Accelerating Approximate Weighted Matching on GPUs MD NAIM*, FREDRIK - - PowerPoint PPT Presentation
Accelerating Approximate Weighted Matching on GPUs MD NAIM*, FREDRIK MANNE*, MAHANTESH HALAPPANAVAR + , ANTONINO TUMEO + , JOHANNES LANGGUTH # *University of Bergen Bergen, Norway + Pacific Northwest National Laboratory Richland, WA, USA #
MD NAIM*, FREDRIK MANNE*, MAHANTESH HALAPPANAVAR+, ANTONINO TUMEO+, JOHANNES LANGGUTH#
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*University of Bergen – Bergen, Norway
+Pacific Northwest National Laboratory – Richland, WA, USA #Simula Research Laboratory – Oslo, Norway
Given an edge weighted graph G(V,E). Select a set M of non-incident edges of maximum weight. Best known algorithm has running time O(|V||E|+|V|2log|V|) [Gabow 90] Often too expensive for real applications
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W(opt) = 22
Add most expensive remaining edge (x,y) to the solution Remove (x,y) and all edges incident on x or y Running time: O(|E| log |V|) (due to sorting) Guarantees a 1⁄2 - approximation, but in practice often much better
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Suitor Algorithm (Manne and Halappanavar 2015)
P(x) = “v: Heaviest neighbor of x that does not already have a better
Lemma: If P(x) is set correctly for every x, then P() defines the same solution as Greedy. Outline of The Suitor Algorithm:
Process the vertices in a linear fashion to find the best candidate for each vertex Set the Suitor value of the candidate Whenever a vertex is dislodged it must be re-processed before moving to the next vertex
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(x,y) in M
x y
a b c d e
END
2 3 6 8 10 1 2 3 4 5 6 7 8 9 b c a a d e c e c d 4 5 4 5 7 9 7 10 9 10
e d c a b 1 7 9 5 4 Compressed Edge List Idx E W
e d c a b 1 7 9 5 4 __:__ __:__ __:__ __:__ __:__ Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 __:__ __:__ __:__ __:__ __:__ Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 __:__ __:__ __:__ a : 5 __:__ Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 __:__ __:__ __:__ a : 5 __:__ Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 __:__ __:__ __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 __:__ __:__ __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 c : 9 __:__ __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
e d c a b 1 7 9 5 4 __:__ __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
c : 9
e d c a b 1 7 9 5 4 __:__ __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
d : 9
e d c a b 1 7 9 5 4 c : 7 __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
d : 9
e d c a b 1 7 9 5 4 c : 7 __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
d : 9
e d c a b 1 7 9 5 4 e : 10 __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
d : 9
e d c a b 1 7 9 5 4 e : 10 __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
d : 9
e d c a b 1 7 9 5 4 e : 10 __:__ a : 5 b : 4 Unprocesse d Current Candidate Processed Suitor:Offer
d : 9
Search for the best possible candidate for each vertex → O(dv) In the worst case, needs to repeat the search dv times Set the Suitor and corresponding Offer, O(1) Running Time: O( Σv∈V (dv)2) ≤ O(∆E) With sorted neighbor lists: O( Σv∈V (dv log dv )2 + V + E) ≤ O(Elog∆+ V + E)
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The order in which vertices are processed has no influence on the final matching The vertices can be processed independently Need to protect the Suitor values as multiple vertices may try to become suitor at the same time
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x v u Mutual Exclusion
Assign a chunk of vertices to each warp Each warp processes its chunk independently Each warp starts with the assigned chunk in shared memory The warp processes its vertices in a linear fashion Search for the best possible candidate for each v ∈ chunk The threads in the warp read the neighbor list of v in an interleaved fashion O(Ceil( |N(v)|/32))
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9
. . . Start Indices . . . . . . . . . W E ….. warp0 warp1 warpL-1 Stored in shared memory of the warp
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1 ... 30 31 w w
1
.. w
30
w
31
w
32
w
33
.. w
62
w
63
1 ... 30 31 time |N(v)| = 64 1 ... 30 31 w w
1
.. w
30
w
31
w
32
w
33
.. w
62
w
63
1 ... 30 31 T
best_t
=max{ wt , wt+32 ,wt+64,...}
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Assign a chunk of vertices to each warp Each warp processes its chunk independently Each warp starts with the assigned chunk in shared memory The warp processes its vertices in a linear fashion Search for the best possible candidate for each v ∈ chunk The threads in the warp read the neighbor list of v in an interleaved fashion O(Ceil( |N(v)|/32)) Saves the best found so far in a local variable Reduction at the end, O( log2(32))
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log2(32) Butterfly Reduction T T
1
T
2
T
3
. . . . T
30
T
31
T
01
T
01
T
23
T
23
.. .. .. ..
0123 0123 0123 0123 …. ....
.... .... …. …. T
best
=max{T
best_0,
....
,
T
best_31
} After 5 Steps
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Assign a chunk of vertices to each warp Each warp processes its chunk independently Each warp starts with the assigned chunk in shared memory The warp processes its vertices in a linear fashion Search for the best possible candidate for each v ∈ chunk The threads in the warp read the neighbor list of v in an interleaved fashion O(Ceil( |N(v)|/32)) Saves the best found so far in a local variable Reduction at the end, O( log2(32)) Set the Suitor and corresponding Offer for the candidate, O(1) Can be buffered in registers Save failed and dislodged vertices in shared memory Processed by the same warp in the next round Redistributed across the warps of the same block
Each warp stores failed and dislodged vertices in programmable cache
Can be redistributed across the warps of the same block Need synchronization among the warps of the same block Synchronization can become a bottleneck
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Dataset
Florida Sparse Matrix Collection: 269 problems ranging from a million to billion of edges
Alternatives
Locally Dominant(LD): Shared Memory and GPU Implementation Suitor: Shared Memory
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Intel CPUs 2 sockets, 64 GB and BW of 51.2 GB/s. Hyper-threaded 8-core Intel Xeon E5 @3.10 GHz GCC 4.9.2, GOMP_CPU_AFFINITY, numactl NVIDIA GPU Tesla K40 with GK110B GPU, 15 SMXes, 2880 cores @ 745 MHz 12 GB of GDDR5 at 3 GHz, BW of 288 GB/sec. nvcc (CUDA 7.0)
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Speedup of GPU-Suitor relative to GPU-LD and OMP-LD (left),and OMP-Suitor (right)
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Presented an implementation of the Suitor Matching algorithm for GPUs Faster than multicore and previous GPU implementations Future works:
Extension to multigpus
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Questions? Md Naim - naim.md@gmail.com Fredrik Manne - ManneFredrik.Manne@ii.uib.no Mahantesh Halappanavar – mahantesh.halappanavar@pnnl.gov Antonino Tumeo – antonino.tumeo@pnnl.gov Johannes Langguth – langguth@simula.no
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