. Coloring Oriented Graphs While there are uncolored nodes –Use Algorithm Color-Random for loglog n rounds –If  =  ((logn) 1/2 loglogn) Use Algorithm Color-Random for (8/  +4) (logn) 1/2 /loglogn rounds –Else Use Algorithm Color-Random for 4(logn) 1/2 rounds –Use Algorithm Color-Wait (logn) 1/2 rounds  constant,  >0,  > log  +1/2 n loglog n Phase I Phase II Phase III

19. Phase I Lemma: After phase I, the number of uncolored neighbors of any node reduces to log n w.h.p.. Use Algorithm Color-Random for loglog n rounds

20. Phase II Lemma: After phase II, every path of length (logn) 1/2 has at least one colored node, w.h.p.. If  =  ((logn) 1/2 loglogn) Use Algorithm Color-Random for (8/  +4) (logn) 1/2 /loglogn rounds Else Use Algorithm Color-Random for 4(logn) 1/2 rounds

21. Phase III After phase III, all nodes will be colored. Use Algorithm Color-Wait (logn) 1/2 rounds

22. Coloring Oriented Graphs Given an -acyclic oriented graph G=(V,E) of maximum degree , for any constant  >0 a (1+  )  - vertex coloring of G can be obtained in O(log  ) + O( log log n) rounds, with high probability.

23. Results for any constant  >0

24. Conclusion & Open Problems Distributed coloring algorithm Acyclic orientations, better bounds Deterministic distributed algorithms for  +1-coloring that run in polylogarithmic number of rounds

25. References K. Kothapalli, C. Scheideler, M. Onus, C. Schindelhauer. Distributed coloring with O(logn) bits. submitted to IPDPS 06. M. Luby. A simple parallel algorithm for the maximal independent set problem. STOC 1985.