distributed verification and hardness of distributed approximation atish das sarma stephan holzer...
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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
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PLANResult summary
Techniques Overview
From communication complexity to distributed algo. lower bound
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Distributed network
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Distributed network A graph G of n nodes, diameter D
n= 4, D=2
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Main issue: LOCALITY and BANDWIDTH
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Time complexity = number of rounds
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log n
log n
log nlog n
log nlog n
log n
log nlog n
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Example: Spanning tree in O(D) time
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Weighted distributed network
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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.
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How fast can we compute distributively?
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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.
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Distributed algorithms for the above problems require
W(n1/2+D) time
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Two main ingredients
1. Verification Approximation2. Connection to communication complexity.
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ShowcaseMinimum Spanning Tree
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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]
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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]
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Implication of our results
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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)
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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.
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PLANResult summary
Techniques Overview
From communication complexity to distributed algo. lower bound
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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
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PLANResult summary
Techniques Overview
From communication complexity to distributed algo. lower bound
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Communication complexity of EQUALITY
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How many bits do they have to exchange?
Alice Bobx {0, 1}100 y {0, 1}100
x=y?
Yes, x=yYes, x=y
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One solution: Alice sends everything ... time=100
Alice Bobx {0, 1}100 y {0, 1}100
x=y?
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Theorem: Any algorithm needs ≥100 bits
Alice Bobx {0, 1}100 y {0, 1}100
x=y?
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Distributed time complexity of EQUALITY
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Alicex {0, 1}100
Boby {0, 1}100
100 green nodes
Alice and Bob are connected by many paths of length 100
∞
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Alicex {0, 1}100
Boby {0, 1}100
100 green nodes
In each step, one edge can carry one bit on each direction
∞
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How many steps do they need to check whether “x=y”?
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Alice Bob
100 green nodes
A: 100 steps because the network diameter is 100
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Let’s make the diameter smaller
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Alice Bob
100 green nodes
10 green nodes 10 green nodes
Now the diameter is 30How many steps do we need?
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Claim: Need > 50 steps.
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Proof: Assume there is a distributed algorithm A that uses
≤ 50 steps
A
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Alice Bobx {0, 1}100 y {0, 1}100
A50 bits
x=y x=y
Contradiction
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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
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Alice
x {0, 1}100
Bob
y {0, 1}100A
x=y x=y
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Alice Bobx {0, 1}100 y {0, 1}100
Alice’s network Bob’s network
Run A Run AA A
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A
Alice Bobx {0, 1}100 y {0, 1}100
x y? ?
Alice’s network Bob’s network
0Step
Run A A Run A
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In step 0, Alice can run A on all machines except Bob’s
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Alice Bob
x y? ?
1Step
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Alice Bob
x y? ?
1Step
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Alice Bob
x y? ?
1Step
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b1
a1
b1 = bit sent by A run on Bob’s machine
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Alice Bob
x y? ?
1Step
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b1
a1
a1
b1
b1 = bit sent by A from Bob’s machine
keep this keep this
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Alice Bob
x y? ?
2Step
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b2
a2
a2
b2
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b2= bit sent by A from Bob’s machine
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Alice Bob
x y? ?
3Step
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b3
a3
a3
b3
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b3 = bit sent by A from Bob’s machine
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Alice Bob
x y? ?
4Step
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b4
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Alice Bob
x y? ?
5Step
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b5
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Alice Bob
x y? ?
50Step
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b50
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A finishes
x=yx=y
x=yx=y
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Alice Bobx {0, 1}100 y {0, 1}100
A50 bits
x=y x=y
Contradiction
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Remarks
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1. By replacing 100 by n1/2, we can reduce distributed EQUALITY to
ST verification
x=y? Do red edges form a spanning tree?
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2. Reduce diameter ...
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Alice Bob
n1/2 p
aths
n1/2 green nodes
n1/4 orange nodes
n1/4 green nodes
Diameter = n1/4
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Alice Bob
Diameter = log nn1/
2 pat
hs
n1/2 green nodes
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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]
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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.
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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?
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Thank you!
Related talk at PODCToday 5:10pm
“A tight unconditional lower bound on distributed random walk computation”