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 Gs University of Toronto Amos Korman University Paris Diderot & CNRS Jukka Suomela University of Helsinki & HIIT
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
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Local algorithm
O(1) communication rounds
O(1) radius neighbourhood
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 2 / 9
❬❋❑P FOCS’11❪
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
Fraigniaud et al. Local decision without IDs 23rd July 2013 3 / 9
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
❬❋❍❑ OPODIS’12❪
(deterministic)
V(G) ⊆ {1, 2, 3, . . .}
Fraigniaud et al. Local decision without IDs 23rd July 2013 4 / 9
❬❋❍❑ OPODIS’12❪
(deterministic)
V(G) ⊆ {1, 2, 3, . . .}
Restriction: Output is invariant under relabelling the nodes
(i.e., depends only on topology)
Fraigniaud et al. Local decision without IDs 23rd July 2013 4 / 9
Under some assumptions:
Proof by simulation. ..
Fraigniaud et al. Local decision without IDs 23rd July 2013 5 / 9
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
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
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
❬❋❍❑ ❪
Main theorem*
(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
* Contrary to a conjecture of ❬❋❍❑’12❪
Main theorem*
(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
* Contrary to a conjecture of ❬❋❍❑’12❪
Main theorem*
(I.e., there is a locally decidable property that cannot be decided ID-obliviously)
Fraigniaud et al. Local decision without IDs 23rd July 2013 6 / 9
Promise problem
Input:
Promise:
Output:
Fraigniaud et al. Local decision without IDs 23rd July 2013 7 / 9
Promise problem
Input:
Promise:
Output:
ID-oblivious Impossible: Must solve the Halting Problem
Fraigniaud et al. Local decision without IDs 23rd July 2013 7 / 9
Promise problem
Input:
Promise:
Output:
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
Promise:
Fraigniaud et al. Local decision without IDs 23rd July 2013 8 / 9
Promise:
Computation table of M
yes instance
Fraigniaud et al. Local decision without IDs 23rd July 2013 8 / 9
Promise:
Computation table of M
yes instance
Interesting bit: Table needs to be obfuscated!
Fraigniaud et al. Local decision without IDs 23rd July 2013 8 / 9
❬❋❍❑ ❪ ❬❍❍❘❙ ❬◆❙ ❪ ❬●❍❙ ❪
For local decision, we proved:
❬❋❑PP ❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9
IDs help IDs don’t help Decision This work
❬❋❍❑ OPODIS’12❪
Search
❬❍❍❘❙
SIROCCO’12]
❬◆❙ Sicomp’95❪ ❬●❍❙ PODC’12❪
For local decision, we proved:
❬❋❑PP ❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9
❬❋❍❑ ❪ ❬❍❍❘❙ ❬◆❙ ❪ ❬●❍❙ ❪
For local decision, we proved:
Randomisation?
Open problems in randomised decision ❬❋❑PP DISC’12❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9
❬❋❍❑ ❪ ❬❍❍❘❙ ❬◆❙ ❪ ❬●❍❙ ❪
For local decision, we proved:
Randomisation?
Open problems in randomised decision ❬❋❑PP DISC’12❪
Fraigniaud et al. Local decision without IDs 23rd July 2013 9 / 9