1 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Distributed Coloring in Õ(  log n) Bit Rounds COST 293 GRAAL and COST 295 DYNAMO Discussion Workshop and MC Meeting Maribor, Christian Schindelhauer joint work with Christian Scheideler Kishore Kothapalli Melih Onus IPDPS 2006

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 2  Def.: Vertex Coloring –Given a graph G=(V,E) –Find a coloring w:V  {1,..,C} –such that for all (u,v)  E: w(u) ≠ w(v)   (G) cannot be approximated unless P=NP = chromatic number of the graph = minimum C s.t. a coloring exists  Vertex Coloring is NP-hard  Every graph can be colored with ∆+1 colors (  (G) ≤ ∆(G)+1) –where ∆ is the maximum degree of a vertex in a graph  in linear time Good Old Vertex Coloring

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 3 Distributed ∆+1-Coloring  Idea –Every node checks the colors of its at most ∆ neigbors –Then at least one color is left –Choose this color  Problem –What if neighbors have not chosen a color so far?  Solutions –Choose sequentially takes n steps –Start with random assignment symmetry breaking necessary –Start with larger number of colors and reduce the number of colors

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 4 Distributed ∆+1-Coloring Results  Complexity Measure: –Maximal number of rounds until a coloring ist etablished –In each round every node can send one bit to a neighbor  Best known deterministic algorithm [Panconesi Srinivasan 96]  Randomized distributed algorithm Luby O(log n)

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 5 Example: The Line Graph End node starts coloring Runtime: n

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 6 Random Algorithm on Unoriented Line Graph  Algorithm:  Repeat –Choose random color –Check with neighbors  until colors are o.k.  Runtime: O(log n) with high probability, –i.e. 1-n -c

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 7 Lower Bound for the Unoriented Line Graph  Choose t = (log n)/2 leads to a probability of 2/3 n -1/2  E[number of neighbored same colored nodes after (log n)/2] ≥ 1

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 8 The Oriented Algorithm  Algorithm Color-Random –C u = {1,...,∆} –While u is not colored do 1.Node u chooses a color c u from the available colors in C u uniformly at random. 2.Node u communicates its choice c u, from step 1, to all of its uncolored neighbors that have a lower priority over u, i.e. to nodes v such that u → v. 3.If node u does not receive a message from any of its neighbors w with w → u and c w = c u, then node u gets colored with color c u. Otherwise node u remains uncolored. 4.If u is colored during step 3 of the current round, then u informs all of its uncolored neighbors about the color of u. 5.Node u updates the list of available colors according to colors taken up by u‘s neighbors.

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 9 Example: Oriented Line Graph

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 10 Analysis of Oriented Line Graph (I)  After steps with high probability no interval of length has an uncolored node

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 11 Analysis of Oriented Line Graph (II)  After further steps all uncolored intervals of length will be colored

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 12 Lower Bound for the Oriented Line Graph  Theorem –Every Las-Vegas-Algorithm needs bit rounds to color the oriented line graph with ∆+1 colors.  Idea: –Describe the state of a node by its input/output behavior –In every step and for every pair node the probability to choose the same state is at least a constant if they have been in the same state before

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 13 Fast Coloring in Constant Degree Oriented Graphs  Theorem –The oriented Line Graph can be distributely colored within bit rounds with high probability.  Algorithm works also for more general graphs  Definition –A k-acyclic graph is a directed graph with no cycles of length smaller than k.

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 14 Fast Coloring of Constant Degree Networks  Proof idea: –Replace line segments by connected components of uncolored nodes

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 15 High Degree Networks  Definition –A k-acyclic graph is a directed graph with no cycles of length smaller than k.  Use same algorithm with more colors  Analysis is more involved  Actual bound:

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 16 Small is Ugly! Big is Beautiful!  A graph with a small cycle needs  (log n) bit rounds  A graph with only large cycles can be colored in bit rounds

University of Freiburg Institute of Computer Science Computer Networks and Telematics Prof. Christian Schindelhauer COST 293 GRAAL and COST 295 DYNAMO Maribor, Distributed Coloring in Õ(  log n) Bit Rounds - 17 Applications for Coloring  Leader Election and Symmetry Breaking  Scheduling  Resource Allocation  Motivation for this work –Symmetry breaking in ad-hoc networks –Problem: small cycles!  Other application –Leader election in peer-to-peer networks –For large networks cycles are not small

18 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Thank you COST 293 GRAAL and COST 295 DYNAMO Discussion Workshop and MC Meeting Maribor, Online-Multipath Routing in a Maze Christian Schindelhauer joint work with Christian Scheideler Kishore Kothapalli Melih Onus IPDPS 2006