6/28/2016
Presenter: Yunfei Shen
Luyao Li
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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Introduction
Plans (1,2 & 3)
Data Analysis
Demo
6/28/2016
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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MapReduce is efficient to parse small files,
but massive scientific raw data…
MapReduce is not optimized for matrix-form
data and linear algebra kernels
6/28/2016
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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Consider a very simple matrix multiplication
problem, A x B=C. Possible plans are [1]:
 1. Simple dot product
Ci,j= Ai,k * Bk,j
 2. Divide A into s t submatrices (blocks) and B
into t u ones.
Cblock=Ablock*Bblock
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[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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Sub-matrix
 SequenceFileWrite ->key:location (x, y)
Mapper:
Input Key1 (index1, index2) , IntWritable (v)
A{(s,t), As,t}
B{(t,u), Bt,u}
Need specially
consider lastBlock in
row or column,
maybe unbalanced:
lastBlockIndex,
lastBlockSize
Output Key2(index1, index2, index3) , Value (index1, index2, v)
{(S,T,U), <(s,t,As,t), (t,u,Bt,u)>}
Reducer:
Input Key2(index1, index2, index3) , Value (index1, index2, v)
1 Cs,t,u=As,t*Bt,u Multiplication
2 Cs,u=∑ Cs,t,u Sum
Output Key1(index1, index2) , Intwritable (v)
C{(s,t), Cs,t}
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Sub-matrix - map
A= 4*4, sub=2*2
0
B=4*4, sub=2*2
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Key1(x,y)
A
B
( 0, 0) ( 0, 2)
( 0, 1) ( 0, 3 )
( 1, 0) ( 1, 2 )
( 1, 1) ( 1, 3 )
IntWritable(v)
A
B
1
5
2 6
3
7
4 8
Key2(S,T,U, m)
( 0, 0, 1, 0)
( 0, 0, 1, 1)
Value(x,y,v)
( 0, 0, 1), (0,1, 2), (1,0,3 ), (1,1, 4 )
( 0, 0, 5), (0,1, 6), ( 1,0,7 ), (1,1, 8 )
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Sub-matrix - reduce
Key2(S,T,U, m)
( 0, 0, 1, 0)
( 0, 0, 1, 1)
Value(x,y,v)
( 0, 0, 1), (0,1, 2), (1,0,3 ), (1,1, 4 )
( 0, 0, 5), (0,1, 6), ( 1,0,7 ), (1,1, 8 )
Key1 (x, y)
(0, 3)
(0, 4)
(1, 3)
(1, 4)
Value (x,y, v)
(0, 0, 19)
(0, 1, 21)
(1, 1, 43)
(1, 2, 50)
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C=4*4
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The simplest strategy is to have each reducer do just one of
the block multiplications.
0, 0, 1
Key2(S, T, U, m)
(0, 0, 1, 0)
(0, 0, 1, 1)
Value(x,y,v)
(0, 0, 1)
(0, 0, 2)
Job1:
Reducer
Job 1
Do multiplication in
Job1 reducer:
Key1(x,y) IntWritable(v)
(0, 2)
2
Job 2
Do sum in Job2 reducer:
Key1(x,y) IntWritable(v)
(0, 2)
{…,2,…}
Too many Reducers:
numS*numT*numU,
heavy network traffic!!!
6/28/2016
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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In this plan, we use a single reducer to multiply a single A
block times a whole row of B blocks. Also two jobs.
Job1
A block (S, T)
e.g. ( 0, 0)
0, 0
Greatly decreasing the
number of Reducers:
numS*numT maximum
reducers are needed for this
0 ,1 plan!
1,0
1, 1
Do multiplication Key2(S, T, U)
Value(x,y,v)
A (0, 0, -1)
(0, 0, 1)
B (0, 0, 1)
(0, 0, 2)
B block (T, K1,2…)
e.g. ( 0, 0) (0, 1)
Job 2: do sum.
6/28/2016
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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In Plan3, we use a single reducer to compute the final C
block, and there's no need for a second MapReduce job.
 A blocks and B blocks are arranged like following order after Mapper:
for 0 <= T < numT, A[S,0] B[0,U] A[S,1] B[1,U] ... A[S,numT-1] B[numT, U]
Reducer number is: numS*numU
Only one job needed!
A 0 row
0, 0
0 ,1
1,0
1, 1
A(0,0)
B(0,1)
A(0,1)
B(1,1)
B 0 column
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Key2(S, U, T)
Value(x,y,v)
A (0, 1,0)
(0, 0, 1)
B (0, 1,0)
(0, 0, 2)
Multiply and sum in one job
if (ms != s || mu != u) write
reducer
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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Plan
Job
Reducer Time (s)
1
4500
2
2
280
60
2
35
58
4000
3
3500
s t u Plan
3000
103 112 119 1
2500
2
2000
3
1500
1
40
32
s t u Plan
1000
504 523 547 1
500
2 0
Job
Reducer Time
2
125
2
25
1 10K
25
s t u
50 68 72
3
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300
Job
200
Reducer
100
0
Job
Reducer Time
2
196
120
2
36
109
1
36
65
4091
3692
25k
3216
200
1
2
Time
3
Job
1
100
0
1
2
2
Reducer
3
3
Time
150
Job
100
Reducer
50
500k
0
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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2
3
Time/100
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A=
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C=A x B=
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Use 2x2 sub-matrix, run with strategy 1, 2, 3
respectively
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, B=
[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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[1] Yi Zhang, Parallel Matrix Layout for MapReduce
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