science.uu.nl project csg - A GPU Algorithm for Greedy Graph …bisse101/Software/match... · 2015....

A GPU Algorithm for Greedy Graph Matching

B. O. Fagginger Auer R. H. Bisseling

Utrecht University

September 29, 2011

Fagginger Auer, Bisseling (UU) GPU Greedy Graph Matching September 29, 2011 1 / 27

Outline

1 Introduction

2 CPU matching

3 GPU matching

4 Implementation

5 Results

6 Conclusion

Introduction

We will discuss generating greedy graph matchings on the GPU.

Graph matching ≈ a pairing of neighbouring vertices within a graph.

Matching has applications in

I minimising wireless network power consumption,I Travelling salesman problem heuristics,I organ donation,I . . .

Our primary interest is graph coarsening, where we contract matchedvertices to obtain a coarser version of the original graph.

Introduction

Matching has applications inI minimising wireless network power consumption,

I Travelling salesman problem heuristics,I organ donation,I . . .

Introduction

Matching has applications inI minimising wireless network power consumption,I Travelling salesman problem heuristics,

I organ donation,I . . .

Introduction

Matching has applications inI minimising wireless network power consumption,I Travelling salesman problem heuristics,I organ donation,

I . . .

Introduction

Matching has applications inI minimising wireless network power consumption,I Travelling salesman problem heuristics,I organ donation,I . . .

Introduction

Matching has applications inI minimising wireless network power consumption,I Travelling salesman problem heuristics,I organ donation,I . . .

Graph Matching

A graph is a pair G = (V ,E ) with vertices V and edges E .

All edges e ∈ E are of the form e = {v ,w} for vertices v ,w ∈ V .

A matching is a collection M ⊆ E of edges that are disjoint.

We will view matchings as a map π : V → N such that

π(v) = π(w) ⇐⇒ {v ,w} ∈ M.

Graph Matching

π(v) = π(w) ⇐⇒ {v ,w} ∈ M.

Graph Matching

π(v) = π(w) ⇐⇒ {v ,w} ∈ M.

Graph Matching

π(v) = π(w) ⇐⇒ {v ,w} ∈ M.

Maximal Matching

A matching is maximal if we cannot enlarge it further by addinganother edge to it.

Maximum Matching

A matching is maximum if it possesses the largest possible number ofedges, compared to all other matchings.

Graph Matching

If the edges are provided with weights ω : E → R>0, finding amatching M which maximises

ω(M) =∑e∈M

ω(e),

is called weighted matching.

Greedy matching provides us with maximal matchings, but notnecessarily of maximum possible weight or maximum number ofvertices/edges.

Graph Matching

If the edges are provided with weights ω : E → R>0, finding amatching M which maximises

ω(M) =∑e∈M

ω(e),

is called weighted matching.

Greedy matching provides us with maximal matchings, but notnecessarily of maximum possible weight or maximum number ofvertices/edges.

CPU matching

We will now look at a serial greedy algorithm which generates amaximal matching.

In random order, vertices v ∈ V select and match neighboursone-by-one.

Here, we can pick

I the first available neighbour w of v (random matching),I the neighbour w for which ω({v ,w}) is maximal (weighted matching).

CPU matching

Here, we can pick

CPU matching

Here, we can pick

CPU matching

Here, we can pickI the first available neighbour w of v (random matching),

I the neighbour w for which ω({v ,w}) is maximal (weighted matching).

CPU matching

Here, we can pickI the first available neighbour w of v (random matching),I the neighbour w for which ω({v ,w}) is maximal (weighted matching).

CPU matching

We will create a random matching for this graph.

CPU matching

Consider the vertices one-by-one.

CPU matching

Select unmatched neighbour. . .

CPU matching

. . . and match.

CPU matching

Skip matched vertices.

CPU matching

Skip already matched neighbours.

CPU matching

Keep matching until we have treated all vertices.

CPU matching

We have obtained a maximal matching (also maximum in this case).

Problematic parallelism

Directly extending this to a parallel algorithm is problematic.

Disjoint edges requirement leads to serialisation.

Directly extending this to a parallel algorithm is problematic.

Disjoint edges requirement leads to serialisation.

Suppose we match vertices simultaneously.

Vertices find an unmatched neighbour. . .

. . . but generate an invalid matching.

GPU matching

To solve this we create two groups of vertices: blue and red.

Blue vertices propose.

Red vertices respond.

Proposals that were responded to are matched.

GPU matching

GPU implementation

The graph (neighbour ranges, indices, and weights) is stored as atriplet of 1D textures on the GPU.

We create one thread for each vertex in V .

Each vertex v ∈ V only updates

I its colour/matching value π(v);I and its proposal/response value σ(v).

Both π and σ are stored in 1D arrays in global memory.

GPU implementation

Each vertex v ∈ V only updates

I its colour/matching value π(v);I and its proposal/response value σ(v).

GPU implementation

Each vertex v ∈ V only updatesI its colour/matching value π(v);I and its proposal/response value σ(v).

GPU implementation

Each vertex v ∈ V only updatesI its colour/matching value π(v);I and its proposal/response value σ(v).

GPU matching

ColourProposeRespondMatch

1 2 3 4 5 6 7 8 9

π - - - - - - - - -σ - - - - - - - - -

GPU matching

1 2 3 4 5 6 7 8 9

π b r r b b r b b rσ - - - - - - - - -

GPU matching

1 2 3 4 5 6 7 8 9

π b r r b b r b b rσ 3 - - 3 6 - 3 2 -

GPU matching

ColourProposeRespond

1 2 3 4 5 6 7 8 9

π b r r b b r b b rσ 3 8 7 3 6 5 3 2 -

GPU matching

1 2 3 4 5 6 7 8 9

π b 2 3 b 5 5 3 2 rσ 3 8 7 3 6 5 3 2 -

GPU matching

1 2 3 4 5 6 7 8 9

π r 2 3 r 5 5 3 2 bσ 3 8 7 3 6 5 3 2 -

GPU matching

1 2 3 4 5 6 7 8 9

π r 2 3 r 5 5 3 2 bσ - - - - - - - - d

GPU matching

1 2 3 4 5 6 7 8 9

π r 2 3 r 5 5 3 2 bσ - - - - - - - - d

GPU matching

1 2 3 4 5 6 7 8 9

π r 2 3 r 5 5 3 2 dσ - - - - - - - - d

GPU matching

1 2 3 4 5 6 7 8 9

π b 2 3 r 5 5 3 2 dσ - - - - - - - - d

GPU matching

1 2 3 4 5 6 7 8 9

π b 2 3 r 5 5 3 2 dσ 4 - - - - - - - -

GPU matching

1 2 3 4 5 6 7 8 9

π b 2 3 r 5 5 3 2 dσ 4 - - 1 - - - - -

GPU matching

1 2 3 4 5 6 7 8 9

π 1 2 3 1 5 5 3 2 dσ 4 - - 1 - - - - -

Matching saturation

0 5 10 15 20Mat

Number of iterations

Saturation of matching size

ecology2 (1,997,996)ecology1 (1,998,000)

G3_circuit (3,037,674)thermal2 (3,676,134)

kkt_power (6,482,320)af_shell9 (8,542,010)

ldoor (22,785,136)af_shell10 (25,582,130)

audikw1 (38,354,076)nlpkkt120 (46,651,696)

cage15 (47,022,346)

Fraction of matched vertices as function of the number of iterations.

Colouring vertices

To colour vertices v ∈ V , we use for a fixed p ∈ [0, 1]

colour(v) =

{blue with probability p,red with probability 1− p.

How to choose p? Maximise the number of matched vertices.

For a large random graphs, the expected fraction of matched verticescan be approximated by (independent of edge density)

2 (1− p)(

1− e−p

). (2)

Colouring vertices

colour(v) =

2 (1− p)(

1− e−p

). (2)

Colouring vertices

colour(v) =

2 (1− p)(

1− e−p

). (2)

Choosing p

0 20 40 60 80 100Fra

Fraction of vertices that are blue (%)

Influence of relative blue/red group size

Matching weightMatching sizeMatching time

0 20 40 60 80 100Fra

Fraction of vertices that are blue (%)

Influence of relative blue/red group size

ObservedEquation (2)

Equation (2): we should choose p ≈ 0.53406.

Results

Created an implementation on the GPU using CUDA and on the CPUusing TBB.

We consider both random and weighted matching.

Vertex orderings are randomised and results are averaged over 32randomisations.

Time only pertains to matching, not I/O or randomisation.

Test set: ongoing 10th DIMACS challenge on graph partitioning andUniversity of Florida Sparse Matrix Collection.

Test hardware: dual quad-core Xeon E5620 and an NVIDIA TeslaC2050 (thanks: the Little Green Machine project).

Results

Results (scaling)

60 70 80 90

1 2 4 8 16

Number of CPU threads

Matching time scaling

ecology2 (1,997,996)ecology1 (1,998,000)

G3_circuit (3,037,674)thermal2 (3,676,134)

kkt_power (6,482,320)af_shell9 (8,542,010)

ldoor (22,785,136)af_shell10 (25,582,130)

audikw1 (38,354,076)nlpkkt120 (46,651,696)

cage15 (47,022,346)ideal scaling

Scaling of TBB implementation (8 physical cores + hyperthreading).

Results (vs. local random matching)

101 102 103 104 105 106 107 108

Number of graph edges

Matching size for random parallel matching (vs. Alg. 1)

CUDATBB

101 102 103 104 105 106 107 108S

Speedup for random parallel matching (vs. Alg. 1)

CUDATBB

Matching size and speedup for parallel vs. serial local random matching.

Results (vs. local weighted matching)

101 102 103 104 105 106 107 108

Matching weight for weighted parallel matching (vs. Alg. 1)

CUDATBB

101 102 103 104 105 106 107 108S

Speedup for weighted parallel matching (vs. Alg. 1)

CUDATBB

Matching weight and speedup for parallel vs. serial local weightedmatching.

Results (vs. global weighted matching)

101 102 103 104 105 106 107 108

Matching weight for weighted parallel matching (vs. Alg. 2)

CUDATBB

101 102 103 104 105 106 107 108S

Speedup for weighted parallel matching (vs. Alg. 2)

CUDATBB

Matching weight and speedup for parallel local vs. serial global weightedmatching.

Conclusion

We have presented a fine-grain, shared-memory parallel greedy graphalgorithm, suited for GPUs.

The algorithm provides similar quality random matching withspeedups up to 6.8 for large graphs.

The algorithm provides better quality than local weighted matchingswith speedups up to 5.6.

Compared to a global greedy weighted matching algorithm quality isworse, but speedups up to 37 are achieved.

We look forward to employ this algorithm in (hyper)graph coarsening.

Conclusion

Questions

∃ any questions?

Choosing p

We should maximise the relative number of matched vertices eachround.

The number of matched vertices equals twice the number of redvertices that receive at least one proposal: maximise 2N

|V | , where

N := number of red vertices receiving at least one proposal.

For a random graph with n vertices, we can approximate(independent of edge density)

limn→∞

2E (N(n))

n≈ 2 (1− p)

(1− e−

p1−p

). (3)

Choosing p

|V | , where

limn→∞

2E (N(n))

n≈ 2 (1− p)

(1− e−

p1−p

). (3)

Choosing p

|V | , where

limn→∞

2E (N(n))

n≈ 2 (1− p)

(1− e−

p1−p

). (3)

Choosing p

Let G = ({1, . . . , n},E ) with P({v ,w} ∈ E ) = d for d ∈]0, 1]. ThenE (N(n)) is given by

∑v∈V

P(π(v) = red)P(v is proposed to | π(v) = red)

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

(1− P(w proposes to v | π(v) = red))

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

(1− P(π(w) = blue)P({v ,w} ∈ E )

nr. of red neighb. of w

)≈ n (1− p)

(1− p d

1 + (1− p) (d (n − 1)− 1)

)n−1).

Choosing p

Let G = ({1, . . . , n},E ) with P({v ,w} ∈ E ) = d for d ∈]0, 1]. ThenE (N(n)) is given by∑v∈V

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

(1− P(π(w) = blue)P({v ,w} ∈ E )

)≈ n (1− p)

(1− p d

1 + (1− p) (d (n − 1)− 1)

)n−1).

Choosing p

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

(1− P(π(w) = blue)P({v ,w} ∈ E )

)≈ n (1− p)

(1− p d

1 + (1− p) (d (n − 1)− 1)

)n−1).

Choosing p

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

(1− P(π(w) = blue)P({v ,w} ∈ E )

≈ n (1− p)

(1− p d

1 + (1− p) (d (n − 1)− 1)

)n−1).

Choosing p

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

=∑v∈V

P(π(v) = red)

1−∏

w∈V\{v}

(1− P(π(w) = blue)P({v ,w} ∈ E )

)≈ n (1− p)

(1− p d

1 + (1− p) (d (n − 1)− 1)

)n−1).

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