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![Page 1: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/1.jpg)
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Shortest Paths in Decremental, Distributed
and Streaming Settings
Danupon Nanongkai KTH Royal Institute of Technology
BIRS, Banff, March 2015
![Page 2: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/2.jpg)
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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
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This talk: Organization
Graph structures
Unweighted, Undirected
UnWeighted, Undirected
UnWeighted, UnDirected
Tools
Approximation
Randomness
(Things are pretty much the same for unweighted directed graphs)
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Model Exact alg. Lower bound Approx + Rand Approx + Rand
Distributed(# rounds)
? ? ? ?
Semi-Stream(# passes) ? ? ? ?
Decremental(total update time) ? ? ? ?
![Page 5: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/5.jpg)
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Preliminaries
Part 0
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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
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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!)
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Introduction
Part 1
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Distributed Setting (CONGEST)
Part 1.1
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Network represented by a weighted graph G with n nodes and diameter D.
n=6D=2
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1
Nodes know only local information
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Time complexity “number of days”
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Days: Exchange O(log n) bit
1Day 1
23
4
5 6
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Nights: Perform local computation
1
23
4
5 6
1Day
1Night
Assume: Any calculation finishes in one night
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1Night
Days: Exchange O(log n) bit
2Day 1
23
4
5 6
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2Day
2Night
Nights: Perform local computation
1
23
4
5 6
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Finish in t days Time complexity = t
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Example
s-t distance
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s
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5 t
1
1
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Goal: Node t knows distance from s
Distance from s = ?
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s
23
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5 t
1
1
1
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Distance from s = 4
Goal: Node t knows distance from s
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s
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5 t
1
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Distance from s = 4 8
2-approximate solution
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Computing s-t distance can be done in O(D) time by using the
Breadth-First Search (BFS) algorithm.
Unweighted Case
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s
23
4
5 t
0
Source node sends its distance to neighbors
1Day
![Page 24: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/24.jpg)
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23
4
5
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Each node updates its distance
1Day
1Day
1Day
1Night
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1
s
t
![Page 25: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/25.jpg)
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Nodes tell new knowledge to neighbors
2Day
11
1
s
t
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23
4
5
0
Each node updates its distance
1Day
1Day
1Day
2Night
11
1
22
s
t
![Page 27: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/27.jpg)
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This algorithm takes O(D) time
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(Multi-pass) Streaming Setting
Part 1.2
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Small RAM
Huge Harddisk
3rd pass
1 2
3 4
(1, 2) (2, 4) (1, 3) (2, 3)
W(n2) space
O(n) space
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Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
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Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
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Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
![Page 33: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/33.jpg)
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Small RAM
Huge Harddisk(1, 2) (2, 4) (1, 3) (2, 3)
![Page 34: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/34.jpg)
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Small RAM
Huge Harddisk
2nd pass
(1, 2) (2, 4) (1, 3) (2, 3)
![Page 35: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/35.jpg)
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Small RAM
Huge Harddisk
3nd pass
(1, 2) (2, 4) (1, 3) (2, 3)
![Page 36: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/36.jpg)
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Complexity = # of passes
Ideally: (polylog n) passesLimitation: (n polylog n) space
![Page 37: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/37.jpg)
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Example
s-t distance
![Page 38: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/38.jpg)
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Huge Harddisk
3rd pass
s 2
3 t
(1, 2) (2, 4) (1, 3) (2, 3)
Small RAM
Initially
0
![Page 39: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/39.jpg)
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Small RAM
Huge Harddisk
3rd pass
s 2
3 t
(1, 2) (2, 4) (1, 3) (2, 3)
1st pass
0 1
1
![Page 40: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/40.jpg)
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Small RAM
Huge Harddisk
3rd pass
s 2
3 t
(1, 2) (2, 4) (1, 3) (2, 3)
2st pass
0 1
1
2
![Page 41: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/41.jpg)
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This algorithm takes O(D)=O(n) passes
![Page 42: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/42.jpg)
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Decremental Setting
Part 1.3
![Page 43: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/43.jpg)
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We start with a graph withof n nodes and m edges.
![Page 44: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/44.jpg)
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Edges are gradually deleted
![Page 45: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/45.jpg)
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Edges are gradually deleted
![Page 46: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/46.jpg)
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GoalMaintain some graph property
under edge deletions
![Page 47: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/47.jpg)
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Total Update Time=
Total time to maintaingraph property after all m deletions
![Page 48: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/48.jpg)
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Example
s-t distance
![Page 49: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/49.jpg)
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Goal
Maintain the distance between s and t after every deletions
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Naive algorithmCompute
Breadth-First Search Tree (BFS)after every deletion
Total update time = O(m2)
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Better Solution
Dynamic BFS Tree(Even-Shiloach Tree [JACM 1981])
O(m2) O(mn)
![Page 52: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/52.jpg)
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Algorithm descriptionas nodes talking to each other
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s
e
b c
f
d
Single-Source Shortest Paths from s
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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
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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
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s
eb
c
f
dlevel=1 level=1
level=2 level=2level=2
![Page 56: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/56.jpg)
b announces its level change
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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
![Page 57: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/57.jpg)
f connects to a new parent. e changes level.
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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
![Page 58: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/58.jpg)
Again, e announces level change
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level(e)=3
s
e
b
c
f
d
level=2level=2
level=3
level=1 level=1
![Page 59: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/59.jpg)
Again, e announces level change
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s
e
b
c
f
d
level=2level=2
level=3
level=1 level=1This is what we obtain after deleting (s,b)
![Page 60: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/60.jpg)
Even-Shiloach tree can be implemented in such a way that
total update time = number of messages
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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
![Page 62: Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.](https://reader035.fdocuments.in/reader035/viewer/2022062523/5a4d1b107f8b9ab05998e835/html5/thumbnails/62.jpg)
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Exercise
Number of messages (thus time complexity) is
O(mD) = O(mn)
Hint
Node v sends degree(v) messages every time level(v) increases.
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63
Unweighted, Undirected Graphs
Part 2
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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]
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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]
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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]
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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]
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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
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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)?
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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
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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]
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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)?
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73
UnWeighted, Undirected Graphs
Part 3
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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]
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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]
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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)
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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]
?
?
?
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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]
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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?
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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]
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(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
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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
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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
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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]
?
?
?
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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)
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Hop Set
Skip
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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
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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.
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Example (1)
Add shortcuts between every pairInput graph
89Picture from Cohen [JACM’00]
4
a
25
6
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Example (1)
Add shortcuts between every pairInput graph
90Picture from Cohen [JACM’00]
4
a
25
6
45
6
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Example (1)
Input graph
Picture from Cohen [JACM’00]
4
a
25
6
45
6
a 6
b
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(1, 0)-hopsetone edge is enoughto get distance no error
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Example (2)
Input graph with (5, 0)-hopsetInput graph
92Picture from Cohen [JACM’00]
11
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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
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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]
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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]
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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]
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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]
?
?
? ?
?
?
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UnWeighted, UnDirected Graphs
Part 4
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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
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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)
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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
?
?
?
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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
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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
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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
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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
?
?
?
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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?
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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
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108
Conclusion
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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)?
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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]
?
?
? ?
?
?
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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
?
?
?
?
?
?
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112
Thank you