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# Counting Triangles and the Curse of the Last Reducer

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### Counting Triangles and the Curse of the Last Reducer

1. 1. Counting Triangles &The Curse of the Last Reducer Siddharth Suri Sergei Vassilvitskii Yahoo! Research
2. 2. Why Count Triangles?WWW 2011 2 Sergei Vassilvitskii
3. 3. Why Count Triangles? 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 2WWW 2011 3 Sergei Vassilvitskii
4. 4. Why Count Triangles? 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 cc ( ) = N/A cc ( ) = 1/3 cc ( )=1 cc ( )=1WWW 2011 4 Sergei Vassilvitskii
5. 5. Why Count Triangles? 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)}| #∆s incident on v = dv = dv 2 2 cc ( ) = N/A cc ( ) = 1/3 cc ( )=1 cc ( )=1WWW 2011 5 Sergei Vassilvitskii
6. 6. Why Clustering Coefficient? Captures how tight-knit the network is around a node. cc ( ) = 0.5 cc ( ) = 0.1 vs.WWW 2011 6 Sergei Vassilvitskii
7. 7. 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]WWW 2011 7 Sergei Vassilvitskii
8. 8. 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 !WWW 2011 8 Sergei Vassilvitskii
9. 9. 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]=0WWW 2011 9 Sergei Vassilvitskii
10. 10. 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]=1 w uWWW 2011 10 Sergei Vassilvitskii
11. 11. 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]=1 w uWWW 2011 11 Sergei Vassilvitskii
12. 12. How to Count Triangles Sequential Version: foreach v in V foreach u,w in Adjacency(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.WWW 2011 12 Sergei Vassilvitskii
13. 13. Parallel Version Parallelize the edge checking phaseWWW 2011 13 Sergei Vassilvitskii
14. 14. Parallel Version Parallelize the edge checking phase – 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) ( , ); ( , ); ( , );WWW 2011 14 Sergei Vassilvitskii
15. 15. Parallel Version Parallelize the edge checking phase – 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 ( , ); −→WWW 2011 15 Sergei Vassilvitskii
16. 16. Data skew How much parallelization can we achieve? - Generate all the paths to check in parallel - The running time becomes max d2 v v∈VWWW 2011 16 Sergei Vassilvitskii
17. 17. 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.WWW 2011 17 Sergei Vassilvitskii
18. 18. “Just 5 more minutes” Running the naive algorithm on LiveJournal Graph – 80% of reducers done after 5 min – 99% done after 35 minWWW 2011 18 Sergei Vassilvitskii
19. 19. 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?WWW 2011 19 Sergei Vassilvitskii
20. 20. 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 overlapWWW 2011 20 Sergei Vassilvitskii
21. 21. How to Count Triangles Better Sequential Version [Schank ’07]: foreach v in V foreach u,w in Adjacency(v) if deg(u) deg(v) deg(w) deg(v) if (u,w) in E Triangles[v]++WWW 2011 21 Sergei Vassilvitskii
22. 22. Does it make a difference?WWW 2011 22 Sergei Vassilvitskii
23. 23. 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 nodeWWW 2011 23 Sergei Vassilvitskii
24. 24. 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 optimalWWW 2011 24 Sergei Vassilvitskii
25. 25. 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.WWW 2011 25 Sergei Vassilvitskii
26. 26. Approach 2: Graph Split 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 inWWW 2011 26 Sergei Vassilvitskii
27. 27. Approach 2: Graph Split Analysis: – Each subgraph has O(m/p2 ) edges in expectation. – Very balanced running timesWWW 2011 27 Sergei Vassilvitskii
28. 28. Approach 2: Graph Split Analysis: – Very balanced running times – p controls memory needed per machineWWW 2011 28 Sergei Vassilvitskii
29. 29. Approach 2: Graph Split Analysis: – Very balanced running times – p controls memory needed per machine 3 – Total work: p · O((m/p2 )3/2 ) = O(m 3/2 ) , independent of pWWW 2011 29 Sergei Vassilvitskii
30. 30. Approach 2: Graph Split Analysis: – Very balanced running times – p controls memory needed per machine 3 – Total work: p · O((m/p2 )3/2 ) = O(m 3/2 ) , independent of p Input too big: Shuffle time paging increases with duplicationWWW 2011 30 Sergei Vassilvitskii
31. 31. Overall Naive Parallelization Doesn’t help with Data SkewWWW 2011 31 Sergei Vassilvitskii
32. 32. 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 approximationWWW 2011 32 Sergei Vassilvitskii
33. 33. ConclusionsWWW 2011 33 Sergei Vassilvitskii
34. 34. Conclusions Think about data skew.... and avoid the curseWWW 2011 33 Sergei Vassilvitskii
35. 35. Conclusions Think about data skew.... and avoid the curse – Get programs to run fasterWWW 2011 33 Sergei Vassilvitskii
36. 36. Conclusions Think about data skew.... and avoid the curse – Get programs to run faster – Publish more papersWWW 2011 33 Sergei Vassilvitskii
37. 37. Conclusions Think about data skew.... and avoid the curse – Get programs to run faster – Publish more papers – Get more sleepWWW 2011 33 Sergei Vassilvitskii
38. 38. Conclusions Think about data skew.... and avoid the curse – Get programs to run faster – Publish more papers – Get more sleep – ..WWW 2011 33 Sergei Vassilvitskii
39. 39. Conclusions Think about data skew.... and avoid the curse – Get programs to run faster – Publish more papers – Get more sleep – .. – The possibilities are endless!WWW 2011 33 Sergei Vassilvitskii
40. 40. Thank You