streaming graph partitioning for large distributed...

Streaming graph partitioning for large distributed graphs

Daniel Spanier

June 5, 2015

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Overview

1 IntroductionMotivationBalanced graph partitioning

2 HeuristicsCompetitorsStream orderingDatasets

3 FENNEL

4 Conclusion

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Motivation

Lets take a look at Pregel:

Pregel has to load the vertices

and distribute them over its machines

then graph processing can start

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Motivation

Pregel uses a hash function h : V ! {1, . . . k}

Now each machine has the same amount of vertices.Daniel Spanier Streaming graph partitioning June 5, 2015 4 / 40

Motivation

doesn’t minimize number of edges cut

higher communication cost between machines afterwards

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Motivation

doesn’t minimize number of edges cut

higher communication cost between machines afterwards

But: Pregel supports customized graph partitioning ! use balanced graphpartitioning

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Balanced k-graph partitioning

Let k be the number of partitions we want to have.The goal is to minimize the cross partition edges and keeping the numberof nodes in each partition even.

But: This is NP-hard, so in practise, only approximations are feasible.

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Graph partitioning variants

We have two di↵erent approaches:

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Graph partitioning

- o✏ine: know all vertices beforehand, then decide their places

- streaming: load the graph vertex after vertex and decide vertex placeimmediately

O✏ine algorithms are better at minimizing the amount of cut edges,streaming algorithm can be used during the same time pregel loads thegraph into main memory.

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O✏ine graph partitioning

METIS is widely used

combines various graph partition algorithms

needs access to all vertices at once, so it cannot be used during theloading phase of Pregel

We will compare the results of the streaming graph partitioningalgorithms with those of METIS

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Online graph partitioning

Let k be the number of partitions we want, P = (P1

, . . .Pk

) the partitionsand G = (V ,E ) the graph with |V | = n, |E | = m

Heuristics decide, in which partition Pi

a vertex v 2 V is put.We will see two kinds of heuristics: with a bu↵er and without one

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Heuristics

Hashing

We use a fairly simple hash-function H: V ! {1, . . . k} :H(v) = (v mod k) + 1

Hashing is mainly used in practise, because every machine knows where anode has been distributed to without the need of a mapping table.

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Heuristics

Chunking

Let C be the capacity of a Partition.Then: divide stream into chunks of size C and fill partitions in orderindex(v) = (v div C ) + 1

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Heuristics

Deterministic Greedy

Assign v to the partition where it has the most edges.

Let t be the time v arrives, N(v) be the set of vertices v neighbours andw a weight function penalizing partitions with too many vertices.index(v) = argmax

i2[k]{|P

i

\ N(v)| ⇤ w(i , t)}

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Heuristics

Deterministic Greedy

Assign v to the partition where it has the most edges.Let t be the time v arrives, N(v) be the set of vertices v neighbours andw a weight function penalizing partitions with too many vertices.index(v) = argmax

i2[k]{|P

i

\ N(v)| ⇤ w(i , t)}

w(i , t) = 1 for Unweighted Deterministic Greedy

w(i , t) = 1� |Pi

|C

for Linear Deterministic Greedy

w(i , t) = 1� exp{|Pi

|� C} for Exponentially Deterministic Greedy

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Heuristics

This heuristic uses a Bu↵er of size C and assumes a way to di↵erentiatebetweeen high-degree and low-degree nodes

Avoid Big

We maintain the bu↵er and a threshold on large nodes.If the bu↵er is filled or the threshold is reached we greedily assign alllow-degree nodes in the bu↵er.If only high-degree nodes remain, assign them with deterministic greedy.

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Stream ordering

Stream ordering means the order in which the vertices arrive at thestreaming algorithm.Depending on the ordering, di↵erent heuristics achieve better or worseresults ! we have to specify the ordering

Random: Standard Ordering in literature

Breadth-First-Search ordering

Depth-First-Search ordering

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Datasets

Marvel Comics social network, which resembles real social networks|V | = 6, 486, |E | = 427, 018

PL, synthetic graphs generated by a power-law graph generator withclusteringexample P1000:|V | = 1000, |E | = 9, 878

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Evaluation

Marvel, k = 8

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Evaluation

PL1000, k = 4

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Evaluation

Name Random BFS DFSAvoid Big -46.4 -27.3 -38.6Chunking 0.7 37.6 35.7Deterministic Greedy 45.4 57.7 54.7Exp. Det. Greedy 47.5 59.4 56.2Hashing -1.7 -1.9 -2.1Linear Det. Greedy 75.3 76 73

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Results

Linear Deterministic Greedy had the highest average gain, no matterwhat streaming ordering was used

Nearly every heuristic is an improvement

Except Avoid Big with negative improvement; this heuristic shouldn’tbe used

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Results

Why is Linear Deterministic Greedy better than the others?

Deterministic Greed

index(v) = argmaxi2[k]

{|Pi

\ N(v)| ⇤ w(i , t)}

w(i , t) = 1 for Unweighted Det. Greedy

w(i , t) = 1� |Pi

|C

for Linear Det. Greedy

w(i , t) = 1� exp{|Pi

|� C} for Exponentially Det. Greedy

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Results

Why is Linear Deterministic Greedy better than the others?

Unweighted Det. Greedy only indicates a already full partition

Exponential Det. Greedy indicates a full partition only nearly at thelimit

Linear Det. Greedy prefers less loaded partitions

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FENNEL

FENNEL combines two widely used families of heuristics:

place newly arrived vertices in Partition with the largest number ofneighbours

place newly arrived vertices in Partition with the least non neighbours

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FENNEL

Let P = (P1

, ...Pk

) be all k > 0 partitions, Pi

Partition i .Then we define a global objective function:

f (P) = cout

(P) + cin

(P)

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FENNEL

cout

(P) = #edges cut

is the inter-partition cost.But what to do with c

in

(P)?

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FENNEL

But what to do with cin

(P)?

Focus on individual partitions: cin

(P) =P

k

i=1

cin

(Pi

)

Size of Pi

should be approximately n

k

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FENNEL

We take a look at function family c(x) = ↵ ⇤ x�

� lets us control the importance of imbalance in the partition size

� = 1 means ignoring imbalance of partition sizes

� = 2 would diminish the importance of cout

(P)

Most of the time, the authors stuck to � = 1.5

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FENNEL

We take a look at function family c(x) = ↵ ⇤ x�

↵ = m ⇤ k

��1

n

� , with |E | = mwith that we get:

f (P) = cout

(P) +P

k

i=1

cin

(Pi

)

= #edges�cut

m

+ 1

k

Pk

i=1

( |Pi

|n/k )

�

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FENNEL

The graph partition problem with this objective function becomes:

minP

(#edges�cut

m

+ 1

k

Pk

i=1

( |Pi

|n/k )

�)

But this isn’t doable in a streaming algorithm!

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FENNEL

Solution: We greedily decide, to which Pi

we assign an incoming vertex v

f ((P1

, . . .Pi

[ {v}, . . .Pk

)) f ((P1

, . . .Pj

[ {v}, . . .Pk

) , 8j 2 {1, . . . k}

To stop overloading, we can add a threshold t:If |P

i

| > t ⇤ n

k

, Pi

can’t be chosen for v

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FENNEL

Transform minimization problem into maximization:g((P

1

, . . .Pi

[ {v}, . . .Pk

)) � g((P1

, . . .Pj

[ {v}, . . .Pk

) , 8j 2 {1, . . . k}

This leads to:G (v ,P

i

) = |N(v) \ Pi

|� ↵|Pi

|��1

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FENNEL

G (v ,Pi

) = |N(v) \ Pi

|� ↵�|Pi

|��1

For � = 1 this is Deterministic Unweighted Greedy.For � = 2 = 1

↵ FENNEL would place vertices to the partitions with theleast number of non-neighbours

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FENNEL

Now we compare FENNEL with Linear Det. Greedy and METIS (k = 32):

Name BFS % edges cut BFS max partition loadLinear Det. Greedy 34 % 1.01FENNEL 14% 1.10METIS 8% 1.00

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FENNEL

Now we compare FENNEL with Linear Det. Greedy and METIS (k = 32):

Name RANDOM % edges cut RANDOM part. loadLinear Det. Greedy 40 % 1.00FENNEL 14% 1.02METIS 8% 1.02

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FENNEL

. . . and with METIS alone:

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FENNEL

Remember Pegel and others use a hash heuristic instead of FENNELduring the graph loading;now if we use FENNEL instead before a PageRank computation onLiveJournal:

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Conclusion

Despite a single pass, FENNEL can achieve results comparable toMETIS, but is much faster.

We have seen that using FENNEL actually improves graph operationslike PageRank.

This improvement stems entirely from the reduced networkcommunication.

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