This document provides an overview and agenda for a lecture on graph processing using MapReduce. It discusses representing graphs as adjacency matrices or lists, and gives examples of single source shortest path and PageRank algorithms. Graph processing in MapReduce typically involves computations at each node and propagating those computations across the graph. Key challenges include representing graph structure suitably for MapReduce and traversing the graph in a distributed manner through multiple iterations.
H2O.ai CEO/Founder: Sri Ambati Keynote at Wells Fargo Day
Data-Intensive Computing for Text Analysis
1. Data-Intensive Computing for Text Analysis
CS395T / INF385T / LIN386M
University of Texas at Austin, Fall 2011
Lecture 7
October 6, 2011
Jason Baldridge Matt Lease
Department of Linguistics School of Information
University of Texas at Austin University of Texas at Austin
Jasonbaldridge at gmail dot com ml at ischool dot utexas dot edu
1
2. Acknowledgments
Course design and slides based on
Jimmy Lin’s cloud computing courses at
the University of Maryland, College Park
Some figures courtesy of the following
excellent Hadoop books (order yours today!)
• Chuck Lam’s Hadoop In Action (2010)
• Tom White’s Hadoop: The Definitive Guide,
2nd Edition (2010)
2
3. Today’s Agenda
• Hadoop Counters
• Graph Processing in MapReduce
– Representing/Encoding Graphs
• Adjacency matrices vs. Lists
– Example: Single Source Shortest Page
– Example: PageRank
• Themes
– No shared memory redundant computation
• More computational capability overcomes less efficiency
– Iterate MapReduce computations until convergence
– Use non-MapReduce driver for over-arching control
• Not just for pre- and post-processing
• Opportunity for global synchronization between iterations
• In-class exercise 3
6. White p. 172
Hadoop Counters & Global State
• Hadoop’s Counters provide its only means for
sharing/modifying global distributed state
– Built-in safeguards for distributed modification
• e.g. two tasks try to increment a counter simultaneously
– Lightweight: only long bytes… per counter
– Limited control
• create, read, and increment
• no destroy, arbitrary set, or decrement
• Advertised use: progress tracking and logging
• To what extent might we “abuse” counters for
tracking/updating interesting shared state?
6
7. How high (and precisely) can you count?
• How precise?
• Integer representation
• To approximate fractional values, scale and truncate (Lin & Dyer p. 99)
• How high?
– “8-byte integers” (Lin & Dyer p. 99 ): really only one byte?
– Old API: org.apache.hadoop.mapred.Counters
• long getCounter(…), incrCounter(…, long amount)
– New API: org.apache.hadoop.mapreduce.Counter
• long getValue(), increment(long incr)
• How many?
– Old API: static int MAX_COUNTER_LIMIT (next slide…)
– New API: ???? (int countCounters() ) 7
10. White p. 173, 227-231
• incrCounter(…)
• getCounters(…)
• getCounter(…)
• findCounter(…)
• http://developer.yahoo.com/hadoop/tutorial/module5.html#metrics
10
11. White p. 172
Counters and Global State
Counter values are definitive only once a job has successfully completed
- White p. 227
What about while a job is running?
• If a task reports progress, it sets a JobTracker flag to indicate a status
change should be sent to the TaskTracker
– The flag is checked in a separate thread every 3s, and if set, the
TaskTracker is notified
– What about counter updates?
• The TaskTracker sends heartbeats to the JobTracker (at least every 5s)
which include the status of all tasks being run by the TaskTracker...
– Counters (which can be relatively larger) are sent less frequently
• JobClient receives the latest status by polling the jobtracker every 1s
• Clients can call JobClient’s getJob() to obtain a RunningJob instance
with the latest status information (at time of the call?)
11
13. What’s a graph?
Graphs are ubiquitous
The Web (pages and hyperlink structure)
Computer networks (computers and connections)
Highways and railroads (cities and roads/tracks)
Social networks
G = (V,E), where
V: the set of vertices (nodes)
E: the set of edges (links)
Either/Both may contain additional information
• e.g. edge weights (e.g. cost, time, distance)
• e.g. node values (e.g. PageRank)
Graph types
Directed vs. undirected
Cyclic vs. acyclic
14. Some Graph Problems
Finding shortest paths
Routing Internet traffic and UPS trucks
Finding minimum spanning trees
Telco laying down fiber
Finding Max Flow
Airline scheduling
Identify “special” nodes and communities
Breaking up terrorist cells, spread of avian flu
Bipartite matching
Monster.com, Match.com
And of course... PageRank
15. Graphs and MapReduce
MapReduce graph processing typically involves
Performing computations at each node
• e.g. using node features, edge features, and local link structure
Propagating computations
• “traversing” the graph
Key questions
How do you represent graph data in MapReduce?
How do you traverse a graph in MapReduce?
16. Graph Representation
How do we encode graph structure suitably for
computation
propagation
Two common approaches
Adjacency matrix 2
Adjacency list
1
3
4
17. Adjacency Matrices
Represent a graph as an |V| x |V| square matrix M
Mjk = w directed edge of weight w from node j to node k
• w=0 no edge exists
• Mii: main diagonal gives self-loop weights from node i to itself
If undirected, use only top-right of matrix (symmetry)
2
1 2 3 4
1 0 1 0 1 1
3
2 1 0 1 1
3 1 0 0 0
4 1 0 1 0 4
18. Adjacency Matrices: Critique
Advantages:
Amenable to mathematical manipulation
Easy iteration for computation over out-links and in-links
• Mj* column over all out-links from node j
• M*k row over all in-links to node k
Disadvantages
Sparsity: wasted computations, wasted space
20. Inverted Index: Boolean Retrieval
Doc 1 Doc 2 Doc 3 Doc 4
one fish, two fish red fish, blue fish cat in the hat green eggs and ham
1 2 3 4
blue 1 blue 2
cat 1 cat 3
egg 1 egg 4
fish 1 1 fish 1 2
green 1 green 4
ham 1 ham 4
hat 1 hat 3
one 1 one 1
red 1 red 2
two 1 two 1
21. Adjacency Lists: Critique
Vs. Adjacency matrix
Sparsity: More compact, fewer wasted computations
Easy to compute over out-links
What about computation over in-links?
1 2 3 4
1 0 1 0 1 1: 2, 4
2 1 0 1 1 2: 1, 3, 4
3 1 0 0 0 3: 1
4: 1, 3
4 1 0 1 0
23. Problem
Find shortest path from a source node to one or more
target nodes
Shortest may mean lowest weight or cost, etc.
Classic approach
Dijkstra’s Algorithm
• Maintain a global priority queue over all (node, distance) pairs
• Sort queue by min distance to reach each node from the source node
• Initialization: distance to source node = 0, all others =
• Visit nodes in order of (monotonically) increasing path length
• Whenever node visited, no shorter path exists
• For each node is visited
• update its neighbours in the queue
• Remove the node from the queue
24. Edsger W. Dijkstra
May 11, 1930 – August 6, 2002
Received the 1972 Turing Award
Schlumberger Centennial Chair of Computer Science at
UT Austin (1984-2000)
http://en.wikipedia.org/wiki/Dijkstra’s_algorithm
Wikipedia has nice animation of it in action
25. Dijkstra’s Algorithm
Maintain global priority queue over all (node, distance) pairs
Sort queue by min distance to reach each node from the source node
Initialization
distance to source node = 0
distance to all other nodes =
While queue not empty
visit next node (i.e. the node with shortest path length in the queue)
• Output distance to it if desired
• Update distance to each of its neighbours in the queue
• Remove it from the queue
32. Problem
Find shortest path from a source node to one or more
target nodes
Shortest may mean lowest weight or cost, etc.
Classic approach
Dijkstra’s Algorithm
33. Problem
Find shortest path from a source node to one or more
target nodes
Shortest may mean lowest weight or cost, etc.
Classic approach
Dijkstra’s Algorithm
MapReduce approach
Parallel Breadth-First Search (BFS)
34. Finding the Shortest Path
Assume unweighted graph (for now…)
General Inductive Approach
Initialization
• DISTANCETO(source s) = 0
• For any node n connected to s, DISTANCETO(n) = 1
• Else DISTANCETO(any other node p) =
For each iteration
• For every node n
• For every neighbor m M(n),
DISTANCETO(m) = 1 + min( DISTANCETO(n) )
d1 m1
…
d2
s … n
m2
… d3
m3
36. From Intuition to Algorithm
Representation
Key: node n
Value: d (distance from start)
• Also: adjacency list (list of nodes reachable from n)
Initialization: d = for all nodes except start node
Mapper
m adjacency list: emit (m, d + 1)
Sort/Shuffle
Groups distances by reachable nodes
Reducer
Selects minimum distance path for each reachable node
Additional bookkeeping needed to keep track of actual path
38. Multiple Iterations Needed
Each iteration advances the “frontier” by one hop
Subsequent iterations find more reachable nodes
Multiple iterations are needed to explore entire graph
Preserving graph structure
Problem: Where did the adjacency list go?
Solution: mapper emits (n, adjacency list) s well
39. Stopping Criterion
How many iterations are needed?
Convince yourself: when a node is first “discovered”,
we’ve found the shortest path
Now answer the question...
Six degrees of separation?
Practicalities of implementation in MapReduce
40. Comparison to Dijkstra
Dijkstra’s algorithm is more efficient
At any step it only pursues edges from the minimum-cost path
inside the frontier
MapReduce explores all paths in parallel
Lots of “waste”
Useful work is only done at the “frontier”
Why can’t we do better using MapReduce?
41. Weighted Edges
Now consider non-unit, positive edge weights
Why can’t edge weights be negative?
Adjacency list now includes a weight w for each edge
In mapper, emit (m, d + wp) instead of (m, d + 1) for each node m
Is that all?
42. Stopping Criterion
How many iterations are needed in parallel BFS (positive
edge weight case)?
Convince yourself: when a node is first “discovered”,
we’ve found the shortest path
43. Additional Complexities
1
search frontier 1
n6 n7 1
n8
r 10
1 n9
n5
n1
s 1 1
q
p 1 n4
n2 1
n3
44. Stopping Criterion
How many iterations are needed in parallel BFS (positive
edge weight case)?
Practicalities of implementation in MapReduce
Unrelated to stopping… where have we seen min/max before?
45. In General: Graphs and MapReduce
Graph algorithms typically involve
Performing computations at each node: based on node features,
edge features, and local link structure
Propagating computations: “traversing” the graph
Generic recipe
Represent graphs as adjacency lists
Perform local computations in mapper
Pass along partial results via outlinks, keyed by destination node
Perform aggregation in reducer on inlinks to a node
Iterate until convergence: controlled by external “driver”
Don’t forget to pass the graph structure between iterations
47. Random Walks Over the Web
Random surfer model
User starts at a random Web page
User randomly clicks on links, surfing from page to page
PageRank
Characterizes the amount of time spent on any given page
Mathematically, a probability distribution over pages
PageRank captures notions of page importance
Correspondence to human intuition?
One of thousands of features used in web search
Note: query-independent
48. PageRank: Defined
Given page x with inlinks t1…tn, where
C(t) is the out-degree of t
is probability of random jump
N is the total number of nodes in the graph
1 n
PR (ti )
PR ( x) (1 )
N i 1 C (ti )
t1
X
t2
…
tn
49. Computing PageRank
Properties of PageRank
Can be computed iteratively
Effects at each iteration are local
Sketch of algorithm:
Start with seed PRi values
Each page distributes PRi “credit” to all pages it links to
Each target page adds up “credit” from multiple in-bound links to
compute PRi+1
Iterate until values converge
50. Simplified PageRank
First, tackle the simple case:
No random jump factor
No dangling links
Then, factor in these complexities…
Why do we need the random jump?
Where do dangling links come from?
55. Complete PageRank
Two additional complexities
What is the proper treatment of dangling nodes?
How do we factor in the random jump factor?
Solution:
Second pass to redistribute “missing PageRank mass” and
account for random jumps
1 m
p' (1 ) p
G G
p is PageRank value from before, p' is updated PageRank value
|G| is the number of nodes in the graph
m is the missing PageRank mass
How to perform bookkeeping for dangling nodes?
How to implement this 2nd pass in Hadoop?
56. PageRank Convergence
Alternative convergence criteria
Iterate until PageRank values don’t change
Iterate until PageRank rankings don’t change
Fixed number of iterations
Convergence for web graphs?
57. Local Aggregation d1 m1
d2
Use combiners m2
n
BFS uses min, PageRank uses sum
d3
• associative and commutative m3
In-mapper combining design pattern also applicable
Opportunity for aggregation when mapper sees multiple nodes
with out-links to same destination node
How do we maximize opportunities for local aggregation?
Partition the dataset into clusters with many internal and few
external links
Chicken-and-egg problem: don’t we need MapReduce to do this?
• Use cheap heuristics
• e.g. social network: zip code or school
• e.g. for web: language or domain name
• etc.
58. Limitations of MapReduce
Amount of intermediate data (to shuffle) is proportional to
number of edges in graph
We have considered sparse graphs (i.e. with few edges),
minimizing such intermediate data
For dense graphs with O(n^2) edges, runtime would be
dominated by copying intermediate data
Consequently, MapReduce algorithms are often
impractical on large, dense graphs
But isn’t data-intensive computing exactly what
MapReduce is supposed to help us with??
See (Lin and Dyer, p. 101)
60. 1: class Mapper 1: class Mapper
2: method Map( Node N ) 2: method Map( sid s, Node N )
3: d = N.Distance 3: d = N[s].Distance
4: Emit( N.id, N ) 4: Emit( Pair(sid, N.id), N )
5: for all (nid m in N.AdjacencyList) do 5: for all (nid m in N.AdjacencyList) do
6: Emit( m, d + 1) 6: Emit( Pair(sid, m), d + 1)
1: class Reducer 1: class Reducer
2: method Reduce(nid m, [d1, d2, ...]) 2: method Reduce( Pair(sid s,nid m), [d1,
3: dmin = 1 d2, ...] )
4: Node M = null 3: dmin = 1
5: for all d in counts [d1, d2, ...] do 4: M = null
6: if IsNode(d) then 5: for all d in counts [d1, d2, ...] do
7: M=d 6: if IsNode(d) then
8: else if d < dmin then 7: M=d
9: dmin = d 8: else if d < dmin then
10: M.Distance = dmin 9: dmin = d
11: Emit( M ) 10: M[s].Distance = dmin
11: Emit( M )
61. 1: class Mapper 1: class Mapper
2: method Map( sid s, Node N ) 2: method Map( sid s, Node N )
3: d = N[s].Distance 3: d = N[s].Distance4:
4: Emit( Pair(sid, N.id), N ) 4: if sid=0 then
5: for all (nid m in N.AdjacencyList) do 5: Emit( Pair(sid, N.id), N )
6: Emit( Pair(sid, m), d + 1) 6: for all (nid m in N.AdjacencyList) do
7: Emit( Pair(sid, m), d + 1)
1: class Reducer
2: method Reduce( Pair(sid s,nid m), [d1, Partition: all pairs with same 2nd nid to same
d2, ...] ) reducer
3: dmin = 1 KeyComp: order by sid, the nid, sort sid=0
4: M = null first
5: for all d in counts [d1, d2, ...] do
6: if IsNode(d) then 1: class Reducer
7: M=d 2: M = null
8: else if d < dmin then 3: method Reduce( Pair(sid s,nid m), [d1,
9: dmin = d d2, ...] )
10: M[s].Distance = dmin 4: dmin = 1
11: Emit( M ) 5: for all d in counts [d1, d2, ...] do
6: if IsNode(d) then
7: M=d
8: else if d < dmin then
9: dmin = d
10: M[s].Distance = dmin
11: Emit( M )