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This talk: Organization 3 Graph structures Unweighted, Undirected UnWeighted, Undirected UnWeighted, UnDirected Tools A pproximation R andomness (Things are pretty much the same for unweighted directed graphs)
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1
Shortest Paths in Decremental, Distributed
and Streaming Settings
Danupon Nanongkai KTH Royal Institute of Technology
BIRS, Banff, March 2015
2
This talk• Focus on single-source shortest paths (SSSP)• 3 Settings: Distributed, Decremental,
Streaming• The three settings seem to share some
common features: All we can do is essentially BFS
• Better guess for the right solution by looking at these settings at the same time
*
* There are exceptions
3
This talk: Organization
Graph structures
Unweighted, Undirected
UnWeighted, Undirected
UnWeighted, UnDirected
Tools
Approximation
Randomness
(Things are pretty much the same for unweighted directed graphs)
4
Model Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
? ? ? ?
Semi-Stream(# passes) ? ? ? ?
Decremental(total update time) ? ? ? ?
5
Preliminaries
Part 0
6
Notations
• n = number of nodes• m = number of edges• W = (max weight) / (min weight)• SSSP = single-source shortest paths problem• APSP = all-pairs shortest paths problem
7
Remarks
• polylog n and polylog W are mostly hidden• Some great results may not be mentioned
(sorry!)• If I seem to miss something, please let me
know (thank you!)
8
Introduction
Part 1
9
Distributed Setting (CONGEST)
Part 1.1
10
1
23
4
5 6
1
1
1
1
1
4
3
74
4
Network represented by a weighted graph G with n nodes and diameter D.
n=6D=2
11
1
23
4
5 6
43
6
1
1
1
1
1
1
4
3
74
4
41
1
Nodes know only local information
12
Time complexity “number of days”
13
Days: Exchange O(log n) bit
1Day 1
23
4
5 6
14
Nights: Perform local computation
1
23
4
5 6
1Day
1Night
Assume: Any calculation finishes in one night
15
1Night
Days: Exchange O(log n) bit
2Day 1
23
4
5 6
16
2Day
2Night
Nights: Perform local computation
1
23
4
5 6
17
Finish in t days Time complexity = t
18
Example
s-t distance
19
s
23
4
5 t
1
1
1
1
1
4
3
74
4
Goal: Node t knows distance from s
Distance from s = ?
20
s
23
4
5 t
1
1
1
1
1
4
3
74
4
Distance from s = 4
Goal: Node t knows distance from s
21
s
23
4
5 t
1
1
1
1
1
4
3
74
4
Distance from s = 4 8
2-approximate solution
22
Computing s-t distance can be done in O(D) time by using the
Breadth-First Search (BFS) algorithm.
Unweighted Case
23
s
23
4
5 t
0
Source node sends its distance to neighbors
1Day
24
23
4
5
0
Each node updates its distance
1Day
1Day
1Day
1Night
11
1
s
t
25
23
4
5
0
Nodes tell new knowledge to neighbors
2Day
11
1
s
t
26
23
4
5
0
Each node updates its distance
1Day
1Day
1Day
2Night
11
1
22
s
t
27
This algorithm takes O(D) time
28
(Multi-pass) Streaming Setting
Part 1.2
29
Small RAM
Huge Harddisk
3rd pass
1 2
3 4
(1, 2) (2, 4) (1, 3) (2, 3)
W(n2) space
O(n) space
30
Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
31
Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
32
Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
33
Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
34
Small RAM
Huge Harddisk
2nd pass
(1, 2) (2, 4) (1, 3) (2, 3)
35
Small RAM
Huge Harddisk
3nd pass
(1, 2) (2, 4) (1, 3) (2, 3)
36
Complexity = # of passes
Ideally: (polylog n) passesLimitation: (n polylog n) space
37
Example
s-t distance
38
Huge Harddisk
3rd pass
s 2
3 t
(1, 2) (2, 4) (1, 3) (2, 3)
Small RAM
Initially
0
39
Small RAM
Huge Harddisk
3rd pass
s 2
3 t
(1, 2) (2, 4) (1, 3) (2, 3)
1st pass
0 1
1
40
Small RAM
Huge Harddisk
3rd pass
s 2
3 t
(1, 2) (2, 4) (1, 3) (2, 3)
2st pass
0 1
1
2
41
This algorithm takes O(D)=O(n) passes
42
Decremental Setting
Part 1.3
43
We start with a graph withof n nodes and m edges.
44
Edges are gradually deleted
45
Edges are gradually deleted
46
GoalMaintain some graph property
under edge deletions
47
Total Update Time=
Total time to maintaingraph property after all m deletions
48
Example
s-t distance
49
Goal
Maintain the distance between s and t after every deletions
50
Naive algorithmCompute
Breadth-First Search Tree (BFS)after every deletion
Total update time = O(m2)
51
Better Solution
Dynamic BFS Tree(Even-Shiloach Tree [JACM 1981])
O(m2) O(mn)
52
Algorithm descriptionas nodes talking to each other
53
s
e
b c
f
d
Single-Source Shortest Paths from s
54
s
e
b c
f
d
Every node v maintains its level in the BFS
level=1 level=1 level=1
level=2 level=2
s
e
b c
f
d
Delete (s,b) b connects to a new parent
level=1 level=1 level=1
level=2 level=2
55
s
eb
c
f
dlevel=1 level=1
level=2 level=2level=2
b announces its level change
56
s
e
b c
f
dlevel=1 level=1 level=1
level=2 level=2
s
eb
c
f
dlevel=1 level=1
level=2 level=2level=2
level(b)=2
level(b)=2
f connects to a new parent. e changes level.
57
s
eb
c
f
dlevel=1 level=1
level=2 level=2level=2
s
e
b
c
f
d
level=2level=2
level=3
level=1 level=1
level(b)=2
level(b)=2
Again, e announces level change
58
level(e)=3
s
e
b
c
f
d
level=2level=2
level=3
level=1 level=1
Again, e announces level change
59
s
e
b
c
f
d
level=2level=2
level=3
level=1 level=1This is what we obtain after deleting (s,b)
Even-Shiloach tree can be implemented in such a way that
total update time = number of messages
60
s
eb
c
f
dlevel=1 level=1
level=2 level=2level=2
level(b)=2
level(b)=261
Takes
3 time steps
Even-Shiloach tree can be implemented in such a way that
total update time = number of messages
62
Exercise
Number of messages (thus time complexity) is
O(mD) = O(mn)
Hint
Node v sends degree(v) messages every time level(v) increases.
63
Unweighted, Undirected Graphs
Part 2
64
Unweighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
Stream(# passes)
O(D)
[BFS]
Decremental(total update time)
O(mD)
[BFS]
65
Unweighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
Stream(# passes)
O(D) O(n)
[BFS]
Decremental(total update time)
O(mD) O(mn)
[BFS]
66
Unweighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
W(D)(even for approx)
[Folklore]
see weighted caseStream(# passes)
O(D) O(n)
[BFS]
W(R)for distance R=O(log n),
exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13]
Decremental(total update time)
O(mD) O(mn)
[BFS]
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
67
Unweighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
W(D)(even for approx)
[Folklore]
see weighted caseStream(# passes)
O(D) O(n)
[BFS]
W(R)for distance R=O(log n),
exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13]
Decremental(total update time)
O(mD) O(mn)
[BFS]
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
68
Lower bounds for streaming SSSP
• Feigenbaum et al. SODA’05: computing the set of vertices at distance p from source s in ≤ p/2 passes requires n1+Ω(1/p) space. – Guruswami, Onak CCC’13: Same space lower bound holds
even for (p−1) passes
• Guruswami, Onak, CCC’13: A p passes algorithm requires n1+W(1/p)/pO(1) space to check if dist(s, t) ≤ 2(p + 1) – Superlinear space when p is small
69
Unweighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
W(D)(even for approx)
[Folklore]
see weighted caseStream(# passes)
O(D) O(n)
[BFS]
W(R)for distance R=O(log n),
exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13]
Decremental(total update time)
O(mD) O(mn)
[BFS]
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
W(n)?
70
Hardness for Decremental SSSP
• Roditty, Zwick, ESA’04: – Assume: no combinatorial O(n3-e)-time algorithm for
Boolean Matrix Multiplication – Then: no combinatorial exact decremental SSSP
algorithm with O(mn1-e) total update time• Henzinger et al. STOC’15: – Assume: no combinatorial O(n3-e)-time algorithm for
Online Boolean Matrix-Vector Multiplication– Then: no combinatorial exact decremental SSSP
algorithm with O(mn1-e) total update time
71
Online Boolean Matrix-Vector Multiplication
• Given an (n x n)-matrix M. • Given an n-vector v1.
• Must answer Mv1. • …• Given an n-vector vn.
• Must answer Mvn. • Conjecture: No O(n3-e)-time algorithm• Current best: O(n3/log2 n) [Williams, SODA’07]
72
Unweighted, Undirected SSSP -- ConclusionModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
W(D)(even for approx)
[Folklore]
see weighted caseStream(# passes)
O(D) O(n)
[BFS]
W(R)for distance R=O(log n),
exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13]
Decremental(total update time)
O(mD) O(mn)
[BFS]
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
W(n)?
73
UnWeighted, Undirected Graphs
Part 3
74
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
Stream(# passes)
O(n)[Bellman-Ford]
Decremental(total update time)
O(m2)[trivial]
75
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx[Das Sarma et al STOC’11]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
Decremental(total update time)
O(m2)[trivial]
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
76
W(n1/2+D) lower bound for distributed weighted SSSP
• W(D) is from the unweighted case.• Das Sarma et al. STOC’11:
There exists a family of O(log n)-diameter graphs s.t. poly(n)-approximating dist(s, t) requires W(n1/2) time(Klauck et al. PODC’14: Also hold for quantum distributed algorithms)
77
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx[Das Sarma et al STOC’11]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
Decremental(total update time) O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
?
?
?
78
Related open problems• Bernstein STOC’13: Exists O(mn) time for
decremental exact APSP on undirected graphs? – Exists: O(mn) time (1+e) approximation on
weighted directed graphs– Interesting even for unweighted undirected case– Weighted case: O(mn2) total update time via fully-
dynamic algorithm [Demetrescu, Italiano, STOC’03]
– Unweighted case: O(n3) total update time [Demetrescu, Italiano FOCS’01] [Baswana et al., STOC’02]
79
Related open problems• Bernstein STOC’13: Exists O(mn) time for
decremental exact APSP on undirected graphs?
• One more here: Getting O(mn) for exact weighted SSSP?
• Also: distributed APSP in O(n) time– Known: O(n)-time (1+e)-approximation
• Also from Bernstein: Can we remove log W?
80
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx
O(n1/2+o(1)+D)(1+e)-approx
[Henzinger et al. ‘15]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
O(no(1)) (1+e)-approx
O(n1+o(1)) space
[Henzinger et al.’15]
Decremental(total update time)
O(m2) W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(m1+o(1)) (1+e)-approx
[Henzinger et al. FOCS’14]
81
(1+e)-approximation for weighted undirected case
• Henzinger et al. FOCS’14: (1+e)-approximation decremental SSSP in O(m1+o(1)) total update time– Hidden in o(1): O(1/elog1/2 n)– Heavily rely on randomization
• Henzinger et al.’15: (1+e)-approximation SSSP in– Streaming: no(1) passes and n1+o(1) space– Distributed: n1/2+o(1) time
82
Key subroutine: BFS Algorithms
Hop set
Thorup-Zwick Clusterspreviously used for distance oracles and spanners
Bounded-depth BFS trees from every nodewith special stopping rules
83
Note: 1-pass streaming algorithm
• Feigenbaum et al. [ICALP’04]: A (2k-1)-spanner can be constructed in one pass, O(kn1/k) space– Implies, e.g., O(log n)-approximation 1-pass O(n)-
space algorithm for SSSP
84
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx
O(n1/2+o(1)+D)(1+e)-approx
[Henzinger et al. ‘15]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
O(no(1)) (1+e)-approx
O(n1+o(1)) space
[Henzinger et al.’15]
Decremental(total update time)
O(m2) W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(m1+o(1)) (1+e)-approx
[Henzinger et al. FOCS’14]
?
?
?
85
Open: Eliminate no(1) terms
• E.g. (1+e)-approx O(n polylog n)-space (polylog n)-pass streaming algorihtm for SSSP?
• Exists an (polylog n, e)-hop set of size polylog n?– Known: (no(1), e)-hop set of size n1+o(1)
86
Hop Set
Skip
a
d
ef
c
b
a
d
ef
c
b
Spanner(Sparsify graph)
a
d
ef
c
b
Hopset(Densify graph)
a
d
ef
c
b
Two orthogonal approaches
Hopset [Cohen, JACM’00]
88
(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.
Example (1)
Add shortcuts between every pairInput graph
89Picture from Cohen [JACM’00]
4
a
25
6
Example (1)
Add shortcuts between every pairInput graph
90Picture from Cohen [JACM’00]
4
a
25
6
45
6
Example (1)
Input graph
Picture from Cohen [JACM’00]
4
a
25
6
45
6
a 6
b
91
(1, 0)-hopsetone edge is enoughto get distance no error
Example (2)
Input graph with (5, 0)-hopsetInput graph
92Picture from Cohen [JACM’00]
11
93
Hopset constructions
References (h, e) Size NoteCohen [JACM’00] (polylog n, e) n1+o(1) PRAM alg
Bernstein [FOCS’09] (no(1),e) n1+o(1) Use Thorup-Zwick ClustersStatic O(m) time alg
Henzinger et al. [FOCS’14]
” ” Decremental O(m1+o(1))-time alg
94
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx
O(n1/2+o(1)+D)(1+e)-approx
[Henzinger et al. ‘15]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
O(no(1)) (1+e)-approx
O(n1+o(1)) space
[Henzinger et al.’15]
Decremental(total update time)
O(m2) W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(m1+o(1)) (1+e)-approx
[Henzinger et al. FOCS’14]
95
UnWeighted, Undirected SSSPModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx
O(n1/2+o(1)+D)(1+e)-approx
[Henzinger et al. ‘15]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
O(no(1))
(1+e)-approxO(n1+o(1)) space
[Henzinger et al.’15]
Decremental(total update time)
O(m2) W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(m1+o(1)) (1+e)-approx
[Henzinger et al. FOCS’14]
?
Derandomization Ideas from [Roditty et al., ICALP’05], [Lenzen, Patt-Shamir’15], [Goldberg et al., STOC’87]
96
Related question: Deterministic weighted APSP
• Deterministic decremental (1+e)-approximation O(mn)-time algorithm for weighted APSP
• Known for unweighted APSP [Henzinger et al., FOCS’13] – Derandomized [Roditty, Zwick, FOCS’04]
– Tight [Dor et al, FOCS’96], [Henzinger et al, STOC’15]
• Randomized decremental (1+e)-approximation O(mn)-time algorithm for weighted directed APSP [Bernstein, STOC’13]
97
UnWeighted, Undirected SSSP -- ConclusionModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx
O(n1/2+o(1)+D)
(1+e)-approx
[Henzinger et al. ‘15]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
O(no(1))
(1+e)-approxO(n1+o(1)) space
[Henzinger et al.’15]
Decremental(total update time)
O(m2) W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(m1+o(1)) (1+e)-approx
[Henzinger et al. FOCS’14]
?
?
? ?
?
?
98
UnWeighted, UnDirected Graphs
Part 4
99
UnWeighted, UnDirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(n)[Bellman-Ford]
W(n1/2+D) even for approx
O(n1/2+o(1) +D)(1+e)-approx
O(n1/2D1/2+D)(1+e)-approx
Stream(# passes)
O(n)[Bellman-Ford]
W(log n) any approx (reachability)
[Guruswami,Onak CCC’13[
r-pass n2/r space[trivial]
Decremental(total update time)
O(mn)(1+e)-approx
O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(mn1+o(1)) (1+e)-approx
O(mn0.9) (1+e)-approx
100
Lower bounds for streaming directed SSSP
• Guruswami, Onak, CCC’13:
A p passes algorithm for s-t reachability requires n1+W(1/p)/pO(1) space
(Superlinear space when p is small)
101
UnWeighted, UnDirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(n)[Bellman-Ford]
W(n1/2+D) even for approx
O(n1/2+o(1) +D)(1+e)-approx
O(n1/2D1/2+D)(1+e)-approx
Stream(# passes)
O(n)[Bellman-Ford]
W(log n) any approx (reachability)
[Guruswami,Onak CCC’13[r-pass n2/r space
[trivial]
Decremental(total update time)
O(mn)(1+e)-approx
O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(mn1+o(1)) (1+e)-approx
O(mn0.9) (1+e)-approx
?
?
?
102
UnWeighted, UnDirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(n)[Bellman-Ford]
W(n1/2+D) even for approx
O(n1/2+o(1) +D)(1+e)-approx
O(n1/2D1/2+D)(1+e)-approx
Stream(# passes)
O(n)[Bellman-Ford]
W(log n) any approx (reachability)
[Guruswami,Onak CCC’13[r-pass n2/r space
[trivial]
Decremental(total update time)
O(mn)(1+e)-approx
O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(mn1+o(1)) (1+e)-approx
O(mn0.9) (1+e)-approx
103
UnWeighted, UnDirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(n)[Bellman-Ford]
W(n1/2+D) even for approx
O(n1/2+o(1) +D)(1+e)-approx
O(n1/2D1/2+D)(1+e)-approx
Stream(# passes)
O(n)[Bellman-Ford]
W(log n) any approx (reachability)
[Guruswami,Onak CCC’13[r-pass n2/r space
[trivial]
Decremental(total update time)
O(mn)(1+e)-approx
O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(mn1+o(1)) (1+e)-approx
O(mn0.9) (1+e)-approx
104
Upper Bounds for Directed SSSP
• Nanongkai STOC’14 (implicit): (1+e)-approximation O(n1/2D1/2+D)-time distributed algorithm
• Henzinger et al. STOC’14: (1+e)-approximation decremental algorithm with O(mn0.99) total update time – Recently improve to O(mn0.9) total update time
105
UnWeighted, UnDirected SSSPModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(n)[Bellman-Ford]
W(n1/2+D) even for approx
O(n1/2+o(1) +D)(1+e)-approx
O(n1/2D1/2+D)(1+e)-approx
Stream(# passes)
O(n)[Bellman-Ford]
W(log n) any approx (reachability)
[Guruswami,Onak CCC’13[r-pass n2/r space
[trivial]
Decremental(total update time)
O(mn)(1+e)-approx
O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(mn1+o(1)) (1+e)-approx
O(mn0.9) (1+e)-approx
?
?
?
106
Hop Set for Directed Graphs?
• k-Transitive-Closure Spanner [Thorup WG’92]:– Has the same transitive closure as in the original
graph– Diameter at most k
• There is a n1/2-TC-spanner of size O(n). How efficient can we compute it in various settings?
107
Conjecture• Two parties each gets part of the directed graph.• Conjecture: There exists no communication
protocol that takes r rounds and o(n2/r) communication that can solve s-t shortest path on n-node directed graphs.
• Might be true even for reachability• Will imply a tight lower bound in the streaming
setting• Will imply a non-trivial (perhaps tight) lower
bound in the distributed setting
108
Conclusion
109
Unweighted, Undirected SSSP -- ConclusionModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(D)[BFS]
W(D)(even for approx)
[Folklore]
see weighted caseStream(# passes)
O(D) O(n)
[BFS]
W(R)for distance R=O(log n),
exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13]
Decremental(total update time)
O(mD) O(mn)
[BFS]
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
W(n)?
110
UnWeighted, Undirected SSSP -- ConclusionModel Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
O(D)O(n)[Bellman-Ford]
W(D) W(n1/2+D) even for approx
O(n1/2+o(1)+D)
(1+e)-approx
[Henzinger et al. ‘15]
Stream(# passes)
O(n)[Bellman-Ford]
W(R) for distance R=O(log n), exact only
[Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[
O(no(1))
(1+e)-approxO(n1+o(1)) space
[Henzinger et al.’15]
Decremental(total update time)
O(m2) W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(m1+o(1)) (1+e)-approx
[Henzinger et al. FOCS’14]
?
?
? ?
?
?
111
UnWeighted, UnDirected SSSP -- ConclusionModel Exact alg. Lower bound Approx + Rand
Distributed(# rounds)
O(n)[Bellman-Ford]
W(n1/2+D) even for approx
O(n1/2+o(1) +D)(1+e)-approx
O(n1/2D1/2+D)(1+e)-approx
Stream(# passes)
O(n)[Bellman-Ford]
W(log n) any approx (reachability)
[Guruswami,Onak CCC’13[r-pass n2/r space
[trivial]
Decremental(total update time)
O(mn)(1+e)-approx
O(m2)
W(mn)conditional, exact only
[Roditty, Zwick, ESA’04][Henzinger et al. STOC’15]
O(mn1+o(1)) (1+e)-approx
O(mn0.9) (1+e)-approx
?
?
?
?
?
?
112
Thank you