1. Parallel Graph Algorithms
Sathish Vadhiyar

2. Graph Traversal
Graph search plays an important role in analyzing large data sets
Relationship between data objects represented in the form of graphs
Breadth first search used in finding shortest path or sets of paths

3. Level-synchronized algorithm
Proceeds level-by-level starting with the source vertex
Level of a vertex – its graph distance from the source
Also, called frontier-based algorithm
The parallel processes process a level, synchronize at the end of the level, before moving to the next level – Bulk Synchronous Parallelism (BSP) model
How to decompose the graph (vertices, edges and adjacency matrix) among processors?

4. Distributed BFS with 1D Partitioning
Each vertex and edges emanating from it are owned by one processor
1-D partitioning of the adjacency matrix
Edges emanating from vertex v is its edge list = list of vertex indices in row v of adjacency matrix A

5. 1-D Partitioning
At each level, each processor owns a set F – set of frontier vertices owned by the processor
Edge lists of vertices in F are merged to form a set of neighboring vertices, N
Some vertices of N owned by the same processor, while others owned by other processors
Messages are sent to those processors to add these vertices to their frontier set for the next level

6. Lvs(v) – level of v, i.e, graph distance from source vs

7. BFS on GPUs

8. BFS on GPUs One GPU thread for a vertex
In each iteration, each vertex looks at its entry in the frontier array
If true, it forms the neighbors and frontiers
Severe load imbalance among the treads
Scope for improvement

9. Depth First Search (DFS)

10. Parallel Depth First Search
Easy to parallelize
Left subtree can be searched in parallel with the right subtree
Statically assign a node to a processor – the whole subtree rooted at that node can be searched independently.
Can lead to load imbalance; Load imbalance increases with the number of processors (more later)

11. Maintaining Search Space
Each processor searches the space depth-first
Unexplored states saved as stack; each processor maintains its own local stack
Initially, the entire search space assigned to one processor
The stack is then divided and distributed to processors

12. Termination Detection
As processors are independently, how will they know when to terminate the program?
Dijikstra’s Token Termination Detection Algorithm
Based on passing of a token in a logical ring; P0 initiates a token when idle; A processor holds a token until it has completed its work, and then passes to the next processor; when P0 receives again, then all processors have completed
However, a processor may get more work after becoming idle due to dynamic load balancing (more later)

13. Tree Based Termination Detection
Uses weights
Initially processor 0 has weight 1
When a processor transfers work to another processor, the weights are halved in both the processors
When a processor finishes, weights are returned
Termination is when processor 0 gets back 1
Goes with the DFS algorithm; No separate communication steps
Figure 11.10

14. Graph Partitioning

15. Graph Partitioning
For many parallel graph algorithms, the graph has to be partitioned into multiple partitions and each processor takes care of a partition
Criteria:
The partitions must be balanced (uniform computations)
The edge cuts between partitions must be minimal (minimizing communications)
Some methods
BFS: Find BFS and descend down the tree until the cumulative number of nodes = desired partition size
Mostly: Multi-level partitioning based on coarsening and refinement (a bit advanced)
Another popular method: Kernighan-Lin

19. Parallel partitioning
Can use divide and conquer strategy
A master node creates two partitions
Keeps one for itself and gives the other partition to another processor
Further partitioning by the two processors and so on…

20. Graph Coloring

21. Graph Coloring Problem
Given G(A) = (V, E)
σ: V {1,2,…,s} is s-coloring of G if σ(i) ≠ σ(j) for every (i, j) edge in E
Minimum possible value of s is chromatic number of G
Graph coloring problem is to color nodes with chromatic number of colors
NP-complete problem

22. Parallel graph Coloring – General algorithm

23. Parallel Graph Coloring – Finding Maximal Independent Sets – Luby (1986)
I = null
V’ = V
G’ = G
While G’ ≠ empty
Choose an independent set I’ in G’
I = I U I’; X = I’ U N(I’) (N(I’) – adjacent vertices to I’)
V’ = V’ \ X; G’ = G(V’)
end
For choosing independent set I’: (Monte Carlo Heuristic)
For each vertex, v in V’ determine a distinct random number p(v)
v in I iff p(v) > p(w) for every w in adj(v)
Color each MIS a different color
Disadvantage:
Each new choice of random numbers requires a global synchronization of the processors.

24. Parallel Graph Coloring – Gebremedhin and Manne (2003)
Pseudo-Coloring

25. Sources/References
Paper: A Scalable Distributed Parallel Breadth-First Search Algorithm on BlueGene/L. Yoo et al. SC 2005.
Paper:Accelerating large graph algorithms on the GPU usingCUDA. Harish and Narayanan. HiPC 2007.
M. Luby. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing. 15(4) (1986)
A.H. Gebremedhin, F. Manne, Scalable parallel graph coloring algorithms, Concurrency: Practice and Experience 12 (2000)