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Two dimensional signed majority
Universidad Adolfo Ibanez- Chile antonio.chacc@gmail.com
Two dimensional signed majority Universidad Adolfo Ibanez- Chile - - PowerPoint PPT Presentation
Two dimensional signed majority Universidad Adolfo Ibanez- Chile - - PowerPoint PPT Presentation
Two dimensional signed majority Universidad Adolfo Ibanez- Chile antonio.chacc@gmail.com Remains constant at 1 If 0 otherwise Decision problem PRE: given an initial configuration and a specific node at value 0. does there
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Decision problem
PRE: given an initial configuration and a specific node at value 0. does there exist T>0 such that this node becomes 1?
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Theorem (P. Montealegre, I. Todinca, E:G (1911)) Given an undirected graph G if the maximum degree ≥ 5, PER is P-complete. Else PRE belongs to NC
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Clearly PRE belongs to P, because in almost O(n) steps the dynamics reaches the steady state.
The proof of P-Completeness consist to simulate the monotone circuits behavior inside the strict majority dynamics.
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1 1
DIODE
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Information only flows to the right
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OR gate And Gate
Diode arc
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For the case maximun degree ≤ 4 one may reduce the problem to compute connected and biconnected components in the graph, which one may do in a PRAM in
See Jaja …………
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1 1 1 1 1 1 1 1 1 1 1 1 1 1 Decision site
Alliances
Max degree ≤ 4 Its vertices never change
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Decision site 0’s Connected component
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The Complexity of the majority vote rule for planar graphs
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Decision problem
PRE: given an initial configuration and a specific node at value 0. Does there exist T>0 such that this node becomes 1?
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We consider the similar decision problem PER
This problem has been studied by C. Moore for d-dimensional regular lattices with nearest interactions
Von Neumann neighborhood in 2D Nearest neighborhood In 3D
PER is P-Complete for d ≥ 3
- pen for d = 2
(C. Moore)
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For planar graphs PRE is P-Complete
(P. Montealegre, E:G, 2012)
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PRE is in P
Majority is a particular case of a threshold network: Since G is undirected W is a nxn symmetric matrix and the threshold: Odd neighborhood Even neighborhood The parallel dynamic is driven by Which is strictly decreasing and bounded
So PRE is in P
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wire Duplicate a signal diode
=
GADGETS FOR CIRCUITS
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AND-gate OR-gate
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The cross-over gadget
diode
(traffic light)
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Cross-over from a to e
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We will study 2D majority with signs
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Symmetric majority Antisymmetric majority Asymmetric majority
Several steps Initial condition attractor
Periodic configuration
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EG,1980 E.G. P. Montealegre, I Todinca, 2013 E.G., P. Montealegre, 2014
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F1 simulation of AND OR gates (no cross over)
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F5
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