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Finding Strongly Connected Components and Topological Sort in Parallel using O(log² n) reachability queries Warren Schudy Brown University Work done while interning at Google Research Mountain View

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A Scheduling Problem To-do List –Topological sort (TS) –Strongly connected components (SCC) –Reachability SCC application in scientific computing requiring parallelism [McLendon et al. 01]

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Previous Results for TS, SCC and Reachability Assume sparse graph with n vertices using 1 ≤ p ≤ n 4/3 processors: Question: is reachability fundamentally easier to parallelize than SCC and TS? (In Transactions of Information Processing Society of Japan ’99 ’04, Akio, Masahiro and Ryozo claim runtime n/p for TS & SCC) Coppersmith and Winograd ‘87 T. Spencer ‘97Ullman & Yannakakis ‘91 Runtime (ignore logs) n 2.38… /pn/p 1/3 n/p 1/2 ProblemsAll Reachability

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Answer: no (up to logs) Our main result: a reduction of SCC and TS to O(log 2 n) reachability queries Remainder of talk focuses on SCC problem This workCoppersmith et al.SpencerUllman et al. Runtimen/p 1/2 n 2.38 /pn/p 1/3 n/p 1/2 ProblemsTS and SCCAll Reachability

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Desc(s) SCC(s) V \ Desc(s) A simple SCC algorithm Choose random vertex s V Determine SCC(s) and output it Determine the vertices Desc(s) reachable from s Recurse (in parallel) on: –Desc(s) \ SCC(s) –V \ Desc(s) s (Similar to [Coppersmith, Fleischer, Hendrickson and Pinar ’05])

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High-runtime instance Desc(s) s

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Algorithmic Idea If this algorithm divided the graph roughly in two, recursion depth would be log n So pick many source vertices instead of 1 (number chosen later) Desc( )

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Make one pivot vertex s special, and output its SCC Outputting SCCs s Desc( ) Desc(s) Recurse on blue, green, yellow and unreached subgraphs SCC(s)

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Our Multipivot Algorithm Permute the vertices randomly Determine the smallest s s.t. {1,2,…s} together reach at least half the edges (binary search) Output SCC(s) Recurse on: –V \ Desc(1…s) –(Desc(1…s-1) Desc(s)) \ SCC(s) –Desc(1…s-1) \ Desc(s) –Desc(s) \ (Desc(1…s- 1) SCC(s)) s= Desc(1…s-1) Desc(s) 1 2 3 4 SCC(s)

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Runtime Analysis k2k2 May contain almost all the edges, but will contain only some of the transitive closure edges (due to random order) s= Desc(1…s-1) Desc(s) 1 2 3 4 Each contains less than half the edges by definition of s SCC(s) k

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Key Lemma 9 Number of edges in transitive closure of Desc(s) \ (Desc(1…s-1) SCC(s)) is at most 3/4 that of the parent subgraph V Correction for proof of Lemma 10: "vertices strictly between g(v) and v" other than v after

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Open question Are there better parallel algorithms for reachability? E.g. can reachability on a 3-regular digraph be computed in o(√n) time using n processors? Ullman & Yannakakis takes O~(√n) time.

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Acknowledgements & Questions D. Sivakumar Claire Mathieu Maurice Herlihy Glencora Borradaile

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**Extra slides**

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Combining Reachability Queries on subgraphs Subgraph 2Subgraph 1 Sources Super-source

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