Parallel Graph Algorithms Sathish Vadhiyar
Jan 20, 2016
Parallel Graph Algorithms
Sathish Vadhiyar
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
Level-synchronized algorithm
Proceeds level-by-level starting with the source vertex
Level of a vertex – its graph distance from the source
How to decompose the graph (vertices, edges and adjacency matrix) among processors?
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
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
Lvs(v) – level of v, i.e, graph distance from source vs
2D Partitioning
P=RXC processor mesh Adjacency matric divided into R.C block rows
and C block columns A(i,j)
(*) denotes a block owned by (i,j) processor; each processor owns C blocks
2D Partitioning
Processor (i,j) owns vertices belonging to block row (j-1).R+i
Thus a process stores some edges incident on its vertices, and some edges that are not
2D Paritioning Assume that the edge list for a given vertex is
the column of the adjacency matrix Each block in the 2D partitioning contains
partial edge lists Each processor has a frontier set of vertices, F,
owned by the processor
2D ParitioningExpand Operation
Consider v in F The owner of v sends messages to
other processors in frontier column telling that v is in the frontier; since any of these processors may have partial edge list of v
2D PartitioningFold Operation
Partial edge lists on each processor merged to form N – potential vertices in the next frontier
Vertices in N sent to their owners to form new frontier set F on those processors
These owner processors are in the same processor row
This communication step referred as fold operation
Analysis Advantage of 2D over 1D – processor-
column and processor-row communications involve only R and C processors
BFS on GPUs
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
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
Dynamic Load Balancing (DLB)
Difficult to estimate the size of the search space beforehand
Need to balance the search space among processors dynamically
In DLB, when a processor runs out of work, it gets work from another processor
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
Work Splitting When a processor receives work request, it
splits its search space Half-split: Stack space divided into two equal
pieces – may result in load imbalance Giving stack space near the bottom of the
stack can lead to giving bigger trees Stack space near the top of the stack tend to
have small trees To avoid sending very small amounts of work –
nodes beyond a specified stack depth are not given away – cutoff depth
Strategies
1. Send nodes near the bottom of the stack
2. Send nodes near the cutoff depth 3. Send half the nodes between the
bottom of the stack and the cutoff depth
Example: Figures 11.5(a) and 11.9
Load Balancing Strategies
Asynchronous round-robin: Each processor has a target processor to get work from; the value of the target is incremented with modulo
Global round-robin: One single target processor variable is maintained for all processors
Random polling: randomly select a donor
Termination Detection
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
Algorithm Continued…. Taken care of by using white and black
tokens Initially, the token is white; a processor j
becomes black if it sends work to i<j If j completes work, it changes token to
black and sends it to next processor; after sending, changes to white.
When P0 receives a black token, reinitiates the ring
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
Minimal Spanning Tree, Single-Source and All-pairs Shortest Paths
Minimal Spanning Tree – Prim’s Algorithm
Spanning tree of a graph, G (V,E) – tree containing all vertices of G
MST – spanning tree with minimum sum of weights
Vertices are added to a set Vt that holds vertices of MST; Initially contains an arbitrary vertex,r, as root vertex
Minimal Spanning Tree – Prim’s Algorithm
An array d such that d[v in (V-Vt)] holds weight of the edge with least weight between v and any vertex in Vt; Initially d[v] = w[r,v]
Find the vertex in d with minimum weight and add to Vt
Update d Time complexity – O(n2)
Parallelization
Vertex V and d array partitioned across P processors
Each processor finds local minimum in d Then global minimum across all d
performed by reduction on a processor The processor finds the next vertex u,
and broadcasts to all processors
Parallelization
All processors update d; The owning processor of u marks u as belonging to Vt
Process responsible for v must know w[u,v] to update v; 1-D block mapping of adjacency matrix
Complexity – O(n2/P) + (OnlogP) for communication
Single Source Shortest Path – Dijikistra’s Algorithm
Finds shortest path from the source vertex to all vertices
Follows a similar structure as Prim’s Instead of d array, an array l that
maintains the shortest lengths are maintained
Follow similar parallelization scheme
Single Source Shortest Path on GPUs
SSSP on GPUs
A single kernel is not enough since Ca cannot be updated while it is accessed.
Hence costs updated in a temporary array Ua
All-Pairs Shortest Paths
To find shortest paths between all pairs of vertices
Dijikstra’s algorithm for single-source shortest path can be used for all vertices
Two approaches
All-Pairs Shortest Paths Source-partitioned formulation: Partition the vertices
across processors Works well if p<=n; No communication Can at best use only n processors Time complexity?
Source-parallel formulation: Parallelize SSSP for a vertex across a subset of processors
Do for all vertices with different subsets of processors
Hierarchical formulation Exploits more parallelism Time complexity?
All-Pairs Shortest PathsFloyd’s Algorithm
Consider a subset S = {v1,v2,…,vk} of vertices for some k <= n
Consider finding shortest path between vi and vj
Consider all paths from vi to vj whose intermediate vertices belong to the set S; Let pi,j
(k) be the minimum-weight path among them with weight di,j
(k)
All-Pairs Shortest PathsFloyd’s Algorithm
If vk is not in the shortest path, then pi,j
(k) = pi,j(k-1)
If vk is in the shortest path, then the path is broken into two parts – from vi to vk, and from vk to vj
So di,j(k) = min{di,j
(k-1) , di,k(k-1) + dk,j
(k-1) } The length of the shortest path from
vi to vj is given by di,j(n).
In general, solution is a matrix D(n)
Parallel Formulation2-D Block Mapping Processors laid in a 2D mesh During kth iteration, each process Pi,j
needs certain segments of the kth row and kth column of the D(k-1) matrix
For dl,r(k): following are needed
dl,k(k-1) (from a process along the same process
row) dk,r
(k-1) (from a process along the same process column)
Figure 10.8
Parallel Formulation2D Block Mapping
During kth iteration, each of the root(p) processes containing part of the kth row sends it to root(p)-1 in same column;
Similarly for the same row Figure 10.8 Time complexity?
APSP on GPUs Space complexity of Floyd’s algorithm is O(V2) –
Impossible to go beyond a few vertices on GPUs Uses V2 threads A single O(V) operation looping over O(V2) threads
- can exhibit slowdown due to high context switching overhead between threads
Use Dijikistra’s – run SSSP algorithm from every vertex in graph
Will require only the final output size to be O(V2) Intermediate outputs on GPU can be O(V) and can
be copied to CPU memory
APSP on GPUs
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.
Speedup Anomalies in DFS
The overall work (space searched) in parallel DFS can be smaller or larger than in sequential DFS
Can cause superlinear or sublinear speedups
Figures 11.18, 11.19
Parallel FormulationPipelining
In the 2D formulation, the kth iteration in all processes start only after k-1(th) iteration completes in all the processes
A process can start working on the kth iteration as soon as it has computed (k-1)th iteration and has relevant parts of the D(k-1) matrix
Example: Figure 10.9 Time complexity