MapReduceAlgorithms.ppt

Algorithms for MapReduce
 Sorting
 Searching
 TF-IDF
 BFS
 PageRank
 More advanced algorithms

MapReduce Jobs
 Tend to be very short, code-wise
IdentityReducer is very common
 “Utility” jobs can be composed
 Represent a data flow, more so than a
procedure

Sort: Inputs
 A set of files, one value per line.
 Mapper key is file name, line number
 Mapper value is the contents of the line

Sort Algorithm
 Takes advantage of reducer properties:
(key, value) pairs are processed in order
by key; reducers are themselves ordered
 Mapper: Identity function for value
(k, v)  (v, _)
 Reducer: Identity function (k’, _) -> (k’, “”)

Sort: The Trick
 (key, value) pairs from mappers are sent to a
particular reducer based on hash(key)
 Must pick the hash function for your data such
that k1 < k2 => hash(k1) < hash(k2)
M1 M2 M3
R1 R2
Partition
and
Shuffle

Final Thoughts on Sort
 Used as a test of Hadoop’s raw speed
 Essentially “IO drag race”
 Highlights utility of GFS

Search: Inputs
 A set of files containing lines of text
 A search pattern to find
 Mapper key is file name, line number
 Mapper value is the contents of the line
 Search pattern sent as special parameter

Search Algorithm
 Mapper:
Given (filename, some text) and “pattern”, if
“text” matches “pattern” output (filename, _)
 Reducer:
Identity function

Search: An Optimization
 Once a file is found to be interesting, we
only need to mark it that way once
 Use Combiner function to fold redundant
(filename, _) pairs into a single one
Reduces network I/O

TF-IDF
 Term Frequency – Inverse Document
Frequency
Relevant to text processing
Common web analysis algorithm

The Algorithm, Formally
•| D | : total number of documents in the corpus
• : number of documents where the term ti appears (that is ).

Information We Need
 Number of times term X appears in a
given document
 Number of terms in each document
 Number of documents X appears in
 Total number of documents

Job 1: Word Frequency in Doc
 Mapper
Input: (docname, contents)
Output: ((word, docname), 1)
 Reducer
Sums counts for word in document
Outputs ((word, docname), n)
 Combiner is same as Reducer

Job 2: Word Counts For Docs
 Mapper
Input: ((word, docname), n)
Output: (docname, (word, n))
 Reducer
Sums frequency of individual n’s in same doc
Feeds original data through
Outputs ((word, docname), (n, N))

Job 3: Word Frequency In Corpus
 Mapper
Input: ((word, docname), (n, N))
Output: (word, (docname, n, N, 1))
 Reducer
Sums counts for word in corpus
Outputs ((word, docname), (n, N, m))

Job 4: Calculate TF-IDF
 Mapper
Input: ((word, docname), (n, N, m))
Assume D is known (or, easy MR to find it)
Output ((word, docname), TF*IDF)
 Reducer
Just the identity function

Working At Scale
 Buffering (doc, n, N) counts while
summing 1’s into m may not fit in memory
How many documents does the word “the”
occur in?
 Possible solutions
Ignore very-high-frequency words
Write out intermediate data to a file
Use another MR pass

Final Thoughts on TF-IDF
 Several small jobs add up to full algorithm
 Lots of code reuse possible
Stock classes exist for aggregation, identity
 Jobs 3 and 4 can really be done at once in
same reducer, saving a write/read cycle
 Very easy to handle medium-large scale,
but must take care to ensure flat memory
usage for largest scale

BFS: Motivating Concepts
 Performing computation on a graph data
structure requires processing at each node
 Each node contains node-specific data as
well as links (edges) to other nodes
 Computation must traverse the graph and
perform the computation step
 How do we traverse a graph in
MapReduce? How do we represent the
graph for this?

Breadth-First Search
• Breadth-First
Search is an
iterated algorithm
over graphs
• Frontier advances
from origin by one
level with each pass
1
2
2 2
3
3
3
3
4
4

Breadth-First Search & MapReduce
 Problem: This doesn't “fit” into MapReduce
 Solution: Iterated passes through
MapReduce – map some nodes, result
includes additional nodes which are fed into
successive MapReduce passes

Breadth-First Search & MapReduce
 Problem: Sending the entire graph to a map
task (or hundreds/thousands of map tasks)
involves an enormous amount of memory
 Solution: Carefully consider how we
represent graphs

Graph Representations
• The most straightforward representation of
graphs uses references from each node to
its neighbors

Direct References
 Structure is inherent
to object
 Iteration requires
linked list “threaded
through” graph
 Requires common
view of shared
memory
(synchronization!)
 Not easily serializable
class GraphNode
{
Object data;
Vector<GraphNode>
out_edges;
GraphNode
iter_next;
}

Adjacency Matrices
 Another classic graph representation.
M[i][j]= '1' implies a link from node i to j.
 Naturally encapsulates iteration over nodes
0
1
0
1
4
0
0
1
0
3
1
1
0
1
2
1
0
1
0
1
4
3
2
1

Adjacency Matrices: Sparse
Representation
 Adjacency matrix for most large graphs
(e.g., the web) will be overwhelmingly full of
zeros.
 Each row of the graph is absurdly long
 Sparse matrices only include non-zero
elements

Sparse Matrix Representation
1: (3, 1), (18, 1), (200, 1)
2: (6, 1), (12, 1), (80, 1), (400, 1)
3: (1, 1), (14, 1)
…

Sparse Matrix Representation
1: 3, 18, 200
2: 6, 12, 80, 400
3: 1, 14
…

Finding the Shortest Path
• A common graph
search application is
finding the shortest
path from a start node
to one or more target
nodes
• Commonly done on a
single machine with
Dijkstra's Algorithm
• Can we use BFS to
find the shortest path
via MapReduce?
This is called the single-source shortest path problem. (a.k.a. SSSP)

Finding the Shortest Path: Intuition
 We can define the solution to this problem
inductively:
DistanceTo(startNode) = 0
For all nodes n directly reachable from
startNode, DistanceTo(n) = 1
For all nodes n reachable from some other set
of nodes S,
DistanceTo(n) = 1 + min(DistanceTo(m), m  S)

From Intuition to Algorithm
 A map task receives a node n as a key, and
(D, points-to) as its value
D is the distance to the node from the start
points-to is a list of nodes reachable from n
 p  points-to, emit (p, D+1)
 Reduce task gathers possible distances to
a given p and selects the minimum one

What This Gives Us
 This MapReduce task can advance the
known frontier by one hop
 To perform the whole BFS, a non-
MapReduce component then feeds the
output of this step back into the
MapReduce task for another iteration
Problem: Where'd the points-to list go?
Solution: Mapper emits (n, points-to) as well

Blow-up and Termination
 This algorithm starts from one node
 Subsequent iterations include many more
nodes of the graph as frontier advances
 Does this ever terminate?
Yes! Eventually, routes between nodes will stop
being discovered and no better distances will
be found. When distance is the same, we stop
Mapper should emit (n, D) to ensure that
“current distance” is carried into the reducer

Adding weights
 Weighted-edge shortest path is more useful
than cost==1 approach
 Simple change: points-to list in map task
includes a weight 'w' for each pointed-to
node
emit (p, D+wp) instead of (p, D+1) for each
node p
Works for positive-weighted graph

Comparison to Dijkstra
 Dijkstra's algorithm is more efficient
because at any step it only pursues edges
from the minimum-cost path inside the
frontier
 MapReduce version explores all paths in
parallel; not as efficient overall, but the
architecture is more scalable
 Equivalent to Dijkstra for weight=1 case

PageRank: Random Walks Over
The Web
 If a user starts at a random web page and
surfs by clicking links and randomly
entering new URLs, what is the probability
that s/he will arrive at a given page?
 The PageRank of a page captures this
notion
More “popular” or “worthwhile” pages get a
higher rank

PageRank: Visually
www.cnn.com
en.wikipedia.org
www.nytimes.com

PageRank: Formula
Given page A, and pages T1 through Tn
linking to A, PageRank is defined as:
PR(A) = (1-d) + d (PR(T1)/C(T1) + ... +
PR(Tn)/C(Tn))
C(P) is the cardinality (out-degree) of page P
d is the damping (“random URL”) factor

PageRank: Intuition
 Calculation is iterative: PRi+1 is based on PRi
 Each page distributes its PRi to all pages it
links to. Linkees add up their awarded rank
fragments to find their PRi+1
 d is a tunable parameter (usually = 0.85)
encapsulating the “random jump factor”
PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))

PageRank: First Implementation
 Create two tables 'current' and 'next' holding
the PageRank for each page. Seed 'current'
with initial PR values
 Iterate over all pages in the graph,
distributing PR from 'current' into 'next' of
linkees
 current := next; next := fresh_table();
 Go back to iteration step or end if converged

Distribution of the Algorithm
 Key insights allowing parallelization:
The 'next' table depends on 'current', but not on
any other rows of 'next'
Individual rows of the adjacency matrix can be
processed in parallel
Sparse matrix rows are relatively small

Distribution of the Algorithm
 Consequences of insights:
We can map each row of 'current' to a list of
PageRank “fragments” to assign to linkees
These fragments can be reduced into a single
PageRank value for a page by summing
Graph representation can be even more
compact; since each element is simply 0 or 1,
only transmit column numbers where it's 1

Map step: break page rank into even fragments to distribute to link targets
Reduce step: add together fragments into next PageRank
Iterate for next step...

Phase 1: Parse HTML
 Map task takes (URL, page content) pairs
and maps them to (URL, (PRinit, list-of-urls))
PRinit is the “seed” PageRank for URL
list-of-urls contains all pages pointed to by URL
 Reduce task is just the identity function

Phase 2: PageRank Distribution
 Map task takes (URL, (cur_rank, url_list))
For each u in url_list, emit (u, cur_rank/|url_list|)
Emit (URL, url_list) to carry the points-to list
along through iterations

Phase 2: PageRank Distribution
 Reduce task gets (URL, url_list) and many
(URL, val) values
Sum vals and fix up with d
Emit (URL, (new_rank, url_list))

Finishing up...
 A subsequent component determines
whether convergence has been achieved
(Fixed number of iterations? Comparison of
key values?)
 If so, write out the PageRank lists - done!
 Otherwise, feed output of Phase 2 into
another Phase 2 iteration

PageRank Conclusions
 MapReduce runs the “heavy lifting” in
iterated computation
 Key element in parallelization is
independent PageRank computations in a
given step
 Parallelization requires thinking about
minimum data partitions to transmit (e.g.,
compact representations of graph rows)
Even the implementation shown today doesn't
actually scale to the whole Internet; but it works
for intermediate-sized graphs

MapReduceAlgorithms.ppt

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