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Extensions of MapReduce. Dataflow Systems Extensions for Graphs Recursion. Jeffrey D. Ullman Stanford University. Dataflow Systems. Arbitrary Acyclic Flow Among Tasks Preserving Fault Tolerance The Blocking Property. Generalization of MapReduce. - PowerPoint PPT Presentation
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Extensions of MapReduceDataflow SystemsExtensions for GraphsRecursion
Jeffrey D. UllmanStanford University
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Dataflow Systems
Arbitrary Acyclic Flow Among TasksPreserving Fault ToleranceThe Blocking Property
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Generalization of MapReduce MapReduce uses only two functions (Map and
Reduce). Each is implemented by a rank of tasks. Data flows from Map tasks to Reduce tasks only.
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Generalization – (2) Natural generalization is to allow any number
of functions, connected in an acyclic network. Each function implemented by tasks that feed
tasks of successor function(s). Key fault-tolerance (blocking) property: tasks
produce all their output at the end. Important point: Map tasks never deliver their
output until completed. Thus, we can restart a Map task that failed without
fear that a Reduce task has already used some output of the failed Map task.
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Many Implementations
1. Clustera – University of Wisconsin.2. Hyracks – Univ. of California/Irvine.3. Dryad/DryadLINQ – Microsoft.4. Nephele/PACT – T. U. Berlin.5. BOOM – Berkeley.6. epiC – N. U. Singapore.
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Example: Join + Aggregation Relations D(emp, dept) and S(emp, salary). Compute the sum of the salaries for each
department. D JOIN S computed by MapReduce.
But each Reduce task can also group its emp-dept-salary tuples by dept and sum the salaries.
A Third function is needed to take the dept-SUM(salary) pairs from each Reduce task, organize them by dept, and compute the final sum for each department.
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3-Layer Dataflow
MapTasks
D
S
Join +GroupTasks
Hash byemp
FinalGroup +Aggre-gate
Hash bydept
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Recursion
Transitive-Closure ExampleFault-Tolerance ProblemEndgame ProblemSome Systems and Approaches
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Applications Requiring Recursion
1. PageRank, the original map-reduce application is really a recursion implemented by many rounds of map-reduce.
2. Analysis of social networks.3. Many machine-learning algorithms, e.g.,
gradient descent.4. PDE’s.
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Transitive Closure
Many recursive applications involving large data are similar to transitive closure :
Path(X,Y) :- Arc(X,Y)Path(X,Y) :- Path(X,Z) & Path(Z,Y)
Path(X,Y) :- Arc(X,Y)Path(X,Y) :- Arc(X,Z) & Path(Z,Y)
Nonlinear. Takeslog n rounds on ann-node graph.
(Right) Linear. Takesn rounds on an n-nodegraph.
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Implementing TC on a Cluster Use k tasks. Nonlinear recursion used here. Hash function h sends each node of the graph to
one of the k tasks. Task i receives and stores Path(a,b) if either h(a)
= i or h(b) = i, or both. Task i must join Path(a,c) with Path(c,b) if h(c) = i.
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TC on a Cluster – Basis
Data is stored as relation Arc(a,b). “Map” tasks read chunks of the Arc relation and
send each tuple Arc(a,b) to recursive tasks h(a) and h(b). Treated as if it were tuple Path(a,b). If h(a) = h(b), only one task receives.
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TC on a Cluster – Recursive Tasks
Task iPath(a,b)received
StorePath(a,b)if new.Otherwise,ignore.
Look upPath(b,c) and/orPath(d,a) forany c and d
Send Path(a,c) totasks h(a) and h(c);send Path(d,b) totasks h(d) and h(b)
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Big Problem: Managing Failure MapReduce depends on the blocking property. Only then can you restart a failed task without
restarting the whole job. But any recursive task has to deliver some
output and later get more input.
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HaLoop (U. Washington)
Iterates Hadoop, once for each round of the recursion. Uses Hadoop blocking-based fault tolerance.
Similar idea: Twister (U. Indiana). HaLoop tries to run each task in round i at a
compute node where it can find its needed output from round i – 1.
Also partitions and stores locally a file that is used at each round. Example: Arc in Path(X,Y) :- Arc(X,Z) & Path(Z,Y)
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Pregel (Google) Views all computation as a recursion on some
graph. Nodes send messages to one another.
Messages bunched into supersteps, where each node processes all data received.
Sending individual messages would result in far too much overhead.
Checkpoint all compute nodes after some fixed number of supersteps.
On failure, rolls all tasks back to previous checkpoint.
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Example: Shortest Paths Via Pregel
Node N
I found a pathfrom node M toyou of length L
5 3 6
I found a pathfrom node M toyou of length L+3
I found a pathfrom node M toyou of length L+5
I found a pathfrom node M toyou of length L+6
Is this theshortest path fromM I know about?
If so …
table ofshortestpathsto N
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Other Graph-Oriented Systems Giraph: open-source Pregel. GraphLab: similar system that deals more
effectively with nodes of high degree. Will split the work for such a graph node among
several compute nodes.
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Using Idempotence
Some recursive applications allow restart of tasks even if they have produced some output.
Example: TC is idempotent; you can send a task a duplicate Path fact without altering the result. But if you were counting paths, the answer would
be wrong.
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Big Problem: The Endgame Some recursions, like TC, take a large number
of rounds, but the number of new discoveries in later rounds drops. T. Vassilakis: searches forward on the Web graph
can take hundreds of rounds. Problem: in a cluster, transmitting small files
carries much overhead.
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Approach: Merge Tasks
Decide when to migrate tasks to fewer compute nodes.
Data for several tasks at the same node are combined into a single file and distributed at the receiving end.
Downside: old tasks have a lot of state to move.
Example: “paths seen so far.”
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Approach: Modify Algorithms Nonlinear recursions can terminate in many
fewer steps than equivalent linear recursions. Avoids the endgame problem.
Example: TC. O(n) rounds on n-node graph for linear. O(log n) rounds for nonlinear.
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Advantage of Linear TC
The communication cost (= sum of input sizes of all tasks) for executing linear TC is generally lower than that for nonlinear TC.
Why? Each path is discovered only once (unique-decomposition property). Note: distinct paths between the same endpoints
may each be discovered.
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Example: Linear TC Arc + Path = Path
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Nonlinear TC Constructs Path + Path = Path in Many Ways
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Smart TC
(Valduriez-Boral, Ioannides) Construct a path from two paths:
1. The first has a length that is a power of 2.2. The second is no longer than the first.
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Example: Smart TC
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Other Nonlinear TC Algorithms You can have the unique-decomposition
property with many variants of nonlinear TC. Example: Balance constructs paths from two
equal-length paths. Favor first path when length is odd.
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Example: Balance
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Incomparability of TC Algorithms
On different graphs, any of the unique-decomposition algorithms – left-linear, right-linear, smart, balanced – could have the lowest data-volume cost.
Other unique-decomposition algorithms are possible and also could win.
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Extension Beyond TC
Can you avoid the endgame problem by converting any linear recursion into an equivalent nonlinear recursion that requires logarithmic rounds?
Answer: Not always, without increasing arity and data size.
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Positive Points
1. (Agarwal, Jagadish, Ness) All linear Datalog recursions reduce to TC.
2. Right-linear chain-rule Datalog programs can be replaced by nonlinear recursions with the same arity, logarithmic rounds, and the unique-decomposition property.
Each subgoal shares variablesonly with the next, in a circularsense that includes the head.
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Example: Alternating-Color Paths
P(X,Y) :- Blue(X,Y)P(X,Y) :- Blue(X,Z) & Q(Z,Y)Q(X,Y) :- Red(X,Z) & P(Z,Y)
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The Case of Reachability
Reach(X) :- Source(X)Reach(X) :- Reach(Y) & Arc(Y,X)
Takes linear rounds as stated. Can compute nonlinear TC to get Reach in
O(log n) rounds. But, then you compute O(n2) facts instead of
O(n) facts on an n-node graph.
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Reachability – (2)
Theorem: If you compute Reach using only unary recursive predicates, then it must take (n) rounds on a graph of n nodes. Proof uses the ideas of Afrati, Cosmodakis, and
Yannakakis from a generation ago.
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Summary: Recursion
Key problems are “endgame” and nonblocking nature of recursive tasks.
In some applications, endgame problem can be handled by using a nonlinear recursion that requires O(log n) rounds and has the unique-decomposition property.
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Summary: Research Questions
1. How do you best support fault tolerance when tasks are nonblocking?
2. How do you manage tasks when the endgame problem cannot be avoided?
3. When can you replace linear recursion with nonlinear recursion requiring many fewer rounds, (roughly) the same communication cost, and (roughly) the same number of facts discovered?