Counting Triangles and the Curse of the Last Reducer

Counting Triangles &
The Curse of the Last Reducer
Siddharth Suri
Sergei Vassilvitskii
Yahoo! Research

Why Count Triangles?

WWW 2011 2 Sergei Vassilvitskii


Clustering Coefficient:
Given an undirected graph G = (V, E)
cc(v) = fraction of v’s neighbors who are neighbors themselves
|{(u, w) ∈ E|u ∈ Γ(v) ∧ w ∈ Γ(v)}|
= dv
2



|{(u, w) ∈ E|u ∈ Γ(v) ∧ w ∈ Γ(v)}|
= dv
2

cc ( ) = N/A

cc ( ) = 1/3

cc ( )=1

cc ( )=1



|{(u, w) ∈ E|u ∈ Γ(v) ∧ w ∈ Γ(v)}| #∆s incident on v
= dv = dv
2 2

cc ( ) = N/A

cc ( ) = 1/3

cc ( )=1

cc ( )=1


Why Clustering Coefficient?

Captures how tight-knit the network is around a node.

cc ( ) = 0.5 cc ( ) = 0.1

vs.


Why Clustering Coefficient?

Captures how tight-knit the network is around a node.

cc ( ) = 0.5 cc ( ) = 0.1

vs.

Network Cohesion:
- Tightly knit communities foster more trust, social norms. [Coleman
’88, Portes ’88]
Structural Holes:
- Individuals benefit form bridging [Burt ’04, ’07]

Why MapReduce?

De facto standard for parallel computation on
large data
– Widely used at: Yahoo!, Google, Facebook,
– Also at: New York Times, Amazon.com, Match.com, ...
– Commodity hardware
– Reliable infrastructure

– Data continues to outpace available RAM !


How to Count Triangles

Sequential Version:
foreach v in V
foreach u,w in Adjacency(v)
if (u,w) in E
Triangles[v]++

v

Triangles[v]=0



Sequential Version:
foreach v in V
if (u,w) in E
Triangles[v]++

v

Triangles[v]=1
w

u


Sequential Version:
foreach v in V
if (u,w) in E
Triangles[v]++

v

Triangles[v]=1

w
u


Sequential Version:
foreach v in V
if (u,w) in E
Triangles[v]++

Running time: d2
v
v∈V

Even for sparse graphs can be quadratic if one vertex has high
degree.


Parallel Version

Parallelize the edge checking phase


Parallel Version

– Map 1: For each v send (v, Γ(v)) to single machine.
– Reduce 1: Input: v; Γ(v)
Output: all 2 paths (v1 , v2 ); u where v1 , v2 ∈ Γ(u)
( , ); ( , ); ( , );


Parallel Version

– Map 1: For each v send (v, Γ(v)) to single machine.
– Reduce 1: Input: v; Γ(v)
Output: all 2 paths (v1 , v2 ); u where v1 , v2 ∈ Γ(u)
( , ); ( , ); ( , );

– Map 2: Send (v1 , v2 ); u and (v1 , v2 ); $ for (v1 , v2 ) ∈ E to same
machine.
– Reduce 2: input: (v, w); u1 , u2 , . . . , uk , $?
Output: if $ part of the input, then: ui = ui + 1/3

( , ); , $ −→ +1/3 +1/3 +1/3
( , ); −→


Data skew

How much parallelization can we achieve?
- Generate all the paths to check in parallel
- The running time becomes max d2 v
v∈V


Data skew

How much parallelization can we achieve?
- Generate all the paths to check in parallel
- The running time becomes max d2 v
v∈V

Naive parallelization does not help with data skew
– Some nodes will have very high degree
– Example. 3.2 Million followers, must generate 10 Trillion (10^13)
potential edges to check.

– Even if generating 100M edges per second, 100K seconds ~ 27 hours.


“Just 5 more minutes”

Running the naive algorithm on LiveJournal Graph
– 80% of reducers done after 5 min
– 99% done after 35 min


Adapting the Algorithm

Approach 1: Dealing with skew directly
– currently every triangle counted 3 times (once per vertex)
– Running time quadratic in the degree of the vertex
– Idea: Count each once, from the perspective of lowest degree vertex
– Does this heuristic work?


Adapting the Algorithm

Approach 1: Dealing with skew directly
– currently every triangle counted 3 times (once per vertex)
– Running time quadratic in the degree of the vertex
– Idea: Count each once, from the perspective of lowest degree vertex
– Does this heuristic work?

Approach 2: Divide Conquer
– Equally divide the graph between machines
– But any edge partition will be bound to miss triangles
– Divide into overlapping subgraphs, account for the overlap


How to Count Triangles Better

Sequential Version [Schank ’07]:

foreach v in V
if deg(u) deg(v) deg(w) deg(v)
if (u,w) in E
Triangles[v]++


Does it make a difference?


Dealing with Skew

Why does it help?
– Partition nodes into two groups:
√
• Low: L = {v : dv ≤ m}
√
• High: H = {v : dv m}
– There are at most n low nodes; each produces at most O(m) paths
√
– There are at most 2 m high nodes
• Each produces paths to other high nodes: O(m) paths per node


Dealing with Skew

Why does it help?
– Partition nodes into two groups:
√
• Low: L = {v : dv ≤ m}
√
• High: H = {v : dv m}
– There are at most n low nodes; each produces at most O(m) paths
√
– There are at most 2 m high nodes
• Each produces paths to other high nodes: O(m) paths per node

– These two are identical !
– Therefore, no mapper can produce substantially more work than
others.
3
– Total work is O(m /2 ) , which is optimal


Approach 2: Graph Split

Partitioning the nodes:
- Previous algorithm shows one way to achieve better parallelization
- But what if even O(m) is too much. Is it possible to divide input into
smaller chunks?

Graph Split Algorithm:
– Partition vertices into p equal sized groups V1 , V2 , . . . , Vp .
– Consider all possible triples (Vi , Vj , Vk ) and the induced subgraph:
Gijk = G [Vi ∪ Vj ∪ Vk ]
– Compute the triangles on each Gijk separately.



Some Triangles present in multiple subgraphs:

in p-2 subgraphs

Vi Vj

in 1 subgraph

Vk
in ~p2 subgraphs

Can count exactly how many subgraphs each triangle will be in



Analysis:
– Each subgraph has O(m/p2 ) edges in expectation.
– Very balanced running times


Analysis:
– p controls memory needed per machine


Analysis:
3
– Total work: p · O((m/p2 )3/2 ) = O(m
3/2
) , independent of p


Analysis:
3
– Total work: p · O((m/p2 )3/2 ) = O(m
3/2
) , independent of p

Input too big: Shuffle time
paging increases with
duplication


Overall

Naive Parallelization Doesn’t help with Data Skew


Related Work

• Tsourakakis et al. [09]:
– Count global number of triangles by estimating the trace of the cube
of the matrix
– Don’t specifically deal with skew, obtain high probability
approximations.

• Becchetti et al. [08]
– Approximate the number of triangles per node
– Use multiple passes to obtain a better and better approximation


Conclusions


Conclusions

Think about data skew.... and avoid the curse


Conclusions

– Get programs to run faster


Conclusions

– Publish more papers


Conclusions

– Get more sleep


Conclusions

– Get more sleep
– ..


Conclusions

– Get more sleep
– ..
– The possibilities are endless!


Counting Triangles and the Curse of the Last Reducer

More Related Content

What's hot (20)

Similar to Counting Triangles and the Curse of the Last Reducer (20)

Counting Triangles and the Curse of the Last Reducer