What can be decided locally without identifiers? Pierre Fraigniaud - PowerPoint PPT Presentation

What can be decided locally without identifiers? Pierre Fraigniaud University Paris Diderot & CNRS Mika Göös University of Toronto Amos Korman University Paris Diderot & CNRS Jukka Suomela University of Helsinki & HIIT Fraigniaud et al. Local decision without IDs 23rd July 2013 1 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Local algorithm ≡ O ( 1 ) communication rounds ≡ O ( 1 ) radius neighbourhood Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: yes / no Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: G is accepted iff all nodes ouput yes Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Local decision ❬❋❑P FOCS’11 ❪ Input: graph G is G ∈ P ? Output: Locally decidable P : triangle-freeness Eulerian graphs line graphs Locally checkable labellings ( G , ℓ ) Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9

Our question We ask: Do node identifiers help in local decision? Fraigniaud et al. Local decision without IDs 23rd July 2013 3 / 9

Our question We ask: Do node identifiers help in local decision? IDs do not seem useful.. . Graph properties do not depend on node labels Symmetry breaking is not needed for decision problems! Fraigniaud et al. Local decision without IDs 23rd July 2013 3 / 9

Our question—formalised ❬❋❍❑ OPODIS’12 ❪ LOCAL model (deterministic) V ( G ) ⊆ { 1, 2, 3, . . . } Fraigniaud et al. Local decision without IDs 23rd July 2013 4 / 9

Our question—formalised ❬❋❍❑ OPODIS’12 ❪ vs. LOCAL model ID-oblivious model (deterministic) Restriction: Output is invariant V ( G ) ⊆ { 1, 2, 3, . . . } under relabelling the nodes (i.e., depends only on topology ) Fraigniaud et al. Local decision without IDs 23rd July 2013 4 / 9

Easy cases Warm up! Under some assumptions: LOCAL = ID-oblivious Proof by simulation. .. Fraigniaud et al. Local decision without IDs 23rd July 2013 5 / 9

Easy cases Let A be a LOCAL decision algorithm ID-oblivious simulation of A Input: local neighbourhood ( H , v ) of G For each ID-assignment f : V ( H ) → { 1, 2, . . . , n } : if A ( f ( H , v ) ) = no then output no . Otherwise output yes . Assumptions: � Nodes know n Fraigniaud et al. Local decision without IDs 23rd July 2013 5 / 9

Easy cases Let A be a LOCAL decision algorithm ID-oblivious simulation of A Input: local neighbourhood ( H , v ) of G For each ID-assignment f : V ( H ) → { 1, 2, . . . } : if A ( f ( H , v ) ) = no then output no . Otherwise output yes . Assumptions: � Nodes do not know n Fraigniaud et al. Local decision without IDs 23rd July 2013 5 / 9

Easy cases Let A be a LOCAL decision algorithm ID-oblivious simulation of A Input: local neighbourhood ( H , v ) of G For each ID-assignment f : V ( H ) → { 1, 2, . . . } : if A ( f ( H , v ) ) = no then output no . Otherwise output yes . Assumptions: � Nodes do not know n � Nodes are Turing computable Fraigniaud et al. Local decision without IDs 23rd July 2013 5 / 9

❬❋❍❑ ❪ Our main result Main theorem* LOCAL � = ID-oblivious (I.e., there is a locally decidable property that cannot be decided ID-obliviously) Assumptions: � Nodes do not know n � Nodes are Turing computable Fraigniaud et al. Local decision without IDs 23rd July 2013 6 / 9

Our main result Main theorem* LOCAL � = ID-oblivious (I.e., there is a locally decidable property that cannot be decided ID-obliviously) * Contrary to a conjecture of ❬❋❍❑ ’12 ❪ Assumptions: � Nodes do not know n � Nodes are Turing computable Fraigniaud et al. Local decision without IDs 23rd July 2013 6 / 9

Our main result Main theorem* LOCAL � = ID-oblivious (I.e., there is a locally decidable property that cannot be decided ID-obliviously) * Contrary to a conjecture of ❬❋❍❑ ’12 ❪ Proof... Fraigniaud et al. Local decision without IDs 23rd July 2013 6 / 9

Separation under promise Promise problem • G = ( G , M ) is a labelled n -cycle Input: • M is a Turing machine Promise: • If M halts in s steps, then n ≥ s • yes if M runs forever Output: • no if M halts Fraigniaud et al. Local decision without IDs 23rd July 2013 7 / 9

Separation under promise Promise problem • G = ( G , M ) is a labelled n -cycle Input: • M is a Turing machine Promise: • If M halts in s steps, then n ≥ s • yes if M runs forever Output: • no if M halts ID-oblivious Impossible: Must solve the Halting Problem Fraigniaud et al. Local decision without IDs 23rd July 2013 7 / 9

Separation under promise Promise problem • G = ( G , M ) is a labelled n -cycle Input: • M is a Turing machine Promise: • If M halts in s steps, then n ≥ s • yes if M runs forever Output: • no if M halts ID-oblivious Impossible: Must solve the Halting Problem LOCAL Possible: Node v simulates M for ID ( v ) steps Fraigniaud et al. Local decision without IDs 23rd July 2013 7 / 9

Getting rid of the promise Promise: • If M halts in s steps, then n ≥ s Fraigniaud et al. Local decision without IDs 23rd July 2013 8 / 9

Getting rid of the promise Promise: • If M halts in s steps, then n ≥ s ⇓ Replace! ⇓ ⊆ G yes instance Computation table of M Fraigniaud et al. Local decision without IDs 23rd July 2013 8 / 9

Getting rid of the promise Promise: • If M halts in s steps, then n ≥ s ⇓ Replace! ⇓ ⊆ G yes instance Computation table of M Interesting bit: Table needs to be obfuscated! Fraigniaud et al. Local decision without IDs 23rd July 2013 8 / 9

❬❋❍❑ ❪ ❬❋❑PP ❪ ❬❍❍❘❙ ❬◆❙ ❪ ❬●❍❙ ❪ Summary For local decision, we proved: LOCAL � = ID-oblivious Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9

❬❋❑PP ❪ Summary For local decision, we proved: LOCAL � = ID-oblivious IDs help IDs don’t help Decision This work ❬❋❍❑ OPODIS’12 ❪ ❬❍❍❘❙ ❬◆❙ Sicomp’95 ❪ Search ❬●❍❙ PODC’12 ❪ SIROCCO’12] Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9

❬❋❍❑ ❪ ❬❍❍❘❙ ❬◆❙ ❪ ❬●❍❙ ❪ Summary For local decision, we proved: LOCAL � = ID-oblivious Randomisation? Open problems in randomised decision ❬❋❑PP DISC’12 ❪ Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9

❬❋❍❑ ❪ ❬❍❍❘❙ ❬◆❙ ❪ ❬●❍❙ ❪ Summary For local decision, we proved: LOCAL � = ID-oblivious Randomisation? Open problems in randomised decision ❬❋❑PP DISC’12 ❪ Cheers! Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9