Finding Strongly Connected Components and Topological Sort in

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

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

Previous Results for TS, SCC and Reachability • Assume sparse graph with n vertices using 1 ≤ p ≤ n 4/3 processors: Coppersmith and Winograd ‘ 87 T. Spencer ‘ 97 Ullman & Yannakakis ‘ 91 Runtime (ignore logs) n 2. 38…/p n/p 1/3 n/p 1/2 Problems All Reachability (In Transactions of Information Processing Society of Japan ’ 99 ’ 04, Akio, Masahiro and Ryozo claim runtime n/p for TS & SCC) • Question: is reachability fundamentally easier to parallelize than SCC and TS?

Answer: no (up to logs) • Our main result: a reduction of SCC and TS to O(log 2 n) reachability queries This work Coppersmith et al. Spencer Ullman et al. Runtime n/p 1/2 n 2. 38/p Problems TS and SCC All n/p 1/3 n/p 1/2 All Reachability • Remainder of talk focuses on SCC problem

A simple SCC algorithm • Choose random vertex s V s • 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) SCC(s) Desc(s) (Similar to [Coppersmith, Fleischer, Hendrickson and Pinar ’ 05]) V Desc(s)

High-runtime instance s Desc(s)

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( )

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

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) 3 1 2 Desc(1…s-1) Desc(s) s= 4

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

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

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.

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

**Extra slides**

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