[PPT] - Challenges in Distributed Shortest Paths Algorithms Danupon PowerPoint Presentation

SLIDE 1

Challenges in Distributed Shortest Paths Algorithms

1

Danupon Nanongkai

KTH Royal Institute of Technology, Sweden

SIROCCO 2016

SLIDE 2

Main focus

s-t Distance

2

Known

(1+e)-approx. in Q(n1/2+D) time

About this talk

Open problem Technical challenge

1. Exact O(n1-e+D) time

Avoid bounded-hop distances!

2. Directed O(n1/2+D) time

Avoid sparse spanner, etc.!

SLIDE 3

Note polylog terms will be hidden most of the time

3

SLIDE 4

Plan

1. Problem & Known Results

4

2. Open Problems
3. Technical Challenges

SLIDE 5

CONGEST Model

5

Part 1.1

SLIDE 6

1 2 3 4 5 6

1 1 1 1 1 4 3 7 4 4

6

Network represented by a weighted graph G with n nodes and hop-diameter D. n=6 D=2

SLIDE 7

1 2 3 4 5 6

4 3 6 1

1 1 1 1 1 4 3 7 4 4 4 1 1

7

Nodes know only local information

SLIDE 8

Time complexity

“number of days”

8

SLIDE 9

Days: Exchange one bit

1

Day

9

1 2 3 4 5 6

SLIDE 10

Nights: Perform local computation

10

1 2 3 4 5 6

1

Day

1

Night

Assume: Any calculation finishes in one night

SLIDE 11

1

Night

Days: Exchange one bit

2

Day

1 2 3 4 5 6

11

SLIDE 12

2

Day

2

Night

Nights: Perform local computation

1 2 3 4 5 6

12

SLIDE 13

Finish in t days  Time complexity = t

13

SLIDE 14

Unweighted s-t distance

14

Part 1.2

SLIDE 15

15

Goal: Node t knows distance from s

s 2 3 4 5 t

Distance from s = ?

SLIDE 16

16

s 2 3 4 5 t

Distance from s = 2

SLIDE 17

O(D) time using Breadth-First Search (BFS) algorithm.

17

There is an W(D) lower bound.

Unweighted Case

SLIDE 18

s 2 3 4 5 t

Source node sends its distance to neighbors

1

Day

18

SLIDE 19

2 3 4 5

Each node updates its distance

1

Day

1

Day

1

Day

1

Night

1 1 1

19

s t

SLIDE 20

2 3 4 5

Nodes tell new knowledge to neighbors

2

Day

1 1 1

20

s t

SLIDE 21

2 3 4 5

Each node updates its distance

1

Day

1

Day

1

Day

2

Night

1 1 1 2 2

21

s t

SLIDE 22

This algorithm takes Q(D) time

22

SLIDE 23

How about weighted graphs?

23

Part 1.3

SLIDE 24

1 2 3 4 5 6

4 3 6 1

1 1 1 1 1 4 3 7 4 4 4 1 1

24

Input: weighted network

Remark: Weights do not affect the communication, edge weights ≤ O(polylog n)

SLIDE 25

s 2 3 4 5 t

1 1 1 1 1 3 7 4 4

25

s-t distance

4

4 7 4

SLIDE 26

s 2 3 4 5 t

1 1 1 1 1 3 7 4 4

26

2-approximation

4

4 7 4

8

SLIDE 27

A naïve solution

27

Aggregate everything into one node. Then solve the problem on that node.

Time = O(# of edges)

(using “pipelining” technique)

SLIDE 28

Can we do better?

28

SLIDE 29

Why distributed s-t distance?

Among active research on distributed

algorithms for basic graph problems

– MST, Connectivity, Matching, etc.

Connection to other distributed algorithmic

problems

– Routing, APSP, Diameter, Eccentricity, Radius, etc.

Provide distributed algorithmic viewpoint

– Complement with data streams, dynamic algorithms, parallel algorithms, etc.

29

SLIDE 30

Known Results for s-t distance

30

Part 1.4

(All results also hold for sing-source distance.)

SLIDE 31

Reference Time Approximation Folklore

W(D) any

31

SLIDE 32

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

32

SLIDE 33

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

Elkin [STOC 2006]

W((n/a)1/2 + D) any a

33

polylog(n/e) factors are hidden

SLIDE 34

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

Elkin [STOC 2006]

W((n/a)1/2 + D) any a

Das Sarma et al [STOC 2011]

Elkin et al. [PODC 2014]

W(n1/2 + D) any a

also quantum

34

polylog(n/e) factors are hidden

SLIDE 35

35

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

Elkin [STOC 2006]

W((n/a)1/2 + D) any a

Das Sarma et al [STOC 2011]

Elkin et al. [PODC 2014]

W(n1/2 + D) any a

also quantum

Lenzen,Patt-Shamir

[STOC 2013]

O(n1/2+e+ D) O(1/e)

polylog(n/e) factors are hidden
Lenzen&Patt-Shamir actually achieve more than computing distances

SLIDE 36

36

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

Elkin [STOC 2006]

W((n/a)1/2 + D) any a

Das Sarma et al [STOC 2011]

Elkin et al. [PODC 2014]

W(n1/2 + D) any a

also quantum

Lenzen,Patt-Shamir

[STOC 2013]

O(n1/2+e+ D) O(1/e) N [STOC 2014] O(n1/2D1/4+ D) 1+e

polylog(n/e) factors are hidden
Lenzen&Patt-Shamir actually achieve more than computing distances

SLIDE 37

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

Elkin [STOC 2006]

W((n/a)1/2 + D) any a

Das Sarma et al [STOC 2011]

Elkin et al. [PODC 2014]

W(n1/2 + D) any a

also quantum

Lenzen,Patt-Shamir

[STOC 2013]

O(n1/2+e+ D) O(1/e) N [STOC 2014] O(n1/2D1/4+ D) 1+e

Henzinger,Krinninger,N

[STOC 2016]

O(n1/2+o(1) + D1+o(1))

(Deterministic)

1+e

37

polylog(n/e) factors are hidden
Lenzen&Patt-Shamir actually achieve more than computing distances

SLIDE 38

Reference Time Approximation Folklore

W(D) any

Bellman&Ford [1950s]

O(n) exact

Elkin [STOC 2006]

W((n/a)1/2 + D) any a

Das Sarma et al [STOC 2011]

Elkin et al. [PODC 2014]

W(n1/2 + D) any a

also quantum

Lenzen,Patt-Shamir

[STOC 2013]

O(n1/2+e+ D) O(1/e) N [STOC 2014] O(n1/2D1/4+ D) 1+e

Henzinger,Krinninger,N

[STOC 2016]

O(n1/2+o(1) + D1+o(1))

(Deterministic)

1+e

Becker, Karrenbauer, Krinninger, Lenzen [2016]

O(n1/2 + D)

(Deterministic)

1+e

38

* *

All previous results except Becker et al. can compute shortest-paths tree

SLIDE 39

Main focus

s-t Distance

39

Known

(1+e)-approx. in Q(n1/2+D) time

Summary of Part 1

Distributed approximate s-t distance are essentially solved.

SLIDE 40

Plan

1. Problem & Known Results

40

2. Open Problems
3. Technical Challenges

SLIDE 41

Exact algorithms

41

Part 2.1

SLIDE 42

Is there a sublinear-time exact algorithm for s-t distance?

Current lower bound: W(n1/2 + D)
(1+e)-approx. algorithms need O(n1/2 + D) time
Exact algorithm: no O(n1-e + D) known

42

SLIDE 43

Exact case also open for many other graph problems.

43

SLIDE 44

Is there a linear-time exact algorithm for all-pairs distances ?

Current lower bound: W(n).
We have linear-time (1+e)-approx. algorithm.

44

SLIDE 45

1 2 3 4 5 6

1 1 1 1 1 4 3 7 4 4

45

? ? ?

All-Pairs Shortest Paths

? ?

Distance from 1, 2, …, 5 = ?

source

SLIDE 46

Directed Case

46

Part 2.2

SLIDE 47

1 2 3 4 5 6

1 1 1 1 1 4 3 7 4 4

47

Directed case

Note: Two-way communication, not affected by weights.

dist(6, 3)=1

1

dist(3, 6)=2

SLIDE 48

48

Directed s-t & single-source distances

Reference Time Approximation N [STOC’14]

O(n1/2D1/2+D) 1+e

Ghaffari, Udwani [PODC’15] O(n1/2D1/4+D)

Reachability Open

O(n1/2+D)-time (any)-approximation algorithm.

SLIDE 49

Congested Cliques

49

Part 2.3

SLIDE 50

50

Congested Clique: The underlying network is fully connected

1 2 3 4 5 6 1 2 3 4 5 6

SLIDE 51

51

s-t distance, congested clique

Reference Time Approximation N [STOC’14]

O(n1/2) exact

Censor-Hillel et al.

[PODC’15]*

O(n1/3) O(n0.15715) exact 1+e

Henzinger,Krinninger,N

[STOC’16]

O(no(1)) 1+e

Becker, Karrenbauer, Krinninger, Lenzen [2016]

polylog(n) 1+e Open: Better exact algorithm? Lower bound?

* Censor-Hillel et al.’s result works for APSP

SLIDE 52

52

All-pairs distances

Reference Time Approximation N [STOC’14]

O(n1/2) 2+e

Censor-Hillel, Kaski, Korhonen, Lenzen, Paz, Suomela [PODC’15]

O(n1/3) O(n0.15715) exact 1+e

Le Gall [DISC’16]

Additional algebraic tools (e.g. determinant)
Applications to, e.g., maximum matching

Connection to matrix multiplication Open:

1. Better exact and approximation algorithm.
2. Explore the power of algebraic techniques on congested cliques.
3. Lower bounds?

SLIDE 53

Lower Bounds on Congested Clique?

Drucker et al [PODC’14]:

Not so easy.
Will imply something big in circuit complexity.

53

SLIDE 54

Other Related Problems

54

Part 2.4

SLIDE 55

1 2 3 4 5 6

Network Diameter = ?

? ? ?

Diameter

?

55

?

SLIDE 56

Algorithm Time Approximation BFS D 2

Holzer et al. [PODC’12]

for small D

W(n) 3/2-e

Holzer et al. [PODC’12] Peleg et al. [ICALP’12]

O(n) exact

Frischknecht et al.

[SODA’12]

W((n/D)1/2+D) 3/2-e

Lenzen-Peleg [PODC’13]

O(n1/2+D) 3/2

Holzer et al. [DISC’14]

O((n/D)1/2+D) 3/2+e

Open (By Holzer)

W(n/D+D) 1+e

Diameter (unweighted)

56

Also: Eccentricity, radius, etc. [Censor-Hillel et al. DISC’16]: Result holds for any D and even for sparse graph

SLIDE 57

57

Algorithm Time Approximation

Holzer et al. [PODC’12]

W(n) 2-e

Becker et al. [2016] O(n1/2 + D)

2+e Open sublinear 2

Diameter (weighted)

Intermediate problem to exact SSSP (Getting a sublinear-time exact algorithm for SSSP will resolve this)

SLIDE 58

1 2 3 4 5 6

T6 T4 T3 T2

Routing

T5

58

T1

small “routing table”

SLIDE 59

Some open problems

Thanks: Christoph Lenzen

Eliminate no(1) term as in the case of s-t

shortest path.

– Techniques from s-t SP usually transfer to the routing problem. – Exception: Becker et al [2016]

Lower bounds on the construction time for

stateful routing.

Further read: Elkin, Neiman [PODC’16] &

Lenzen, Patt-Shamir [STOC’13]

59

SLIDE 60

60

Summary of Part 2

Open problem

1. Exact O(n1-e+D) time
2. Directed O(n1/2+D) time

SLIDE 61

Plan

1. Problem & Known Results

61

2. Open Problems
3. Technical Challenges

SLIDE 62

Framework for approximate s-t shortest paths

62

SLIDE 63

63

1. Input graph
2. Skeleton
3. Sparse Spanner, etc.

Additional work bounded-hop distances

SLIDE 64

64

1. Input graph

SLIDE 65

65

2. Skeleton

SLIDE 66

66

3. Sparse spanner, etc.

SLIDE 67

67

1. Input graph
2. Skeleton

Additional work bounded-hop distances

Challenge for exact case Challenge for directed case

3. Sparse Spanner, etc.

SLIDE 68

Exact case challenge: bounded-hop distance

68

Part 3.1

SLIDE 69

Reference Time Approximation Bellman&Ford [1950s]

O(n) exact

Das Sarma et al [STOC 2011] W(n1/2 + D)

any a

OPEN O(n1-e+D) exact

69

polylog(n/e) factors are hidden

Recall

SLIDE 70

Definition: h-hop distance

disth(u,v) := smallest total weight among u-v

paths containing at most h edges.

70

1 2 3 4 5 6

1 1 1 1 1 4 3 7 4 4

dist(1, 6) = 3 dist1(1, 6) = 4

SLIDE 71

Definition: h-hop distance

disth(u,v) := smallest total weight among u-v

paths containing at most h edges.

k-sources h-hop distances: find disth(si, v) for

all k sources s1, …, sk, and all nodes v.

71

Theorem We can find k-sources (1+e)-approx. h-hop distances in

O(k+h/e) time

SLIDE 72

\begin{technical}

72

Skip

SLIDE 73

Approximating k-sources h-hop distances in the weighted case is as easy as computing a BFS tree on unweighted graphs

73

Key idea: Weight rounding

SLIDE 74

1. Pretend that the graph is unweighted

74

v1 v2 v3

300 400 G:

v0

100

1,000

v1 v2 v3

100

G:

v0

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

3-hop Shortest paths

SLIDE 75

2. Round weight -- ignore small errors

75

v1 v2 v3

301 405 G:

v0

99

1,010

v1 v2 v3

100

G:

v0

3-hop Shortest paths

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

SLIDE 76

Approximating k-sources h-hop distances

1. Pretend that the graph is unweighted.
2. Round weight -- ignore small errors.
3. With appropriate rounding, we get distance

O(h) and (1+e) approximation.

4. Run BFS algorithms from k sources in parallel.

See N [STOC’14] for more details.

76

SLIDE 77

\end{technical}

77

SLIDE 78

78

Theorem (recall) We can find k-sources (1+e)-approx. h-hop distances in

O(k+h/e) time

SLIDE 79

Answer (Lenzen, Patt-Shamir [PODC’15]):

No. o(kh) time is impossible.

79

Question Can we find k-sources (1+e)-approx. exact h-hop distances in

O(k+h) time?

If so, we will be able to solve SSSP exactly in sublinear time and APSP exactly in linear time

SLIDE 80

Challenge for exact computation k-sources h-hop distances – avoid it to get O(n1-e+D) time!

80

SLIDE 81

Directed case challenge: sparse spanner

81

Part 3.2

SLIDE 82

82

Recall: Directed s-t & single-source distances

Reference Time Approximation N [STOC’14]

O(n1/2D1/2+D) 1+e

Ghaffari, Udwani [PODC’15] O(n1/2D1/4+D) Reachability OPEN

O(n1/2+D) 1+e or just reachability

SLIDE 83

83

1. Input graph
2. Skeleton
3. Sparse structure

(spanner/hopset)

Additional work bounded-hop distances

Challenge for directed case

SLIDE 84

Definition: Spanner

p-spanner: Subgraph that preserves distances

with multiplicative error

84

a d e f c b a d e f c b

input graph 2-spanner

SLIDE 85

Computing spanner on distributed networks

Baswana-Sen [Rand. Struct & Alg. 2007]:

(2p-1)-spanner of size O(n1+1/p) in O(p) rounds for any p.

There’s a huge literature on this.

– See, e.g., Pettie [Dist. Comp. 2010]

85

* It was pointed out by Pettie that the size of Baswana-Sen’s spanner is O(kn+(log n)n1+1/k)

SLIDE 86

Exists sparse directed spanner?

86

No:

SLIDE 87

\begin{technical}

87

Skip

SLIDE 88

Definitions

p-spanner: Subgraph that preserves distances

with multiplicative error p

p-emulators: Graph on the same set of

vertices that preserves distances

88

a d e f c b a d e f c b

input graph 2-spanner

a d e f c b

2-emulator

SLIDE 89

Hopset [Cohen, JACM’00]

89

(h,e)-hopset of a network G = (V,E)

is a set E* of new weighted edges such that h-edge paths in H=(V, E∪E*) give (1+ε) approximation to distances in G.

SLIDE 90

Example (1)

Add shortcuts between every pair Input graph

90

Picture from Cohen [JACM’00]

4

a

2 5 6

SLIDE 91

Example (1)

Add shortcuts between every pair Input graph

91

Picture from Cohen [JACM’00]

4

a

2 5 6 4 5 6

SLIDE 92

Example (1)

Input graph

Picture from Cohen [JACM’00]

4

a

2 5 6 4 5 6

a

6

b

92

(1, 0)-hopset

ne edge is enough

to get distance

no error

SLIDE 93

Example (2)

Input graph with (5, 0)-hopset Input graph

93

Picture from Cohen [JACM’00]

11

SLIDE 94

Exists sparse directed emulator/hopset? (No)

94

SLIDE 95

\end{technical}

95

SLIDE 96

Challenge for directed case Can we avoid the use of sparse spanner and related structures?

96

SLIDE 97

Main focus

s-t Shortest Paths

97

Known

(1+e)-approx. in Q(n1/2+D) time

Summary

Open problem Technical challenge

1. Exact O(n1-e+D) time

Avoid bounded-hop distances!

2. Directed O(n1/2+D) time

Avoid sparse spanner, etc.!

SLIDE 98

Thank you

98