1. Implementing Parallel Graph Algorithms: Graph coloring Created by: Avdeev Alex, Blakey Paul.

2. Criteria for good graph coloring algorithm: A B

3. Serial Graph coloring Approach: o Greedy Multicoloring Numbering the vertexes order in Lexicographical / BFS / Cuthill Mckee / etc… Iteration through the mesh: each vertex visited, colored by the minimal color available. o Luby Jones Random Numbering of vertexes. Iterate through the mesh. If possible assign the current color to vertex. Pick new color every iteration. o Distance-k Graph Coloring All vertices within the distance-K from each other has unique colors.

4. Parallel Graph Coloring: The last decades a lot of work has been invested into the process of parallelizing graph coloring algorithms. Can be divided into two kinds of algorithms: Graphs divided into sub-sets, where each sub-set sequentially colored by Multicolor Algorithm. Luby Jones approach Algorithms. Yet, none of this left the Greedy Multicoloring Algorithm obsolete.

5. Parallelizing categories: Two kinds of approaches to Parallelizing Coloring Algorithm: o Local Parallelization: The adjacent vertices of the current vertex can be target for parallelization. o Global Parallelization: Divide the graph into sub-sets and color each one of them. Each sub-set is colored in parallel.

6. Implementation optimizations done: Numbering: o We wanted to number the graph so we have less adjacent vertices between different sub-sets, trying to reduce wait time for acquiring locks belonging to different sets. Lexicographical Breadth First Search Random Cuthill Mckee Numbering

7. Parallel Greedy Multicoloring: Research & Implementation

8. Our First Algorithm: Global Cuthill Mckee Numbering o Initialize graph and Number the vertices with Cuthill Mckee Numbering approach. o Divide graph to equal chunks. o For each chunk do in parallel: o Color the vertices with Multicolor algorithm: Iteration through the mesh of chunk: each vertex visited, is colored by the minimal color possible. The synchronization is done by the Galois system.

9. Second Algorithm: Local Breadth First Search Graph Coloring o Go on the graph vertexes on Breadth First Search order by pushing the first vertex to a worklist o And for each vertex in worklist: o Color the vertex with the first available color (not used by adjacent vertex) o Push adjacent vertexes to the worklist if not already pushed or colored

10. Third Algorithm: Global Random Multicoloring o Go over the graph vertexes o For each vertex do: Color the vertex with the first available color (not used by adjacent vertex)

11. Runtime Results: o We received worse running time in algorithm scheduled by Mckee Numbering (Sorting was required) o RndLoc. And BFS received much better running times o We can see in general that as we increase the number of Active threads for RndLoc. And BFS, we receive better performance (until some upper limit)

12. Runtime Results: o BFS Algorithm and the Random Greedy Algorithm receives good running time even on BIG graphs.

13. Runtime Results: o Mckee received best coloring quality on real and dense graphs among the algorithms.

14. Conclusions: Serial Implementations vs.Parallel Implementations: o Color quality vary to both sides with thread number, graph structure, scheduling and chunk size, only guarantee is that the number of colors will still be lower than graph's max degree + 1. o BFS and Random seem to perform on the same level. On the tested graphs, BFS got lower colors and more graphs than Random, but Random usually performed a bit faster. Cuthill Mckee gave a better color quality than both other algorithmswhen it actually done coloring on the real world (large/non-random) graphs. o BFS and Random improved the running with increasing the thread count as we wanted and expected, while Cuthill Mckee sometimes did sometimes didn't. o We concluded that the time cost needed to sort our graph vertex numbers for the "ordered" Cuthill Mckee algorithm so we can divide chunks to threads was too much in Galois system. o Number of colors change from run to run using the same parameters and graph.