Massive Graph Triangulation
by X. Hu, Y. Tao, and C. Chung, SIGMOD’13
Ilias Giechaskiel
Cambridge University, R212
[email protected]
February 21, 2014
Conclusions
Takeaway Messages
I Triangle listing important input for graph properties
I I/O becomes bottleneck for massive graphs
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Obvious approach doesn’t work
MGT algorithm
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Total order of vertices guarantees unique triangle orientation
Near optimal asymptotic I/O + CPU performance
Much faster than alternatives in practice
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Triangle Listing
Definition
Given a graph G = (V , E ), list exactly once all
∆v1 v2 v3 = {v1 , v2 , v3 } such that vi ∈ V and (vi , vj ) ∈ E
Motivation
I Triangle = shortest non-trivial cycle and clique
I Various metrics
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Dense neighborhood discovery
Triangular connectivity
k-truss
Clustering coefficient
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In-Memory Triangle Listing [CC12]
The Algorithm
procedure list(G )
∆(G ) ← ∅
loop u ∈ V
loop v ∈ adjG (u) & v > u
loop w ∈ adjG (u) ∩ adjG (v ) & w > v
∆(G ) ← ∆(G ) ∪ {∆uvw }
return ∆(G )
The Problem
I Random access to adjG (v ) for v ∈ adjG (u)
I O (|E | · scan(dmax )) I/Os in the worst case
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When it doesn’t fit in the memory of size M
Recall: scan(N) = Θ(N/B) where B is the disk block size
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Motivation
Previous Approaches
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External Memory Compact Forward (EM-CF)
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External Memory Node Iterator (EM-NI)
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O |E | + |E |1.5 /B I/Os
|E | I/O reads
Output insensitive
O |E |1.5 /B · logM/B (|E |/B) I/Os
Almost insensitive to M
Output insensitive
Graph Partition [CC12]
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O |E |2 /(MB) + K /B I/Os where K triangles
p
In practice, M > |E |
If M = c|E |, asymptotically optimal
But under a set of assumptions...
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Contributions
This Approach
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O |E |2 /(MB) + K /B I/Os in all settings
O |E | log |E | + |E |2 /M + α|E | CPU time
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α is the arboricity of the graph
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Both optimal up to constants
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Key idea: total order for unique triangle orientation
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Side note: also improves analysis of previous work
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Orienting G
Defining G ∗
I Define ≺ on V by u ≺ v iff
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G ∗ is G with edges oriented by ≺
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d(u) < d(v ) or d(u) = d(v ) and id(u) < id(v )
Is a total order
Takes O (sort(|E |)) I/Os
Recall: sort(N) = Θ N/BlogM/B N/B
Every triangle {u, v , w } has unique orientation u ≺ v ≺ w
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The Algorithm
Initial Idea
1. Load next cM edges of G ∗ into memory (Emem )
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All-or-nothing requirement (small-degree assumption)
2. Find all triangle with pivot edges in Emem
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The Algorithm
Step 2 (Initial)
procedure list(G , Emem )
loop u ∈ V
Vmem (u) ← N + (u) ∩ Vmem
Find triangles with u cone in Emem (u) ∪ Emem
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The Algorithm
Step 2 (Details)
procedure list(G ∗ , Emem )
Build hash structures
loop u ∈ V
Vmem (u) ← N + (u) ∩ Vmem
+ (u)
loop v ∈ Vmem
loop w ∈ Vmem (u)
if v 6= w & (v , w ) ∈ Emem then
Output ∆uvw
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The Algorithm
Analysis
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O |E |2 /(MB) + K /B I/O
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O |E | log |E | + |E |2 /M + α|E | CPU
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Θ (|E |/M) iterations
O (|E |/B) I/Os for scanning
O (K /B) for listing
O (|E | log |E |) for G ∗ sorting
Θ (|E |/M) iterations
+
O (|N + (u)| + |N + (u)| · |Vmem
(u)|)
+
Σ|N (u)| = |E |
Σv ∈V d + (v )2 = O(α|E |)
Optimality comes from considering the complete graph
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The Algorithm
Small-Degree Assumption
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What if ∃v such that d + (v ) > cM/2?
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2.
3.
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5.
Find one
Load a set S of cM/2 of its out-edges
Report all triangles involving one of the edges in S
Remove S from the graph
Repeat
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The Algorithm
Small-Degree Assumption
I How to implement step 3
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Create hash table of loaded vertices
Scan all |E | edges
Also scan N(v ) for each v 6= u with u ∈ N(v )
Does not change complexity
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Evaluation
Experimental Setup
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8GB memory (but memory conscious)
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Graphs unoriented
Real data
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364MB to 7.5GB
4.8 to 165 million vertices
28 to 938 million edges
|E |/|V | from 1.2 to 15.1
Varied M from 5% to 25% of disk size
Synthetic data
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Random, Recursive Matrix, Small World
m = 16n, n from 16 to 80 million
2.1GB to 10.6GB
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Evaluation
Real Data
I MGT always better for CPU
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MGT almost always better for I/O
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RGP higher hidden constant in complexity!
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Evaluation
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Evaluation
Criticism
I I/O analysis excludes cost of sorting
I Algorithm does not exploit parallelism
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Is inherently sequential
Not applicable to distributed environment
Or across cores
RGP ideas applied in this case [PC13]
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Block I/O model for SSDs and parallel environment?
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Behavior for large-degree vertices
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Experiments lacking when M bigger percentage of graph
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Conclusions
Key Insights
I Total order of vertices guarantees unique triangle orientation
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Key idea simple, but multiple tricks
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Near optimal asymptotic I/O + CPU performance
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Much faster than alternatives in practice
Key Questions
I Can you parallelize the algorithms non-trivially on a single PC?
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How can you extend the I/O model to different environments?
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How can you minimize data transfers in a distr. environment?
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Your questions?
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Bibliography I
Shumo Chu and James Cheng, Triangle listing in massive
networks, ACM Trans. Knowl. Discov. Data 6 (2012), no. 4,
17:1–17:32.
Xiaocheng Hu, Yufei Tao, and Chin-Wan Chung, Massive
graph triangulation, Proceedings of the 2013 ACM SIGMOD
International Conference on Management of Data (New York,
NY, USA), SIGMOD ’13, ACM, 2013, pp. 325–336.
Ha-Myung Park and Chin-Wan Chung, An efficient mapreduce
algorithm for counting triangles in a very large graph,
Proceedings of the 22Nd ACM International Conference on
Conference on Information &#38; Knowledge Management
(New York, NY, USA), CIKM ’13, ACM, 2013, pp. 539–548.
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