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Filtering: a method for solving graph problems in map reduce
1. Filtering: A Method for Solving
Graph Problems in
MapReduce
Benjamin Moseley Silvio Lattanzi Siddharth Suri Sergei Vassilvitski
UIUC Google Yahoo! Yahoo!
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
2. Overview
• Introduction to MapReduce model
• Our settings
• Our results
• Open questions
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
3. Introduction to the MapReduce
model
Large Scale Distributed Computation, Jan 2012
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4. XXL Data
• Huge amount of data
• Main problem is to analyze information quickly
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
5. XXL Data
• Huge amount of data
• Main problem is to analyze information quickly
• New tools
• Suitable efficient algorithms
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
6. MapReduce
• MapReduce is the platform of choice for processing
massive data
INPUT
MAP
PHASE
REDUCE
PHASE
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
7. MapReduce
• MapReduce is the platform of choice for processing
massive data
INPUT
• Data are represented as tuple
< key, value >
MAP
PHASE
REDUCE
PHASE
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
8. MapReduce
• MapReduce is the platform of choice for processing
massive data
INPUT
• Data are represented as tuple
< key, value >
MAP
PHASE
• Mapper decides how data is
distributed
REDUCE
PHASE
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
9. MapReduce
• MapReduce is the platform of choice for processing
massive data
INPUT
• Data are represented as tuple
< key, value >
MAP
PHASE
• Mapper decides how data is
distributed
REDUCE
• Reducer performs non-trivial PHASE
computation locally
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
10. How can we model MapReduce?
PRAM
•No limit on the number of processors
•Memory is uniformly accessible from any
processor
•No limit on the memory available
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
11. How can we model MapReduce?
STREAMING
•Just one processor
•There is a limited amount of memory
•No parallelization
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Tuesday, January 17, 2012
12. MapReduce model
[Karloff, Suri and Vassilvitskii]
• N is the input size and 0 is some fixed
constant
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13. MapReduce model
[Karloff, Suri and Vassilvitskii]
• N is the input size and 0 is some fixed
constant
1−
•Less than N machines
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
14. MapReduce model
[Karloff, Suri and Vassilvitskii]
• N is the input size and 0 is some fixed
constant
1−
•Less than N machines
1−
•Less than N memory on each machine
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
15. MapReduce model
[Karloff, Suri and Vassilvitskii]
• N is the input size and 0 is some fixed
constant
1−
•Less than N machines
1−
•Less than N memory on each machine
i i
MRC : problem that can be solved in O(log N )
rounds
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16. Combining map and reduce phase
•Mapper and Reducer work only on a subgraph
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17. Combining map and reduce phase
•Mapper and Reducer work only on a subgraph
•Keep time in each round polynomial
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18. Combining map and reduce phase
•Mapper and Reducer work only on a subgraph
•Keep time in each round polynomial
•Time is constrained by the number of rounds
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20. Algorithmic challenges
•No machine can see the entire input
•No communication between machines during each
phase
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21. Algorithmic challenges
•No machine can see the entire input
•No communication between machines during each
phase
2−2
•Total memory is N
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22. MapReduce vs MUD algorithms
•In MUD framework each reducer operates on a
stream of data.
•In MUD, each reducer is restricted to only using
polylogarithmic space.
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
23. Our settings
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
24. Our settings
•We study the Karloff, Suri and Vassilvitskii model
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
25. Our settings
•We study the Karloff, Suri and Vassilvitskii model
0
•We focus on class MRC
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26. Our settings
•We assume to work with dense graph m = n, 1+c
for some constant c 0
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27. Our settings
•We assume to work with dense graph m = n, 1+c
for some constant c 0
•Empirical evidences that social networks are dense
graphs
[Leskovec, Kleinberg and Faloutsos]
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28. Dense graph motivation
[Leskovec, Kleinberg and Faloutsos]
•They study 9 different social networks
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29. Dense graph motivation
[Leskovec, Kleinberg and Faloutsos]
•They study 9 different social networks
•They show that several graphs from different
1+c
domains have n edges
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
30. Dense graph motivation
[Leskovec, Kleinberg and Faloutsos]
•They study 9 different social networks
•They show that several graphs from different
1+c
domains have n edges
•Lowest value of c founded .08 and four graphs
have c .5
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31. Our results
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32. Results
Constant rounds algorithms for
•Maximal matching
•Minimum cut
•8-approx for maximum weighted matching
•2-approx for vertex cover
•3/2-approx for edge cover
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Tuesday, January 17, 2012
33. Notation
• G = (V, E) : Input graph
• n: number of nodes
• m : number of edges
• η : memory available on each machine
• N : input size
Large Scale Distributed Computation, Jan 2012
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34. Filtering
•Part of the input is dropped or filtered on the first
stage in parallel
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35. Filtering
•Part of the input is dropped or filtered on the first
stage in parallel
•Next some computation is performed on the filtered
input
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36. Filtering
•Part of the input is dropped or filtered on the first
stage in parallel
•Next some computation is performed on the filtered
input
•Finally some patchwork is done to ensure a proper
solution
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37. Filtering
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38. Filtering
FILTER
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39. Filtering
FILTER
COMPUTE
SOLUTION
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40. Filtering
FILTER
COMPUTE
SOLUTION
PATCHWORK
ON REST OF THE
GRAPH
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41. Filtering
FILTER
COMPUTE
SOLUTION
PATCHWORK
ON REST OF THE
GRAPH
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42. Warm-up: Compute the minimum
spanning tree
m=n 1+c
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43. Warm-up: Compute the minimum
spanning tree
m=n 1+c
PARTITION
EDGES
nc/2
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44. Warm-up: Compute the minimum
spanning tree
m=n 1+c
PARTITION
EDGES
nc/2
n1+c/2 edges
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Tuesday, January 17, 2012
45. Warm-up: Compute the minimum
spanning tree
n1+c/2 edges
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46. Warm-up: Compute the minimum
spanning tree
n1+c/2 edges
COMPUTE
MST
nc/2
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Tuesday, January 17, 2012
47. Warm-up: Compute the minimum
spanning tree
n1+c/2 edges
COMPUTE
MST
nc/2
SECOND
ROUND
nc/2
n1+c/2 edges
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48. Show that the algorithm is in MRC0
•The algorithm is correct
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49. Show that the algorithm is in MRC0
•The algorithm is correct
•No edge in the final solution is discarded in partial
solution
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50. Show that the algorithm is in MRC0
•The algorithm is correct
•No edge in the final solution is discarded in partial
solution
•The algorithm runs in two rounds
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51. Show that the algorithm is in MRC0
•The algorithm is correct
•No edge in the final solution is discarded in partial
solution
•The algorithm runs in two rounds
c
•No more than O n 2 machines are used
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52. Show that the algorithm is in MRC0
•The algorithm is correct
•No edge in the final solution is discarded in partial
solution
•The algorithm runs in two rounds
c
•No more than O n 2 machines are used
•No machine uses memory greater than O(n
1+c/2
)
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53. Maximal matching
•Algorithmic difficulty is that each machine can only
see edges assigned to the machine
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54. Maximal matching
•Algorithmic difficulty is that each machine can only
see edges assigned to the machine
•Is a partitioning based algorithm feasible?
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55. Partitioning algorithm
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56. Partitioning algorithm
PARTITIONING
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57. Partitioning algorithm
PARTITIONING
COMBINE
MATCHINGS
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58. Partitioning algorithm
PARTITIONING
COMBINE
MATCHINGS
It is impossible to build a maximal matching using only
red edges!!
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59. Algorithmic insight
•Consider any subset of the edges E
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60. Algorithmic insight
•Consider any subset of the edges E
•Let M be a maximal matching on
G[E ]
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61. Algorithmic insight
•Consider any subset of the edges E
•Let M be a maximal matching on
G[E ]
•The unmatched vertices form a
independent set
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62. Algorithmic insight
•We pick each edge with probability p
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63. Algorithmic insight
•We pick each edge with probability p
•Find a matching on a sample and then find a
matching on unmatched vertices
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64. Algorithmic insight
•We pick each edge with probability p
•Find a matching on a sample and then find a
matching on unmatched vertices
•For dense portions of the graph, many
edges should be sampled
Large Scale Distributed Computation, Jan 2012
Tuesday, January 17, 2012
65. Algorithmic insight
•We pick each edge with probability p
•Find a matching on a sample and then find a
matching on unmatched vertices
•For dense portions of the graph, many
edges should be sampled
•Sparse portions of the graph are small
and can be placed on a single machine
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Tuesday, January 17, 2012
66. Algorithm
•Sample each edge independently with probability
10 log n
p=
nc/2
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67. Algorithm
•Sample each edge independently with probability
10 log n
p=
nc/2
•Find a matching on the sample
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68. Algorithm
•Sample each edge independently with probability
10 log n
p=
nc/2
•Find a matching on the sample
•Consider the induced subgraph on unmatched
vertices
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69. Algorithm
•Sample each edge independently with probability
10 log n
p=
nc/2
•Find a matching on the sample
•Consider the induced subgraph on unmatched
vertices
•Find a matching on this graph and output the union
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70. Algorithm
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71. Algorithm
SAMPLE
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72. Algorithm
SAMPLE
FIRST
MATCHING
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73. Algorithm
SAMPLE
FIRST
MATCHING
MATCHING ON
UNMATCHED NODES
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74. Algorithm
SAMPLE
FIRST
MATCHING
MATCHING ON
UNMATCHED NODES
UNION
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75. Correctness
•Consider the last step of the algorithm
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76. Correctness
•Consider the last step of the algorithm
•All unmatched vertices are placed on a machine
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77. Correctness
•Consider the last step of the algorithm
•All unmatched vertices are placed on a machine
•All unmatched vertices or are matched at the last
step or have only matched neighbors
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78. Bounding the rounds
Three rounds:
•Sampling the edges
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79. Bounding the rounds
Three rounds:
•Sampling the edges
•Find a matching on a single machine for the sampled
edges
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80. Bounding the rounds
Three rounds:
•Sampling the edges
•Find a matching on a single machine for the sampled
edges
•Find a matching for the unmatched vertices
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81. Bounding the memory
10 log n
•Each edge was sampled with probability p =
nc/2
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82. Bounding the memory
10 log n
•Each edge was sampled with probability p =
nc/2
˜ 1+c/2 )
•Using Chernoff the sampled graph has size O(n
with high probability
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83. Bounding the memory
10 log n
•Each edge was sampled with probability p =
nc/2
˜ 1+c/2 )
•Using Chernoff the sampled graph has size O(n
with high probability
•Can we bound the size of the induced subgraph on
the unmatched vertices?
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84. Bounding the memory
1+c/2
•For a fixed induced subgraph with at least n
edges the probability an edge is not sampled is:
n1+c/2
10 log n
1− ≤ exp(−10n log n)
nc/2
Large Scale Distributed Computation, Jan 2012
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85. Bounding the memory
1+c/2
•For a fixed induced subgraph with at least n
edges the probability an edge is not sampled is:
n1+c/2
10 log n
1− ≤ exp(−10n log n)
nc/2
•Union bounding over all 2 induced subgraphs shows n
that at least one edge is sampled from every dense
subgraph with probability
1
1 − exp(−10 log n) ≥ 1 − 10
n
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86. Using less memory
˜ 1+c/2 )
•Can we run the algorithm with less then O(n
memory?
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87. Using less memory
˜ 1+c/2 )
•Can we run the algorithm with less then O(n
memory?
•We can amplify our sampling technique!
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88. Using less memory
•Sample as many edges as possible that fits in memory
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89. Using less memory
•Sample as many edges as possible that fits in memory
•Find a matching
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90. Using less memory
•Sample as many edges as possible that fits in memory
•Find a matching
•If the edges between unmatched vertices fit onto a
single machine, find a matching on those vertices
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91. Using less memory
•Sample as many edges as possible that fits in memory
•Find a matching
•If the edges between unmatched vertices fit onto a
single machine, find a matching on those vertices
•Otherwise, recurse on the unmatched nodes
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92. Using less memory
•Sample as many edges as possible that fits in memory
•Find a matching
•If the edges between unmatched vertices fit onto a
single machine, find a matching on those vertices
•Otherwise, recurse on the unmatched nodes
1+
•With n memory each iteration a factor of n edges
are removed resulting in O(c/) rounds
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93. Parallel computation power
•Maximal matching algorithm does not use
parallelization
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94. Parallel computation power
•Maximal matching algorithm does not use
parallelization
•We use a single machine in every step
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95. Parallel computation power
•Maximal matching algorithm does not use
parallelization
•We use a single machine in every step
•When parallelization is used?
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96. Maximum weighted matching
8
2 2
7
1
4
3
4 2
2
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97. Maximum weighted matching
8
2 2
7
1
4
3
4 2
2
SPLIT THE GRAPH
8
2 2
7
1
4 3
2 4
2
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98. Maximum weighted matching
8
2 2
7
1
4 3
2 4
2
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99. Maximum weighted matching
8
2 2
7
1
4 3
2 4
2
MATCHINGS
2
7
3
2 4
2
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100. Maximum weighted matching
2
7
3
2 4
2
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101. Maximum weighted matching
2
7
3
2 4
2
SCAN THE EDGE
SEQUENTIALLY
2
7
3
4 2
2
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102. Bounding the rounds and memory
Four rounds:
•Splitting the graph and running the maximal
matching:
˜ 1+c/2 ) memory
3 rounds and O(n
•Compute the final solution:
1 round and O(n log n) memory
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103. Approximation guarantee (sketch)
•An edge in the solution can block at most 2 edges in
each subgraph of smaller weight
8
2
7
3
2 4
2
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104. Approximation guarantee (sketch)
•An edge in the solution can block at most 2 edges in
each subgraph of smaller weight
8
2
7
3
2 4
2
•We loose a factor or 2 because we do not consider
the weight
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105. Other algorithms
Based on the same intuition:
•2-approx for vertex cover
•3/2-approx for edge cover
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106. Minimum cut
•Partition does not work, because we loose structural
informations
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107. Minimum cut
•Partition does not work, because we loose structural
informations
•Sampling does not seem to work either
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108. Minimum cut
•Partition does not work, because we loose structural
informations
•Sampling does not seem to work either
•We can use the first steps of Karger algorithm as a
filtering technique
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109. Minimum cut
•Partition does not work, because we loose structural
informations
•Sampling does not seem to work either
•We can use the first steps of Karger algorithm as a
filtering technique
•The random choices made in the early rounds
succeed with high probability, whereas the later rounds
have a much lower probability of success
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110. Minimum cut
1
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111. Minimum cut
RANDOM
WEIGHT
n δ1
3 1 5
1 2 3 2 2 3 1 1 3
4 4
4 3 4
4 3
1 3 2
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112. Minimum cut
RANDOM
WEIGHT
n δ1
3 1 5
1 2 3 2 2 3 1 1 3
4 4
4 3 4
4 3
1 3 2
COMPRESS
EDGES WITH
SMALL WEIGTH
2
2 2
2
2 2 2
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113. Minimum cut
RANDOM
WEIGHT
n δ1
3 1 5
1 2 3 2 2 3 1 1 3
4 4
4 3 4
4 3
1 3 2
COMPRESS
EDGES WITH
SMALL WEIGTH
2
2 2
2
2 2 2
From Karger at least one graph will contain a min cut w.h.p.
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114. Minimum cut
2
2 2
2
2 2 2
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115. Minimum cut
2
2 2
2
2 2 2
COMPUTE
MINIMUM
CUT
2
2 2
2
2 2 2
We will find the minimum cut w.h.p.
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116. Empirical result (matching)
%#!
'()(**+*,*-.)/012
30)+(2/4-
%!!
!#$%'(%)#*+,'
$#!
$!!
#!
!
!!!$ !!!# !!$ !!% !!# !$ $
-.%/0'1234.4)0)*5'(/,'
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117. Open problems
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118. Open problems 1
•Maximum matching
•Shortest path
•Dynamic programming
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119. Open problems 2
•Algorithms for sparse graph
•Does connected components require more than 2
rounds?
•Lower bounds
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120. Thank you!
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