distributed verification and hardness of distributed approximation atish das sarma stephan holzer...
Post on 27-Dec-2015
229 Views
Preview:
TRANSCRIPT
1
Distributed Verification and Hardness of Distributed Approximation
Atish Das Sarma Stephan Holzer
Danupon Nanongkai
Gopal Pandurangan David Peleg
WeizmannGoogle ResearchLiah Kor
Roger Wattenhofer
ETH Zurich
U. of Vienna & Georgia Tech
Nanyang Technological University& Brown University
ETH ZurichWeizmann
Amos KormanU. Paris 7
2
PLANResult summary
Techniques Overview
From communication complexity to distributed algo. lower bound
3
Distributed network
4
Distributed network A graph G of n nodes, diameter D
n= 4, D=2
1
2
3
4
5
Main issue: LOCALITY and BANDWIDTH
?
1
2
3
4
42
3
6
Time complexity = number of rounds
1
2
3
4
log n
log n
log nlog n
log nlog n
log n
log nlog n
7
Example: Spanning tree in O(D) time
1
2
3
4
8
Weighted distributed network
?
10
2
1
9
59
5
1
2
3
4
42
3
9
Fundamental problems
• Spanning Tree – Broadcasting, Aggregation, etc• Minimum Spanning Tree – Efficient
broadcasting, leader election, etc. • Shortest path – Routing, etc.• Steiner tree – Multicasting, etc. • Many other graph problems.
10
How fast can we compute distributively?
11
Three points of this work
1. A new generic technique to prove lower bounds of distributed algorithms. – Works for approximation algorithms. – Connection to communication complexity
2. New bounds for many problems. Tight in some cases.
3. A systematic study of distributed verification.
12
Distributed algorithms for the above problems require
W(n1/2+D) time
13
Two main ingredients
1. Verification Approximation2. Connection to communication complexity.
14
ShowcaseMinimum Spanning Tree
15
Time of Distributed AlgorithmsProblems Upper bound Lower bound
Spanning tree (ST)
O(D) W(D)
MST O(D + n1/2) W(D + n1/2)
a-approx. MST W(D + (n /a)1/2)
MST Verification O(D + n1/2) W(D + n1/2)
[trivial] [trivial]
[Garay, Kutten, Peleg FOCS’93] [Peleg, Rubinovich FOCS’99]
[Elkin STOC’04]
[Kor, Korman, Peleg STACS’11][Kor, Korman, Peleg STACS’11]
16
Time of Distributed AlgorithmsProblems Upper bound Lower bound
Spanning tree (ST)
O(D) W(D)
MST O(D + n1/2) W(D + n1/2)
a-approx. MST W(D + (n /a)1/2)
MST Verification O(D + n1/2) W(D + n1/2)
ST Verification O(D + n1/2)
[trivial] [trivial]
[Garay, Kutten, Peleg FOCS’93] [Peleg, Rubinovich FOCS’99]
[Elkin STOC’04]
[Kor, Korman, Peleg STACS’11][Kor, Korman, Peleg STACS’11]
17
Implication of our results
18
Time of Distributed AlgorithmsProblems Upper bound Lower bound
Spanning tree (ST)
O(D) W(D)
MST O(D + n1/2) W(D + n1/2)
a-approx. MST W(D + (n /a)1/2)
MST Verification O(D + n1/2) W(D + n1/2)
ST Verification O(D + n1/2)
[trivial] [trivial]
[Garay, Kutten, Peleg FOCS’93] [Peleg, Rubinovich FOCS’99]
[Elkin STOC’04]
[Kor, Korman, Peleg STACS’11][Kor, Korman, Peleg STACS’11]
W(D + n 1/2)
19
Previous lower bound proofs• Deterministic : Count the number of states.
Argue that the number is not enough. • Randomized: Come up with a good input
distributions.
Our proof• Simple reduction from communication
complexity.• Avoid complication in proving randomized lower
bounds.
20
PLANResult summary
Techniques Overview
From communication complexity to distributed algo. lower bound
21
Approx MST lower bound W(n1/2)
Distributed equality verificationlower bound W(n1/2)
ST verification lower bound W(n1/2)
Distributed equality verificationlower bound W(n1/2)
Direct equality verificationlower bound W(n1/2)
Well-known result in communication complexity
Similar to hardness of TSP
Similar to lower bounds of graph streaming algorithms
Three steps of reductionDistributed AlgorithmsCommunication Complexity
simulationtheorem
22
PLANResult summary
Techniques Overview
From communication complexity to distributed algo. lower bound
23
Communication complexity of EQUALITY
24
How many bits do they have to exchange?
Alice Bobx {0, 1}100 y {0, 1}100
x=y?
Yes, x=yYes, x=y
25
One solution: Alice sends everything ... time=100
Alice Bobx {0, 1}100 y {0, 1}100
x=y?
26
Theorem: Any algorithm needs ≥100 bits
Alice Bobx {0, 1}100 y {0, 1}100
x=y?
27
Distributed time complexity of EQUALITY
28
Alicex {0, 1}100
Boby {0, 1}100
100 green nodes
Alice and Bob are connected by many paths of length 100
∞
29
Alicex {0, 1}100
Boby {0, 1}100
100 green nodes
In each step, one edge can carry one bit on each direction
∞
30
How many steps do they need to check whether “x=y”?
31
Alice Bob
100 green nodes
A: 100 steps because the network diameter is 100
32
Let’s make the diameter smaller
33
Alice Bob
100 green nodes
10 green nodes 10 green nodes
Now the diameter is 30How many steps do we need?
34
Claim: Need > 50 steps.
35
Proof: Assume there is a distributed algorithm A that uses
≤ 50 steps
A
36
Alice Bobx {0, 1}100 y {0, 1}100
A50 bits
x=y x=y
Contradiction
37
Proof: Assume there is a distributed algorithm A that uses
≤ 50 steps
AGoal: Show that Alice & Bob can
use A to compute EQUALITY using 50 bits
38
Alice
x {0, 1}100
Bob
y {0, 1}100A
x=y x=y
?
?
39
Alice Bobx {0, 1}100 y {0, 1}100
Alice’s network Bob’s network
Run A Run AA A
40
A
Alice Bobx {0, 1}100 y {0, 1}100
x y? ?
Alice’s network Bob’s network
0Step
Run A A Run A
41
In step 0, Alice can run A on all machines except Bob’s
42
Alice Bob
x y? ?
1Step
43
Alice Bob
x y? ?
1Step
44
Alice Bob
x y? ?
1Step
?
?
?
? ?
?
?
?
b1
a1
b1 = bit sent by A run on Bob’s machine
45
Alice Bob
x y? ?
1Step
?
?
?
? ?
?
?
?
b1
a1
a1
b1
b1 = bit sent by A from Bob’s machine
keep this keep this
46
Alice Bob
x y? ?
2Step
?
?
?
?
?
?
b2
a2
a2
b2
?
?
?
?
?
?
? ?
b2= bit sent by A from Bob’s machine
47
Alice Bob
x y? ?
3Step
?
?
?
?
?
?
b3
a3
a3
b3
?
?
?
?
?
?
?
?
?
?
?
?
b3 = bit sent by A from Bob’s machine
48
Alice Bob
x y? ?
4Step
?
?
?
?
?
?
b4
a4
a4
b4
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
49
Alice Bob
x y? ?
5Step
?
?
?
?
?
?
b5
a5
a5
b5
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
50
Alice Bob
x y? ?
50Step
?
?
?
?
?
?
b50
a50
a50
b50
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
A finishes
x=yx=y
x=yx=y
51
Alice Bobx {0, 1}100 y {0, 1}100
A50 bits
x=y x=y
Contradiction
52
Remarks
53
1. By replacing 100 by n1/2, we can reduce distributed EQUALITY to
ST verification
x=y? Do red edges form a spanning tree?
54
2. Reduce diameter ...
55
Alice Bob
n1/2 p
aths
n1/2 green nodes
n1/4 orange nodes
n1/4 green nodes
Diameter = n1/4
56
Alice Bob
Diameter = log nn1/
2 pat
hs
n1/2 green nodes
57
3. Getting randomized lower bound
• EQAULITY does not give randomized lower bound.
• Simulation theorem holds for all functions.• Reduce from communication complexity of
HAMILTONIAN CYCLE [Spieker, Raz FOCS’93]
58
Recap
1. A new generic technique to prove lower bounds of distributed algorithms. – Works for approximation algorithms.
2. New bounds for many problems. Tight in some cases.
3. A systematic study of distributed verification.
59
Open problems
• Tight bounds of shortest paths, mincut, minimum routing cost spanning tree, Steiner forest, ...
• Lower bounds of algorithms on complete graphs?
• Complexity theory of distributed computing?
60
Thank you!
Related talk at PODCToday 5:10pm
“A tight unconditional lower bound on distributed random walk computation”
top related