1. Experimental analysis of simple, distributed vertex coloring algorithms Irene Finocchi Alessandro Panconesi Riccardo Silvestri DSI, University of Rome “La Sapienza”

2. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 The vertex coloring problem  (u,v)  E c(u)  c(v) Given G = (V,E) find c :V  N such that Goal: finding “good” colorings, i.e., using few colors Finding or even approximating the minimum number of colors is hard

3. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Distributed vertex coloring Each graph is  +1-colorable –Sequential algorithm: trivial –Deterministic distributed algorithm working in O(polylog n) time: open problem! Brooks-Vizing colorings = colorings using “many fewer” than  colors G square or triangle free   (  log  )-colorable “Good” colorings in a distributed setting: O(  )-colorings, where  = maximum degree to be computed in O(polylog n) time

4. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Distributed coloring algorithms Characteristics of the algorithms –Each vertex manages a list of colors (palette) –The computation proceeds in rounds Model of computation: synchronous, message-passing –Vertices = processors operate in parallel –Routing messages costs order of magnitude more than performing local computations

5. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 The basic round r for vertex u 1. Wake up!Wake up with probability w r 2. Try!Pick a tentative color c from palette L u (uniformly at random) c-color iff no awaken neighbor has selected c as tentative color 3. Conflict resolution 4. Deliverance?If 3 succeded exit, otherwise remove from L u the final colors of the neighbors If L u =  introduce fresh new colors (up to the greatest color of a neighbor +1) 5. Feed the hungry! Go back to sleep6. Back to square one

6. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Important parameters Palette size: can it be <<  ? Wake up probability: how do we choose it? Can it be constant? Conflict resolution: the give up rule is not very smart... can we improve it? Algorithms with very different characteristics can be obtained changing these parameters

7. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002  +1 colorings: theory Key lemma: At any round an uncolored vertex colors with probability ≥1/4 [Johansson’99,Luby’93] w r =1 Palette = [1,  +1] Trivial algorithm w r =1/2 Palette = [1,  +1] Luby’s algorithm A  +1-coloring is computed in O(log n) rounds with high probability

8. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002  +1 colorings: Luby vs. Trivial How fast are the  +1-coloring algorithms for different wake up probabilities?

9. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002  +1 colorings: Luby vs. Trivial  +1-coloring

10. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Brooks-Vizing colorings: theory Grable & Panconesi 2000 (GP) Reduced palette:  /s (s = shrinking factor) palette size w r = uncolored neighbors G triangle-free  -regular  >> log n With high probability GP colors G with O(  /log  ) colors within O(log n) rounds

11. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 A new conflict resolution rule For each color c: G c (conflict graph) = graph induced by vertices with tentative color c Hungarian step (HS) conflict resolution by computing an independent set on G c The hungarian approach for independent set computation: given a random permutation  of the vertices of G c, a vertex enters the independent set iff it comes before its neighbors in 

12. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Expected number of colored vertices n = number of vertices of G c  = average degree of G c E[|colored vertices|] = n/(s+1) E[|independent set|]=n/(  +1) 1. [Hungarian folklore] s = palette shrinking factor  ] = s (if all vertices wake up) 2. [Easy to prove]

13. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Parameter settings GP Grable Panconesi Give up! palette size uncolored neighbors Reduced:  /s HGP Hungarian GP Hungarian palette size uncolored neighbors Reduced:  /s HungarianConstant Reduced:  /s CH Constantly hungarian Hungarian1 Reduced:  /s TH Trivially hungarian PaletteWake up prob.Conflict res. Algorithms

14. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Wake up probability & rounds

15. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Shrinking factor & colors How good are colorings? (As a function of the initial palette size)

16. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Shrinking factor & colors

17. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Shrinking factor & rounds TH saves approx. 70-80% rounds over GP and HGP

18. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Conclusions & open problems  +1-colorings Trivial is the algorithm of choice Tight analysis? Brooks-Vizing colorings TH is the algorithm of choice Rigorous theoretical analysis? Deterministic algorithms?

19. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 The vertex coloring problem  (u,v)  E c(u)  c(v) Given G = (V,E) find c :V  N such that Each graph is  +1-colorable (greedy approach)  [  = maximum degree] Brooks-Vizing colorings = colorings using “many fewer” than  colors G square or triangle free   (  log  )-colorable Minimize the number of colors

20. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Luby vs. Trivial: intuitive explanation w = wake up probability l = palette size d = # of uncolored neighbors Pr[v colors] ~ w e -d w/l Trivial: w=1, l~dPr[u colors] ~ 1/e Luby: w=1/2, l~dPr[u colors] ~ 1/(2√e) Pr[v c-colors] ~ w/l (1-w/l) d (1-  )  = (1+o(1)) e -  ~ w/l e -d w/l Only for the very first rounds...

21. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Luby vs. Trivial: intuitive explanation w = wake up probability p = palette size d = # of uncolored neighbors ~ w e -d w/p Trivial: w=1, p~dPr[v colors] ~ 1/e Luby: w=1/2, p~dPr[v colors] ~ 1/(2√e) Pr[v colors] = Only for the very first rounds... cLvcLv ∑ Pr[v c-colors] cLvcLv ~ ∑ w/p (1-w/p) d cLvcLv ~ ∑ w/p e -d w/p

22. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002  +1 colorings: Luby vs. Trivial

23. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Distributed coloring algorithms Characteristics of the algorithms –Randomized –Each vertex manages a list of colors (palette) –The computation proceeds in rounds –All the algorithms can be cast into a general framework Model of computation: synchronous, message-passing –Vertices = processors operate in parallel –Routing messages costs order of magnitude more than performing local computations

24. 13th ACM-SIAM Symposium on Discrete AlgorithmsSan Francisco, January 2002 Our questions: Wake up probability & rounds How does the wake up probability affect the running time? Can it be constant? Which is the best choice?