Local Checkability, No Strings Attached Klaus-Tycho Frster , Thomas - - PowerPoint PPT Presentation

ETH Zurich – Distributed Computing – www.disco.ethz.ch

Local Checkability, No Strings Attached

Klaus-Tycho Förster, Thomas Lüdi, Jochen Seidel, Roger Wattenhofer January 06, 2016 @ ICDCN 2016 - Singapore

Deciding vs Checking

Prove

Deciding vs Checking

Prove Verify

Complexity Theory

P NP

Prove

In polynomial time

Verify

In polynomial time

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝑡 − 𝑢 reachability
• Conclusion

Let's get Distributed

• Is 𝑜 even?
• Ω(𝑜) rounds, even with unique identifiers in the
• model

Let's get Distributed

• Is 𝑜 even?
• Ω(𝑜) rounds, even with unique identifiers in the
• model
• rover assigns 1 bit

1 1 1

Let's get Distributed

• Is 𝑜 even?
• Θ(𝑜) rounds in the
• model
• rover assigns 1 bit ->

erify in 1 round 1 1 1

Yes Yes Yes Yes Yes Yes

Let's get Distributed

• Is 𝑜 even?
• Θ(𝑜) rounds in the
• model
• rover assigns 1 bit ->

erify in 1 round

• Other way to think of it: 1 bit of non-determinism
• General question: How many bits necessary/sufficient?

1 1 1

Yes Yes Yes Yes Yes Yes

Accepting a proof

• Every node outputs Yes -> Proof accepted
• One node outputs No -> Proof rejected

1 1 1

Yes Yes Yes Yes Yes Yes

Accepting a proof

• Every node outputs Yes -> Proof accepted
• One node outputs No -> Proof rejected

– rover chose the wrong proof

1 1 1 1

Yes Yes No Yes No No

Accepting a proof

• Every node outputs Yes -> Proof accepted
• One node outputs No -> Proof rejected

– rover chose the wrong proof – Property does not hold

1 1 1

Yes Yes Yes No

1

Yes Yes No

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝑡 − 𝑢 reachability
• Conclusion

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝑡 − 𝑢 reachability
• Conclusion

Some Related Work

• [Naor and Stockmeyer, STOC 1993]:

What can be computed locally?

• [Göös and Suomela, PODC 2011]:

Locally Checkable Proofs (LCP)

• [Korman et al., ICDCN 2006, …]:

Proof Labeling Schemes (PLS)

• [Fraigniaud et al., FOCS 2011,…]:

Nondeterministic Local Decision (NLD) – [Fraigniaud et al., DISC 2012,…]: “Randomization”

• Another way to think of it [Blin et al., SSS 2014]:

– “any mechanism insuring silent self-stabilization is essentially equivalent to a proof-labeling scheme”

“No Strings attached”

• No knowledge of 𝑜
• No identifiers
• No port numbers
• No relaying of messages - just one round

Graphs and Communication

• (Weakly) Connected graphs 𝐻 = (𝑊, 𝐹) with 𝑊 = 𝑜

– Yes instances G ∈ Y & No instances G ∉ Y

• Undirected: U(v) for every v ∈ 𝑊

– multiset of labels of all neighbors

• Directed: D1(v) for every v ∈ 𝑊

– Multiset I of labels of all incoming-neighbors

• Directed: D2(v) for every v ∈ 𝑊

– two multisets (I,O) of labels of all

• incoming-neighbors
• outgoing-neighbors

1 1 1

[1] [0] [0] [ ] 0 , [ ] , [1]

Local Checkability

• rover

gets as input G ∈ Y

– Assigns a labels ℓ(v) for every v ∈ 𝑊

• erifier

is a distributed algorithm that gets as input at node v both ℓ(v) & U(v) (or D1(v) / D2(v))

– Outputs either Yes or No

• A Prover-Verifier pair ( ,

is correct for Y if:

– G ∈ Y & labels from :

• utputs Yes at all nodes

– G ∉ Y:

• utputs No for at least one node

Prover-Verifier Pairs

• We investigate if there are correct ( ,

for some Y

– (abbreviated by U-PVP, D1-PVP, D2-PVP)

• The quality of a PVP is its proof size

– 𝑔 𝑜 , if the PVP uses at most 𝑔 𝑜 bits for each label in any Yes instance with at most 𝑜 nodes

• The U-proof size of Y is the smallest proof size for

which there exists a correct U-PVP

– Analogous for D1-proof size / D2-proof size

• In this talk: All logarithms are of base 2 and rounded up to be of integer value

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝑡 − 𝑢 reachability
• Outlook

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝑡 − 𝑢 reachability
• Outlook

Undirected vs Directed Communication

• The different models can induce different

amount of bits required in the proof size

– Or might even render a problem impossible

• Example problem Y : CYCLE

– U-CYCLE: all undirected graphs containing a cycle – D-CYCLE: all directed graphs containing a directed cycle

D-CYCLE: Is there a D1-PVP?

A B

𝐻:

c1 c2 a b

D-CYCLE: Is there a D1-PVP?

A B

𝐻:

c1 c2 a b Yes Yes Yes Yes

D-CYCLE: Is there a D1-PVP?

A B

𝐻:

Yes Yes Yes Yes

𝐼:

A B

B c1 c2 a b a b b’

D-CYCLE: Is there a D1-PVP?

A B

𝐻:

Yes Yes Yes Yes

𝐼:

A B

B Yes Yes Yes c1 c2 a b a b b’

D-CYCLE: Is there a D1-PVP?

A B

𝐻:

Yes Yes Yes Yes

𝐼:

A B

B Yes Yes Yes

There is no D1-PVP for D-CYCLE

c1 c2 a b a b b’

CYCLE

Problem Directed one-way Directed two-way Undirected CYCLE Impossible

D-CYCLE: Is there a D2-PVP?

D-CYCLE: Is there a D2-PVP?

• rover

labels nodes as follows:

• In a directed cycle? -> 0
• Else: Minimum distance to a cycle

– (in the underlying undirected graph)

• Proof size: log 𝑜 bits

D-CYCLE: Is there a D2-PVP?

• rover

labels nodes as follows:

• In a directed cycle? -> 0
• Else: Minimum distance to a cycle

– (in the underlying undirected graph)

• Proof size: log 𝑜 bits

4

1 1 2 3

5

D-CYCLE: Is there a D2-PVP?

• erifier

returns Yes

– For nodes vc with label ℓ(vc)=0 if for (I,O) holds:

• 0 ∈ O and 0 ∈ I

– For the other nodes v with label ℓ(v) if

• 1. There is a label ℓ(u) in (I,O) with ℓ(v)=ℓ(u)+1, and
• 2. There is no label ℓ(u′) in (I,O) with ℓ(v)>ℓ(u′)+1

D-CYCLE: Is there a D2-PVP?

• erifier

returns Yes

– For nodes vc with label ℓ(vc)=0 if for (I,O) holds:

• 0 ∈ O and 0 ∈ I

– For the other nodes v with label ℓ(v) if

• 1. There is a label ℓ(u) in (I,O) with ℓ(v)=ℓ(u)+1, and
• 2. There is no label ℓ(u′) in (I,O) with ℓ(v)>ℓ(u′)+1

4

1 1 2 3

5

Is the described D2-PVP correct?

• Yes instances labeled by :

– Only nodes in directed cycles labeled with 0 -> Yes – All other nodes: Label is defined by minimum distance to a directed cycle -> Yes

• No instances:

– Is there a node with label 0? Follow “0-path” -> No – No node with label 0, but one with label k?

• Follow “descending path” -> No

D2-proof size: Ω log 𝑜 bits

𝐻:

B A B A vi vi+1 vi+2 vj-1 vj vj+1 vj+2 vi-1 v1 vn vn-1 vn-2

D2-proof size: Ω log 𝑜 bits

𝐻:

B A B A vi vi+1 vi+2 vj-1 vj vj+1 vj+2 vi-1 v1 vn vn-1 vn-2

Yes

D2-proof size: Ω log 𝑜 bits

𝐻: 𝐼:

B A B A B A B A

Yes

vi vi+1 vi+2 vj-1 vj vj+1 vj+2 ui ui+1 ui+2 uj-1 uj uj+1 u'i+2 u'j-1 vi-1 v1 vn vn-1 vn-2

D2-proof size: Ω log 𝑜 bits

𝐻: 𝐼:

B A B A B A B A

Yes

vi vi+1 vi+2 vj-1 vj vj+1 vj+2 ui ui+1 ui+2 uj-1 uj uj+1 u'i+2 u'j-1 vi-1 v1 vn vn-1 vn-2

Yes

CYCLE

Problem Directed one-way Directed two-way Undirected CYCLE Impossible Θ(log 𝑜)

U-proof size: At least 2 Bits

1 1 1 1

𝐻1: 𝐼1: 𝐻2: 𝐻3: 𝐻4: 𝐼2: 𝐼4: 𝐼3:

1 1 1

U-PVP for CYCLE with 2 bits

• rover

labels nodes as follows:

• In a cycle? -> 3
• Else: Remove all cycles, remaining graph is a forest

– For each tree T: » Create a root r adjacent to a cycle in 𝐻 with label 0 » Other nodes: Distance to r modulo 3

• Proof size: 2 bits

3 3

1

3 3 3 3 3 1 2

2

U-PVP for CYCLE with 2 bits

• erifier

returns Yes

– For nodes vc with label ℓ(vc)=3 if holds:

• Two neighbors with label 3 exist

– For the other nodes v with label ℓ(v) ∈ 0,1,2 if

• 1. There is no neighbor with label ℓ(v) , and
• 2. Exactly one neighbor exists with label ℓ(v)−1 mod 3
• r at least one neighbor with label of 3

Is the described U-PVP correct?

• Yes instances labeled by :

– Only nodes in cycles labeled with 3 -> Yes – Without the cycles, all other nodes are in a tree with labels as distance to root mod 3, and root is adjacent to a cycle -> Yes

3 3

1

3 3 3 3 3 1 2

2

Is the described U-PVP correct?

• Yes instances labeled by :

– Only nodes in cycles labeled with 3 -> Yes – Without the cycles, all other nodes are in a tree with labels as distance to root mod 3, and root is adjacent to a cycle -> Yes

• No instances (without a cycle):

– Is there a node with label 3? They form a forest, consider any leaf-> No – Else: follow “descending path” -> No

CYCLE, ACYCLIC, TREE

Problem Directed one-way Directed two-way Undirected CYCLE Impossible Θ(log 𝑜) 2

CYCLE, ACYCLIC, TREE

Problem Directed one-way Directed two-way Undirected CYCLE Impossible Θ(log 𝑜) 2 TREE Θ(log 𝑜)* Θ(log 𝑜) Θ(log 𝑜)* ACYCLIC Θ(log 𝑜) Θ(log 𝑜) same as Tree *: [Korman et al., Distributed Computing 2010]: Proof labeling schemes Idea for Tree:

• Label root as 0
• Other nodes: Label is distance from root

Idea for Acyclicity:

• Label nodes without incoming edges as 0
• Other nodes: Max. incoming label plus 1

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝑡 − 𝑢 reachability
• Conclusion

Overview

• Introduction
• Background & model
• Undirected vs directed communication
• Study of 𝒕 − 𝒖 reachability
• Conclusion

𝑡 − 𝑢 Reachability

• Is there a (directed) path from 𝑡 to 𝑢?

“To ask meaningful questions about connectivity […] we have the promise that there is exactly one node with label 𝑡 and exactly one node with label 𝑢.“ [Göös and Suomela, PODC 2011]

• We thus assume that there are two nodes with the unique

labels 𝑡 and 𝑢

• U-proof size of 1 bit (e.g., [Immermann, 1999]):

– Label nodes along a shortest 𝑡 − 𝑢 path with 1, else 0

Directed 𝑡 − 𝑢 Reachability

• D2-PVP with port numbers: 𝑃 log Δ bits

– With Δ being max degree – Idea: “Point at successor and predecessor” along a shortest 𝑡 − 𝑢 path

• Open question:

“Is there a proof labelling scheme with O(1)-bit proofs?”

[Göös and Suomela, PODC 2011]

D1-PVP for 𝑡 − 𝑢 Reachability

• We don’t have port numbers…
• Idea: Take a shortest 𝑡 − 𝑢 path 𝑡, v1, … vj, 𝑢

– Label according to distance to 𝑡 along the path – All other nodes: Label of 0

• Proof size of log 𝑜

D1-proof size: Ω log 𝑜 bits

𝑡 𝑢

D1-proof size: Ω log 𝑜 bits

𝑡 𝑢

Yes

D1-proof size: Ω log 𝑜 bits

𝑡 𝑢 𝑡 𝑢 A A B C

𝐻:

D1-proof size: Ω log 𝑜 bits

𝑡 𝑢 𝑡 𝑢 A A B C

𝐻:

D1-proof size: Ω log 𝑜 bits

𝑡 𝑢 𝑡 𝑢 𝑡 𝑢 A A A A B B C C

𝐻: 𝐼:

D1-proof size: Ω log 𝑜 bits

𝑡 𝑢 𝑡 𝑢 𝑡 𝑢 A A A A B B C C

𝐻: 𝐼:

There is no D1-PVP with 𝒈(∆) bits!

D2-PVP for 𝑡 − 𝑢 Reachability

• As we don’t have port numbers, we could use

the D1-PVP with log 𝑜 bits

• With port numbers: 𝑃 log Δ bits
• Let us create port numbers!

D2-PVP for 𝑡 − 𝑢 Reachability

• Idea: A 2-hop coloring needs ≤ ∆²+1 colors

– Encoding each color: 𝑃 log Δ bits

• 2-hop coloring can be checked locally

– All colors in the 1-hop neighborhood different?

• Thus, we can point “back and forth” along edges, by

emulating port numbers with 𝑃 log Δ bits

𝑡 𝑢

Conclusion

• Summary

– All three models of communication differ – Our lower bound examples have constant degree

• Can drop the 1 round restriction and go local

– Directed 𝑡 − 𝑢 reachability:

• One-Way: Proof size of Θ(log 𝑜) bits, 𝑔(Δ) bits don’t suffice
• Two-Way: Emulating port numbers -> 𝑃(log Δ) bits proof size
• Open Questions

– What happens in biologically inspired systems?

• E.g., no multisets but sets & finite automata verifier?

– What is the correct answer to D2 𝑡 − 𝑢 reachability? – Can similar techniques be deployed in production networks?

ETH Zurich – Distributed Computing – www.disco.ethz.ch

Thank you

Klaus-Tycho Förster, Thomas Lüdi, Jochen Seidel, Roger Wattenhofer January 06, 2016 @ ICDCN 2016 - Singapore