Algorithmic aspects of rotor-routing and the notion of linear equivalence

Tóthmérész, Lilla

doi:10.1016/j.dam.2017.11.008

Cited by 4 publications

(2 citation statements)

References 13 publications

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“…For rotor-routing, reachability was known to be in P in the special case when the target configuration is recurrent [13]. Here we show that rotorrouting reachability is also decidable in polynomial time in the general case, and we give a combinatorial characterization for the reachability.…”

Section: Introductionmentioning

confidence: 82%

Rotor-routing reachability is easy, chip-firing reachability is hard

Tóthmérész

2022

European Journal of Combinatorics

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Section: Introductionmentioning

confidence: 82%

Rotor-routing reachability is easy, chip-firing reachability is hard

Tóthmérész

2022

European Journal of Combinatorics

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“…We will need the following technical result from [16], that gives a more easily checkable condition for x v0 A = A ′ . Proposition 6.1.…”

Section: Rotor-routingmentioning

confidence: 99%

The Sandpile Group of a Trinity and a Canonical Definition for the Planar Bernardi Action

Kálmán¹,

Lee²,

Tóthmérész³

2022

Combinatorica

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Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if the ribbon structure is planar, moreover, in this case the Bernardi action is compatible with planar duality. Earlier, Chan, Church and Grochow and Chan, Glass, Macauley, Perkinson, Werner and Yang proved analogous results about the rotor-routing action. Baker and Wang moreover showed that the Bernardi and rotor-routing actions coincide for plane graphs.We clarify this still confounding picture by giving a canonical definition for the planar Bernardi/rotor-routing action, and also a canonical isomorphism between sandpile groups of planar dual graphs. Our canonical definition implies the compatibility with planar duality via an extremely short argument. We also show hidden symmetries of the problem by proving our results in the slightly more general setting of balanced plane digraphs.Any balanced plane digraph gives rise to a trinity, i.e., a triangulation of the sphere with a three-coloring of the 0-simplices. Our most important tool is a group associated to trinities, introduced by Cavenagh and Wanless, and a result of a subset of the authors characterizing the Bernardi bijection in terms of a dissection of a root polytope.

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Abelian networks IV. Dynamics of nonhalting networks

Chan¹,

Levine²

2022

Memoirs of the AMS

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An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. This paper is a continuation of the abelian networks series of Bond and Levine (2016), for which we extend the theory of abelian networks that halt on all inputs to networks that can run forever. A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the torsion group of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as \[ ∑ n ≥ 1 r n z n = det ( 1 1 − z D − A ) \sum _{n \geq 1} r_n z^n = \det (\frac {1}{1-z}D - A ) \] where r n r_n is the number of recurrent chip-and-rotor configurations with n n chips; D D is the diagonal matrix of outdegrees, and A A is the adjacency matrix. A consequence is that the sequence ( r n ) n ≥ 1 (r_n)_{n \geq 1} completely determines the spectrum of the simple random walk on the network.

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Algorithmic aspects of rotor-routing and the notion of linear equivalence

Cited by 4 publications

References 13 publications

Rotor-routing reachability is easy, chip-firing reachability is hard

Rotor-routing reachability is easy, chip-firing reachability is hard

The Sandpile Group of a Trinity and a Canonical Definition for the Planar Bernardi Action

Abelian networks IV. Dynamics of nonhalting networks

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