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Distributed Directed SSSP in Sublinear Time Jason Li Carnegie - - PowerPoint PPT Presentation

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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

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SLIDE 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

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SLIDE 2

Distributed Computing, CONGEST Model

Distributed Model

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 3

Distributed Computing, CONGEST Model

Distributed Model

Network graph

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 13

Distributed SSSP

Trivial n − 1 upper bound (Bellman-Ford)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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 n2

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 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 n2

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 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 n2

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 21

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time?

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 22

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 23

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 24

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 25

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 26

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 27

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 28

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound Elkin, 2004: Exact SSSP in sublinear time?

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 29

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound Elkin, 2004: Exact SSSP in sublinear time? [Elk17, STOC’17] ˜ O(max{n5/6, n2/3D1/3}) undirected

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 30

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound Elkin, 2004: Exact SSSP in sublinear time? [Elk17, STOC’17] ˜ O(max{n5/6, n2/3D1/3}) undirected This talk: ˜ O(n3/4D1/4) directed

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 31

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound Elkin, 2004: Exact SSSP in sublinear time? [Elk17, STOC’17] ˜ O(max{n5/6, n2/3D1/3}) undirected This talk: ˜ O(n3/4D1/4) directed [KN18, independent] ˜ O(n1/2D1/2) directed

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 32

Prior Work

Elkin, 2004: Approximate SSSP in sublinear time? [Nan14, STOC’14] ˜ O(n1/2D1/4 + D) undirected, slightly worse for directed [HKN16, STOC’16] O(n1/2+o(1) + no(1)D) undirected [BKKL17, DISC’17] ˜ O(n1/2 + D) undirected [SHK+11, STOC’11] ˜ O(n1/2 + D) lower bound Elkin, 2004: Exact SSSP in sublinear time? [Elk17, STOC’17] ˜ O(max{n5/6, n2/3D1/3}) undirected This talk: ˜ O(n3/4D1/4) directed [KN18, independent] ˜ O(n1/2D1/2) directed [HNS17, FOCS’17] ˜ O(n5/4) time APSP on directed graphs

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 33

Sampling Centers

[UY91]: Sample a set of centers

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 34

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 35

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 36

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 37

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s t

  • • • • • • • • • • • •
  • O(n/k · log n)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 38

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s t

  • • • • • • • • • • • •
  • O(n/k · log n)

Proof?

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-39
SLIDE 39

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s t

  • • • • • • • • • • • •
  • • • • • •
  • O(n/k · log n)

Θ(n/k · log n)

  • Proof?

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-40
SLIDE 40

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s t

  • • • • • • • • • • • •
  • • • • • •
  • O(n/k · log n)

Θ(n/k · log n)

  • Proof?

Pr[ no centers within block of C · n/k · log n ] = (1 − k/n)C∙n/k∙log n ≈ n−C

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-41
SLIDE 41

Sampling Centers

[UY91]: Sample a set of centers Each node declares itself a center w.p. k/n (think k = √n) Chernoff: w.h.p. Θ(k) centers Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s t

  • • • • • • • • • • • •
  • • • • • •
  • O(n/k · log n)

Θ(n/k · log n)

  • Proof?

Pr[ no centers within block of C · n/k · log n ] = (1 − k/n)C∙n/k∙log n ≈ n−C Union bound: all O(n) values of t, all O(n) positions

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-42
SLIDE 42

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-43
SLIDE 43

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • Joint work with Mohsen Ghaffari (ETH Zurich)

Distributed Directed SSSP in Sublinear Time

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SLIDE 44

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • 1

2 1 2 2 2 2 Connect centers c1, c2 if their SP has ˜ O(n/k) hops

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-45
SLIDE 45

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • 1

2 1 2 2 2 2 Connect centers c1, c2 if their SP has ˜ O(n/k) hops

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-46
SLIDE 46

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • 1

2 1 2 2 2 2 Connect centers c1, c2 if their SP has ˜ O(n/k) hops Skeleton graph G′

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-47
SLIDE 47

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • 1

2 1 2 2 2 2 Connect centers c1, c2 if their SP has ˜ O(n/k) hops Skeleton graph G′

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-48
SLIDE 48

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • 1

2 1 2 2 2 2 Connect centers c1, c2 if their SP has ˜ O(n/k) hops Skeleton graph G′ Claim: ∀t, exists shortest s → t path in G ∪ G′ using ˜ O(n/k + k) hops

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-49
SLIDE 49

Sampling Centers

Key property: for all nodes t, exists shortest s → t path with centers spaced at most O(n/k · log n) nodes apart s

  • 1

2 1 2 2 2 2 Connect centers c1, c2 if their SP has ˜ O(n/k) hops Skeleton graph G′ Claim: ∀t, exists shortest s → t path in G ∪ G′ using ˜ O(n/k + k) hops G′ shortcuts the graph G

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 50

Sampling Centers

  • s
  • Blue edges: weight 1

Red edges: weight n2

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-51
SLIDE 51

Sampling Centers

  • s
  • Blue edges: weight 1

Red edges: weight n2

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-52
SLIDE 52

ShortRange [HNS17]

s

  • •
  • Joint work with Mohsen Ghaffari (ETH Zurich)

Distributed Directed SSSP in Sublinear Time

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SLIDE 53

ShortRange [HNS17]

s

  • •
  • Joint work with Mohsen Ghaffari (ETH Zurich)

Distributed Directed SSSP in Sublinear Time

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SLIDE 54

ShortRange [HNS17]

s

  • •
  • Purple edges divided into blue edges (d(c, c′) ≤ ℓ) and red

edges (d(c, c′) > ℓ).

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-55
SLIDE 55

ShortRange [HNS17]

s

  • •
  • Purple edges divided into blue edges (d(c, c′) ≤ ℓ) and red

edges (d(c, c′) > ℓ). ShortRange [HNS17]: fast distributed algorithm, computes blue edges but not red.

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

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SLIDE 56

ShortRange [HNS17]

s

  • •
  • Purple edges divided into blue edges (d(c, c′) ≤ ℓ) and red

edges (d(c, c′) > ℓ). ShortRange [HNS17]: fast distributed algorithm, computes blue edges but not red. Our contribution: compute red edges.

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-57
SLIDE 57

Our Contribution: Bucketing

Red edges: ≤ h hops but distance > ℓ s

  • c1
  • c2

· · · ∈ C ∈ C

  • ≤ h hops

> ℓ distance · · ·

  • t

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-58
SLIDE 58

Our Contribution: Bucketing

Red edges: ≤ h hops but distance > ℓ s

  • c1
  • c2

· · · ∈ C ∈ C

  • ≤ h hops

> ℓ distance · · ·

  • t

B-F works for any distance, so try B-F?

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-59
SLIDE 59

Our Contribution: Bucketing

Red edges: ≤ h hops but distance > ℓ s

  • c1
  • c2

· · · ∈ C ∈ C

  • ≤ h hops

> ℓ distance · · ·

  • t

B-F works for any distance, so try B-F? Idea: bucket the nodes, single B-F for each bucket

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-60
SLIDE 60

Our Contribution: Bucketing

Red edges: ≤ h hops but distance > ℓ s

  • c1
  • c2

· · · ∈ C ∈ C

  • ≤ h hops

> ℓ distance · · ·

  • t

B-F works for any distance, so try B-F? Idea: bucket the nodes, single B-F for each bucket

Ensure: Red edges relevant to SSSP must connect different buckets

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-61
SLIDE 61

Our Contribution: Bucketing

Red edges: ≤ h hops but distance > ℓ s

  • c1
  • c2

· · · ∈ C ∈ C

  • ≤ h hops

> ℓ distance · · ·

  • t

B-F works for any distance, so try B-F? Idea: bucket the nodes, single B-F for each bucket

Ensure: Red edges relevant to SSSP must connect different buckets

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-62
SLIDE 62

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-63
SLIDE 63

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-64
SLIDE 64

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

Blue edges: ≤ h hops and distance ≤ ℓ, known Red edges: ≤ h hops and distance > ℓ, unknown

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-65
SLIDE 65

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-66
SLIDE 66

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-67
SLIDE 67

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-68
SLIDE 68

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-69
SLIDE 69

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F B-F B-F B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-70
SLIDE 70

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F B-F B-F B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-71
SLIDE 71

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F B-F B-F B-F B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-72
SLIDE 72

Processing Buckets

Process buckets in increasing order, using blue edges Run B-F to depth h after finishing distances in one bucket s

  • •
  • B1

B2 B3 B5 B4

  • Blue edges: ≤ h hops and distance ≤ ℓ, known

Red edges: ≤ h hops and distance > ℓ, unknown B-F B-F B-F B-F B-F

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-73
SLIDE 73

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-74
SLIDE 74

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-75
SLIDE 75

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

3

ShortRange(centers c, ≤ h hops, distance ≤ ℓ)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-76
SLIDE 76

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

3

ShortRange(centers c, ≤ h hops, distance ≤ ℓ)

4

for each bucket in order do

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-77
SLIDE 77

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

3

ShortRange(centers c, ≤ h hops, distance ≤ ℓ)

4

for each bucket in order do

1

Process bucket using blue edges

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-78
SLIDE 78

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

3

ShortRange(centers c, ≤ h hops, distance ≤ ℓ)

4

for each bucket in order do

1

Process bucket using blue edges

2

Run B-F for h rounds

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-79
SLIDE 79

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

3

ShortRange(centers c, ≤ h hops, distance ≤ ℓ)

4

for each bucket in order do

1

Process bucket using blue edges

2

Run B-F for h rounds

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-80
SLIDE 80

Entire Algorithm

Parameters k, ℓ

1

Every node joins center w.p. k/n

2

h ← Θ(n log n/k)

3

ShortRange(centers c, ≤ h hops, distance ≤ ℓ)

4

for each bucket in order do

1

Process bucket using blue edges

2

Run B-F for h rounds

Running time: optimize k, ℓ → ˜ O(n3/4D1/4) time

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-81
SLIDE 81

Open Questions

Even approximate SSSP on directed graphs still unresolved

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-82
SLIDE 82

Open Questions

Even approximate SSSP on directed graphs still unresolved

Beat ˜ O(min{n2/3, n1/2D1/2}) for (1 + ǫ)-approximate

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-83
SLIDE 83

Open Questions

Even approximate SSSP on directed graphs still unresolved

Beat ˜ O(min{n2/3, n1/2D1/2}) for (1 + ǫ)-approximate

Exact SSSP

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-84
SLIDE 84

Open Questions

Even approximate SSSP on directed graphs still unresolved

Beat ˜ O(min{n2/3, n1/2D1/2}) for (1 + ǫ)-approximate

Exact SSSP

˜ O(n1/2D1/2) [KN18] is optimal for D = polylog(n)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-85
SLIDE 85

Open Questions

Even approximate SSSP on directed graphs still unresolved

Beat ˜ O(min{n2/3, n1/2D1/2}) for (1 + ǫ)-approximate

Exact SSSP

˜ O(n1/2D1/2) [KN18] is optimal for D = polylog(n) Lower bound: ˜ O(√n + D)

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-86
SLIDE 86

Open Questions

Even approximate SSSP on directed graphs still unresolved

Beat ˜ O(min{n2/3, n1/2D1/2}) for (1 + ǫ)-approximate

Exact SSSP

˜ O(n1/2D1/2) [KN18] is optimal for D = polylog(n) Lower bound: ˜ O(√n + D) For higher D, [KN18] get ˜ O(√nD1/4 + n3/5) time

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time

slide-87
SLIDE 87

Open Questions

Even approximate SSSP on directed graphs still unresolved

Beat ˜ O(min{n2/3, n1/2D1/2}) for (1 + ǫ)-approximate

Exact SSSP

˜ O(n1/2D1/2) [KN18] is optimal for D = polylog(n) Lower bound: ˜ O(√n + D) For higher D, [KN18] get ˜ O(√nD1/4 + n3/5) time

Joint work with Mohsen Ghaffari (ETH Zurich) Distributed Directed SSSP in Sublinear Time