Presentation of a paper by Manuel Then, about efficiently executing many BFS traversals concurrently, on a single core

1. The More the Merrier
Efficient Multi-Source Graph Traversal

2. BFS as algorithm building block
Graph algorithms are often based on BFS
Many BFS traversals from many starting points
Example: closeness centrality

3. Multi-Source BFS (MS-BFS)
Multiple BFS traversals
One graph
One CPU core

4. MS-BFS - TL;DR;
Share common computation across traversals
Reduce number of random accesses
Do no synchronization

5. It’s a small world
Real world graphs are often small world graphs
Power law vertex degree distribution
Low diameter relative to vertex count
“Six degrees of separation”

6. It’s a small world

7. It’s a small world

8. BFS

9. MS-BFS
B={b1
,...,bω
}: bi
is a BFS, ω is # of BFSs
S={s1
,...,sω
}: si
is source vertex for BFS bi
every vertex has seenv
⊆B
visit & visitNext contain tuples (v,B’)
B’⊆B → BFSs that must explore v in next level

10. MS-BFS
mark source of each BFS as discovered
which vertex to explore for each BFS
merge all BFSs that need to explore v into B’v
BFSs that need to explore n in next level
> belong to current level (bi
∈B’v
)
> have not discovered n yet (bi
∈/seenn
)
mark n as discovered & process it for all BFSs bi
∈D

11. MS-BFS Example
> initialize all the things

12. MS-BFS Example
> v1
& v2
are explored
> both are adjacent to v3
& v4
> 2nd
level visit populated with
tuples for v3
& v4
in BFSs b1
& b2

13. MS-BFS Example
> v3
& v4
are explored
> BFSs to explore v3
are merged into
B’3
={b1,b2}
> v3
explored only once, for both b1
& b2
> v5
discovered simultaneously by b1
& b2
> similar process happens for v4

14. MS-BFS ♡ CPU - Bit Fields
● Replace expensive set operations with bit operations
○ A ∪ B --> A | B
○ A B --> A & ∼B
● Bit fields stored in arrays sized to |V|
○ Constant-time access
○ visitv
= visit[v]
● Leverages native bit instructions

15. MS-BFS ♡ CPU - Graph

16. MS-BFS ♡ CPU

17. MS-BFS ♡ CPU

18. MS-BFS ♡ CPU

19. MS-BFS ♡ CPU

20. MS-BFS CPU
> random access to visitNext & seen
> multiple executions for same n

21. MS-BFS ♡ CPU
Aggregated Neighbor Processing (ANP)

22. MS-BFS ♡ CPU - ANP

23. Other Optimizations
● DOT: Direction-Optimized Traversal
○ Scan seen instead of visit
● PF: Neighbor Prefetching
○ Prefetch neighbors of visitNext entries
● CL: Cache Line
○ Use multiple registers per bit field to fill entire cache line
● SHR: Heuristic for Maximum Sharing
○ Grouping vertices with similar degree in the same run

24. Results...

25. Memory VS Concurrency

26. Vertex# VS Runtime

27. Core# VS Throughput

28. BFS# VS Throughput

29. Optimization VS Speedup

30. Reading
● The more the merrier: Efficient multi-source graph traversal
Then, Manuel, et al.
VLDB, 2014
http://www.vldb.org/pvldb/vol8/p449-then.pdf
● Parallel Array-Based Single-and Multi-Source BFS on Large Dense Graphs
Kaufmann, Moritz, et al.
EDBT, 2017
https://openproceedings.org/2017/conf/edbt/paper-107.pdf
● Prefetching with Helper Threads for Loosely Coupled Multiprocessor Systems
Lee, Jaejin, et al.
TPDS, 2009
http://ieeexplore.ieee.org/abstract/document/4641920/