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.

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

4. Hadoop Counters

5. Lam p. 98, White pp. 226-227
5

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

8. 8

9. 9

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

12. Representing Graphs

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

19. Adjacency Lists
Take adjacency matrices… and throw away all the zeros
 Hmm… look familiar…?
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

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

22. Single Source Shortest Path

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

26. Dijkstra’s Algorithm Example
1
 
10
0 2 3 9 4 6
5 7
 
2
Example from CLR

27. Dijkstra’s Algorithm Example
1
10 
10
0 2 3 9 4 6
5 7
5 
2
Example from CLR

28. Dijkstra’s Algorithm Example
1
8 14
10
0 2 3 9 4 6
5 7
5 7
2
Example from CLR

29. Dijkstra’s Algorithm Example
1
8 13
10
0 2 3 9 4 6
5 7
5 7
2
Example from CLR

30. Dijkstra’s Algorithm Example
1
1
8 9
10
0 2 3 9 4 6
5 7
5 7
2
Example from CLR

31. Dijkstra’s Algorithm Example
1
8 9
10
0 2 3 9 4 6
5 7
5 7
2
Example from CLR

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

35. Visualizing Parallel BFS
n7
n0 n1
n3 n2
n6
n5
n4
n8
n9

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

37. BFS Pseudo-Code
What type should we use for the values?

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

46. PageRank

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?

51. Sample PageRank Iteration (1)
Iteration 1 n2 (0.2) n2 (0.166)
0.1
n1 (0.2) 0.1 0.1 n1 (0.066)
0.1
0.066
0.066 0.066
n5 (0.2) n5 (0.3)
n3 (0.2) n3 (0.166)
0.2 0.2
n4 (0.2) n4 (0.3)

52. Sample PageRank Iteration (2)
Iteration 2 n2 (0.166) n2 (0.133)
0.033 0.083
n1 (0.066) 0.083 n1 (0.1)
0.033
0.1
0.1 0.1
n5 (0.3) n5 (0.383)
n3 (0.166) n3 (0.183)
0.3 0.166
n4 (0.3) n4 (0.2)

53. PageRank in MapReduce
n1 [n2, n4] n2 [n3, n5] n3 [n4] n4 [n5] n5 [n1, n2, n3]
Map
n2 n4 n3 n5 n4 n5 n1 n2 n3
n1 n2 n2 n3 n3 n4 n4 n5 n5
Reduce
n1 [n2, n4] n2 [n3, n5] n3 [n4] n4 [n5] n5 [n1, n2, n3]

54. PageRank Pseudo-Code

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)

59. In-class Exercise:
All Pairs PBFS

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 )