Using Cascalog and Hadoop for rapid graph processing and exploration
Recommendation and graph algorithms in Hadoop and SQL
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Transcript of Recommendation and graph algorithms in Hadoop and SQL
Recommendation and graph algorithms in Hadoop and SQL
DAVID F. GLEICH ASSISTANT PROFESSOR "COMPUTER SCIENCE "PURDUE UNIVERSITY
David Gleich · Purdue 1
Code github.com/dgleich/matrix-hadoop-tutorial
Ancestry.com
@dgleich [email protected]
Matrix computations
A =
2
66664
A1,1 A1,2 · · · A1,n
A2,1 A2,2 · · ·...
.... . .
. . . Am�1,nAm,1 · · · Am,n�1 Am,n
3
77775
Least squares Eigenvalues
Ax Ax = b min kAx � bk Ax = �x
Operations Linear "systems David Gleich · Purdue 2 Ancestry.com
Outcomes Recognize relationships between matrix methods and things you’ve already been doing" Example SQL queries as matrix computations See how to work with big graphs as large edge lists in Hadoop and SQL" Example Connected components Understand how to use Hadoop to compute these matrix methods at scale for BigData" Example Recommenders with social network info
David Gleich · Purdue 3 Ancestry.com
matrix computations "≠"
linear algebra
David Gleich · Purdue 4 Ancestry.com
World’s simplest recommendation system.
Suggest the average rating.
David Gleich · Purdue 5 Ancestry.com
A SQL statement as a "matrix computation
http://stackoverflow.com/questions/4217449/returning-average-rating-from-a-database-sql
How do I find the average rating for each product?
David Gleich · Purdue 6 Ancestry.com
A SQL statement as a "matrix computation
http://stackoverflow.com/questions/4217449/returning-average-rating-from-a-database-sql
SELECT ! p.product_id, ! p.name, ! AVG(pr.rating) AS rating_average!FROM products p !INNER JOIN product_ratings pr!ON pr.product_id = p.product_id!GROUP BY p.product_id!ORDER BY rating_average DESC !
How do I find the average rating for each product?
David Gleich · Purdue 7 Ancestry.com
This SQL statement is a "matrix computation!
8 Image from rockysprings, deviantart, CC share-alike Ancestry.com David Gleich · Purdue
SELECT ! ... ! AVG(pr.rating) !... !GROUP BY p.product_id!
product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4 pid9 uid9 1
Is a matrix!
pid1 pid2 pid3 pid4 pid5 pid6 pid7 pid8 pid9
David Gleich · Purdue 9 Ancestry.com
product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4 pid9 uid9 1
Is a matrix!
pid1 pid2 pid3 pid4 pid5 pid6 pid7 pid8 pid9
But it’s a weird matrix"
Missing entries!
David Gleich · Purdue 10
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product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4
Is a matrix!
pid1 pid2 pid3 pid4 pid5 pid6 pid7 pid8 pid9
4
4
4
4 5 4
But it’s a weird matrix"
Matrix
SELECT AVG(r) ... GROUP BY pid
Vector
Average"of ratings
David Gleich · Purdue 11
Ancestry.com
But it’s a weird matrix"and not a linear operator
A =
2
66664
A1,1 A1,2 · · · A1,n
A2,1 A2,2 · · ·...
.... . .
. . . Am�1,nAm,1 · · · Am,n�1 Am,n
3
77775
avg(A) =
2
6664
Pj A1,j/
Pj “A1,j 6= 0”P
j A2,j/P
j “A2,j 6= 0”...P
j Am,j/P
j “Am,j 6= 0”
3
7775
David Gleich · Purdue 12
product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4 pid9 uid9 1
Is a matrix!
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matrix computations "≠"
linear algebra
David Gleich · Purdue 13
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Hadoop, MapReduce, and Matrix Methods
David Gleich · Purdue 14
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MapReduce
David Gleich · Purdue 15
dataMap
dataMap
dataMap
dataMap
keyvalue
keyvalue
keyvalue
keyvalue
keyvalue
keyvalue
()
Shuffle
keyvaluevalue
dataReduce
keyvaluevaluevalue
dataReduce
keyvalue dataReduce
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The MapReduce Framework Originated at Google for indexing web pages and computing PageRank.
Express algorithms in "“data-local operations”. Implement one type of communication: shuffle. Shuffle moves all data with the same key to the same reducer.
MM R
RMM
Input stored in triplicate
Map output"persisted to disk"before shuffle
Reduce input/"output on disk
1 MM R
RMMM
Maps Reduce
Shuffle
2
3
4
5
1 2 M M
3 4 M M
5 M
Data scalable
Fault-tolerance by design
16
David Gleich · Purdue Ancestry.com
wordcount "is a matrix computation too
map(document) :
for word in document
emit (word, 1)
reduce(word, counts) :
emit (word, sum(counts))
1 2 D D
3 4 D D
5 D
matrix,1 matrix,1 matrix,1 matrix,1
bigdata,1 bigdata,1 bigdata,1 bigdata,1 bigdata,1 bigdata,1 bigdata,1 bigdata,1
hadoop,1 hadoop,1 hadoop,1 hadoop,1 hadoop,1 hadoop,1 hadoop,1
David Gleich · Purdue 17
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wordcount "is a matrix computation too
A =
2
66664
A1,1 A1,2 · · · A1,n
A2,1 A2,2 · · ·...
.... . .
. . . Am�1,nAm,1 · · · Am,n�1 Am,n
3
77775
doc1
doc2
docm
= A
colsum(A) = AT e word count = e is the vector of all ones
David Gleich · Purdue 18
Ancestry.com
inverted index"is a matrix computation too
A =
2
66664
A1,1 A1,2 · · · A1,n
A2,1 A2,2 · · ·...
.... . .
. . . Am�1,nAm,1 · · · Am,n�1 Am,n
3
77775
doc1
doc2
docm
= A
David Gleich · Purdue 19
Ancestry.com
2
66664
A1,1 A2,1 · · · Am,1
A1,2 A2,2 · · ·...
.... . .
. . . Am,n�1A1,n · · · Am�1,n Am,n
3
77775= AT
term1
term2
termm
inverted index"is a matrix computation too
David Gleich · Purdue 20
Ancestry.com
A recommender system "with social info
David Gleich · Purdue 21
product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4 pid9 uid9 1
friends_links
uid6 uid1 uid8 uid9 uid7 uid7 uid7 uid4 uid6 uid2 uid7 uid1 uid3 uid1 uid1 uid8 uid7 uid3 uid9 uid1
Ancestry.com
A recommender system "with social info
David Gleich · Purdue 22
product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4 pid9 uid9 1
friends_links
uid6 uid1 uid8 uid9 uid7 uid7 uid7 uid4 uid6 uid2 uid7 uid1 uid3 uid1 uid1 uid8 uid7 uid3 uid9 uid1
pid1
pid2
2
64A1,1 A2,1 · · ·A1,2 A2,2 · · ·...
. . .. . .
3
75uid1
uid2
2
64A1,1 A2,1 · · ·A1,2 A2,2 · · ·...
. . .. . .
3
75
Ancestry.com
A recommender system "with social info
David Gleich · Purdue 23
product_ratings
pid8 uid2 4 pid9 uid9 1 pid2 uid9 5 pid9 uid5 5 pid6 uid8 4 pid1 uid2 4 pid3 uid4 4 pid5 uid9 2 pid9 uid8 4 pid9 uid9 1
friends_links
uid6 uid1 uid8 uid9 uid7 uid7 uid7 uid4 uid6 uid2 uid7 uid1 uid3 uid1 uid1 uid8 uid7 uid3 uid9 uid1
R S
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A recommender system "with social info
David Gleich · Purdue 24
Recommend each item based on the average rating of all trusted users
“X = S RT” with something that is"almost a matrix-matrix"product
R pid1
pid2
2
64A1,1 A2,1 · · ·A1,2 A2,2 · · ·...
. . .. . .
3
75 S uid1
uid2
2
64A1,1 A2,1 · · ·A1,2 A2,2 · · ·...
. . .. . .
3
75
Xuid,pid =
X
uid2
Suid,uid2Ruid2,pid
!· X
uid2
“Suid,uid2 and Ruid2,pid 6= 0”
!�1
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Tools I like
hadoop streaming dumbo mrjob hadoopy C++
David Gleich · Purdue 25
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Tools I don’t use but other people seem to like …
pig java hbase mahout Eclipse Cassandra
David Gleich · Purdue 26
Mahout is the closest thing to a library for matrix computations in Hadoop. If you like Java, you should probably start there. I’m a low-level guy
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hadoop streaming
the map function is a program"(key,value) pairs are sent via stdin"output (key,value) pairs goes to stdout the reduce function is a program"(key,value) pairs are sent via stdin"keys are grouped"output (key,value) pairs goes to stdout
David Gleich · Purdue 27
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mrjob from
a wrapper around hadoop streaming for map and reduce functions in python
class MRWordFreqCount(MRJob): def mapper(self, _, line): for word in line.split(): yield (word.lower(), 1) def reducer(self, word, counts): yield (word, sum(counts)) if __name__ == '__main__': MRWordFreqCount.run()
David Gleich · Purdue 28
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Connected components in SQL and Hadoop
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Connected components
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3 “components” in this graph How can we find them algorithmically … … on a huge network?
Connected components
Ancestry.com David Gleich · Purdue 31
Algorithm!Assign each node a random component id. For each node, take the minimum component id of itself and all neighbors.
DEMO
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Computing Connected Components in SQL
Graph!Edges : id | head | tail !
!“Vector”!v : id | comp ! initialized to random ! component!
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!CREATE TABLE v2 AS ( !SELECT ! e.tail AS id, ! MIN(v.comp) as COMP !FROM edges e !INNER JOIN vector v !ON e.head = v.id!GROUP BY e.tail!); !!DROP TABLE v; !ALTER TABLE v2 ! RENAME TO v; !!... Repeat ... !!
Matrix-vector product and connected components in Hadoop
David Gleich · Purdue 34
Ax = y
y
i
=X
k
A
ik
x
k
A x
See example! ���matrix-hadoop/codes/smatvec.py!
Ancestry.com
Google’s PageRank Word count, average rating!
“AT
x = y”y
i
= min(xi
, mink
A
ki
x
k
)
Connected components
Matrix-vector product
David Gleich · Purdue 35
Ax = y
y
i
=X
k
A
ik
x
k
A x
A is stored by “node”
$ head samples/smat_5_5.txt !0 0 0.125 3 1.024 4 0.121 !1 0 0.597 !2 2 1.247 !3 4 -1.45 !4 2 0.061 !
v initially random !
$ head samples/vec_5.txt !0 0.241 !1 -0.98 !2 0.237 !3 -0.32 !4 0.080 !
Follow along! ���matrix-hadoop/codes/smatvec.py!
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Matrix-vector product"(in pictures)
David Gleich · Purdue 36
Ax = y
y
i
=X
k
A
ik
x
k
A x
A x
Input Map 1!Align on columns"
Reduce 1!Output Aik xk"keyed on row i
A
x Reduce 2!Output sum(Aik xk)"
y
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Matrix-vector product"(in pictures)
David Gleich · Purdue 37
Ax = y
y
i
=X
k
A
ik
x
k
A x
A x
Input Map 1!Align on columns"
def joinmap(self, key, line): ! vals = line.split() ! if len(vals) == 2: ! # the vector ! yield (vals[0], # row ! (float(vals[1]),)) # xi ! else: ! # the matrix ! row = vals[0] ! for i in xrange(1,len(vals),2): ! yield (vals[i], # column ! (row, # i,Aij! float(vals[i+1]))) !
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“Matrix-vector” for connected components
David Gleich · Purdue 38
A x
A x
Input Map 1!Align on columns"
def joinmap(self, key, line): ! vals = line.split() ! if len(vals) == 2: ! # the vector ! yield (vals[0], # row ! (float(vals[1]),)) # vi ! else: ! # the matrix ! row = vals[0] ! for i in xrange(1,len(vals),2): ! yield (row, # head ! (vals[i], # tail)) !
Ancestry.com
“AT
x = y”y
i
= min(xi
, mink
A
ki
x
k
)
Matrix-vector product"(in pictures)
David Gleich · Purdue 39
Ax = y
y
i
=X
k
A
ik
x
k
A x
A x
Input Map 1!Align on columns"
Reduce 1!Output Aik xk"keyed on row i
A
x def joinred(self, key, vals): ! vecval = 0. ! matvals = [] ! for val in vals: ! if len(val) == 1: ! vecval += val[0] ! else: ! matvals.append(val) ! for val in matvals: ! yield (val[0], val[1]*vecval) !
Note that you should use a secondary sort to avoid reading both in memory
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“Matrix-vector” for connected components
David Gleich · Purdue 40
A x
A x
Input Map 1!Align on columns"
Reduce 1!Output Aik xk"keyed on row i
A
x def joinred(self, key, vals): ! vecval = 0. ! matvals = [] ! for val in vals: ! if len(val) == 1: ! vecval += val[0] ! else: ! matvals.append(val) ! for val in matvals: ! yield (val[0], vecval) !
Note that you should use a secondary sort to avoid reading both in memory
Ancestry.com
“AT
x = y”y
i
= min(xi
, mink
A
ki
x
k
)
Matrix-vector product"(in pictures)
David Gleich · Purdue 41
Ax = y
y
i
=X
k
A
ik
x
k
A x
A x
Input Map 1!Align on columns"
Reduce 1!Output Aik xk"keyed on row i
A
x Reduce 2!Output sum(Aik xk)"
y
def sumred(self, key, vals): ! yield (key, sum(vals)) !
Ancestry.com
Our social recommender
David Gleich · Purdue 42
RT S
Follow along! ���matrix-hadoop/recsys/recsys.py!
R is stored entry-wise !
$ gunzip –c data/rating.txt.gz!139431556 591156 5 !139431556 1312460676 5 !139431556 204358 4 139431556 368725 5 !Object ID! User ID! Rating!
S is stored entry-wise !
$ gunzip –c data/rating.txt.gz!3287060356 232085 -1 !3288305540 709420 1 !3290337156 204418 -1 !3294138244 269243 -1 !Other ID! Trust!My ID!
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Matrix-matrix product
David Gleich · Purdue 43
A B
Follow along! ���matrix-hadoop/codes/matmat.py!
AB = CCij =
X
k
Aik Bkj
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Conceptually, the first step is the same as the matrix-vector product with a block of vectors.
Matrix-matrix product "(in pictures)
David Gleich · Purdue 44
A B
AB = CCij =
X
k
Aik Bkj
A Map 1!Align on columns"
B Reduce 1!Output Aik Bkj"keyed on (i,j)
A
B Reduce 2!Output sum(Aik Bkj)"
C
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Social recommender "(in code)
David Gleich · Purdue 45
A B
A Map 1!Align on columns"
B
def joinmap(self, key, line): ! parts = line.split('\t') ! if len(parts) == 8: # ratings ! objid = parts[0].strip() ! uid = parts[1].strip() ! rat = int(parts[2]) ! yield (uid, (objid, rat)) ! else len(parts) == 4: # trust ! myid = parts[0].strip() ! otherid = parts[1].strip() ! value = int(parts[2]) ! if value > 0: ! yield (otherid, (myid,)) !
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Matrix-matrix product "(in pictures)
David Gleich · Purdue 46
A B
A Map 1!Align on columns"
B Reduce 1!Output Aik Bkj"keyed on (i,j)
A
B
def joinred(self, key, vals): ! tusers = [] # uids that trust key ! ratobjs = [] # objs rated by uid=key ! for val in vals: ! if len(val) == 1: ! tusers.append(val[0]) ! else: ! ratobjs.append(val) !! for (objid, rat) in ratobjs: ! for uid in tusers: ! yield ((uid, objid), rat) !
Conceptually, the second step
is the same as the matrix-
matrix product too, we “map”
the ratings from each trusted
user back to the source.
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Matrix-matrix product "(in pictures)
David Gleich · Purdue 47
A B
AB = CCij =
X
k
Aik Bkj
A Map 1!Align on columns"
B Reduce 1!Output Aik Bkj"keyed on (i,j)
A
B Reduce 2!Output sum(Aik Bkj)"
C def avgred(self, key, vals): ! s = 0. ! n = 0 ! for val in vals: ! s += val! n += 1 ! # the smoothed average of ratings ! yield key, ! (s+self.options.avg)/float(n+1) ! !
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Better ways to store "matrices in Hadoop
David Gleich · Purdue 48
A B
A B
Block matrices minimize the number of intermediate keys and values used. I’d form them based on the first reduce No need for “integer” keys that
fall between 1 and n!
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49
A
From tinyimages"collection
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Tall-and-Skinny matrices (m ≫ n) Many rows (like a billion) A few columns (under 10,000)
regression and general linear models"with many samples
block iterative methods panel factorizations
simulation data analysis !
big-data SVD/PCA!
Used in
David Gleich · Purdue
Questions?
50
Image from rockysprings, deviantart, CC share-alike Ancestry.com David Gleich · Purdue