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Distributed Directed SSSP in Sublinear Time Jason Li Carnegie Mellon University Joint work with Mohsen Ghaffari (ETH Zurich) STOC 2018 Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time Distributed

  1. Distributed Directed SSSP in Sublinear Time Jason Li Carnegie Mellon University Joint work with Mohsen Ghaffari (ETH Zurich) STOC 2018 Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  2. Distributed Computing, CONGEST Model Distributed Model Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  3. Distributed Computing, CONGEST Model Distributed Model Network graph Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  4. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  5. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  6. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  7. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Every node sends an O ( log n ) -bit message to each neighbor Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  8. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Every node sends an O ( log n ) -bit message to each neighbor The running time is the number of rounds Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  9. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Every node sends an O ( log n ) -bit message to each neighbor The running time is the number of rounds SSSP Problem Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  10. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Every node sends an O ( log n ) -bit message to each neighbor The running time is the number of rounds SSSP Problem Input graph and network graph have same edges Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  11. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Every node sends an O ( log n ) -bit message to each neighbor The running time is the number of rounds SSSP Problem Input graph and network graph have same edges Beginning: every node knows weights of its edges Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  12. Distributed Computing, CONGEST Model Distributed Model Network graph Vertices are called nodes Algorithm runs in rounds. In each round: Every node performs unbounded local computation Every node sends an O ( log n ) -bit message to each neighbor The running time is the number of rounds SSSP Problem Input graph and network graph have same edges Beginning: every node knows weights of its edges End: every node knows its distance from source Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  13. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  14. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  15. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Unweighted, undirected: BFS in D rounds Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  16. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Unweighted, undirected: BFS in D rounds Weighted, undirected? Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  17. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Unweighted, undirected: BFS in D rounds Weighted, undirected? BFS, Bellman-Ford? Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  18. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Unweighted, undirected: BFS in D rounds Weighted, undirected? BFS, Bellman-Ford? • s • • Blue edges: weight 1 Red edges: weight n 2 • • • • • • • • • Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  19. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Unweighted, undirected: BFS in D rounds Weighted, undirected? BFS, Bellman-Ford? • s • • Blue edges: weight 1 Red edges: weight n 2 • • • • • • • • • Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  20. Distributed SSSP Trivial n − 1 upper bound (Bellman-Ford) Trivial Ω( D ) lower bound Unweighted, undirected: BFS in D rounds Weighted, undirected? BFS, Bellman-Ford all Θ( n ) time on a D = 2 graph! • s • • Blue edges: weight 1 • • Red edges: weight n 2 • • • • • • • Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  21. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  22. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  23. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  24. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected O ( n 1 / 2 + D ) undirected [BKKL17, DISC’17] ˜ Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  25. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected O ( n 1 / 2 + D ) undirected [BKKL17, DISC’17] ˜ O ( n 1 / 2 + D ) lower bound [SHK+11, STOC’11] ˜ Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  26. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected O ( n 1 / 2 + D ) undirected [BKKL17, DISC’17] ˜ O ( n 1 / 2 + D ) lower bound [SHK+11, STOC’11] ˜ Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  27. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected O ( n 1 / 2 + D ) undirected [BKKL17, DISC’17] ˜ O ( n 1 / 2 + D ) lower bound [SHK+11, STOC’11] ˜ Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  28. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected O ( n 1 / 2 + D ) undirected [BKKL17, DISC’17] ˜ O ( n 1 / 2 + D ) lower bound [SHK+11, STOC’11] ˜ Elkin, 2004: Exact SSSP in sublinear time? Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

  29. Prior Work Elkin, 2004: Approximate SSSP in sublinear time? O ( n 1 / 2 D 1 / 4 + D ) undirected, slightly [Nan14, STOC’14] ˜ worse for directed [HKN16, STOC’16] O ( n 1 / 2 + o ( 1 ) + n o ( 1 ) D ) undirected O ( n 1 / 2 + D ) undirected [BKKL17, DISC’17] ˜ O ( n 1 / 2 + D ) lower bound [SHK+11, STOC’11] ˜ Elkin, 2004: Exact SSSP in sublinear time? [ Elk17, STOC’17] ˜ O ( max { n 5 / 6 , n 2 / 3 D 1 / 3 } ) undirected Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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