The lattice structure of rotor-router modelLe Manh Ha

Hue University’s College of Education andInstitute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam

Email: lemanhha@dhsphue.edu.vn

Abstract—In this paper 1, we study the rotor router model inthe relation with the famous discrete dynamical system - ChipFiring Game. We consider the rotor router model as a discretedynamical system defined on digraph and we use order theoryto show that its state space started from any state is a lattice,which implies strong structural properties. The lattice structureof the state space of a dynamical system is of great interest sinceit implies convergence (and more) if the state space is finite.Moreover, we also attempt to define the class L(R) of latticesthat are state space of a rotor router model, and compare it withthe class of distributive lattices and the class of ULD lattices.

Index Terms—Chip Firing Game, Discrete dynamical system,state space, rotor router, order and lattice structure.

I. INTRODUCTION

The abelian sandpile or Chip Firing Game (CFG) androtor-router models were discovered independently in differentcontexts. The Chip Firing Game (CFG) was introduced byBjorner, Lovasz and Shor in 1991 to illustrate the behaviors ofdistributed jobs in networks [3]. Then this model has became avery famous one which can be used to illustrate many systemsin different science domains. For example, in complex systemsresearch, CFG was considered as a paradigm for the so-calledself organized criticality [1], [4], [19]; in economy or computerscience, CFG was studied as a resource distribution systems[5], [13]. Because of this important role, many approachesto investigate the behavior of CFG were developed, fromphysics experimental techniques [1], [19] to other methodsusing algebraic structures [4], formal languages [3], [2] orenumerative combinatorics [17], [20].

Related ideas were explored earlier by Engel [10] in theform of a pedagogical tool (the ”probabilistic abacus”), andby Lorenzini [18] in connection with arithmetic geometry.

The rotor-router model was first introduced by Priezzhev etal. [16] (under the name Eulerian walkers model) in connectionwith self-organized criticality. It was rediscovered severaltimes: by Rabani, Sinclair and Wanka [24] as an approach toload-balancing in multiprocessor systems, by Propp [15] as away to derandomize models such as internal diffusion-limitedaggregation (IDLA) [7], and by Dumitriu, Tetali, and Winkleras part of their analysis of a graph-based game [14].

To define the rotor-router model on a directed graph G,for each vertex of G, fix a cyclic ordering of the outgoingedges. To each vertex v we associate a rotor ρ(v) chosen fromamong the outgoing edges from v. A chip performs a walk on

1This work is supported in part by Vietnam’s National Foundationfor Science and Technology Development (NAFOSTED).

G according to the rotor-router rule: if the chip is at v, wefirst increment the rotor ρ(v) to its successor e = (v, w) inthe cyclic ordering of outgoing edges from v, and then routethe chip along e to w. If the chip ever reaches a sink, i.e. avertex of G with no outgoing edges, the chip will stop there;otherwise, the chip continues walking forever.

In another context, the Conflicting Chip Firing Game(CCFG) model [23], an extension of CFG, has many closeconnection. In this model, the condition of firing rule isreduced and the transferring of chips similar to the rotor-router model. The vertex v is firable if it contains at least onechip and its firing is carried out by sending one chip alongone edge from v to one of its neighbors. We also denote byCCFG(G,n) the set of all configurations of CCFG(G,n)and call the configuration space of this game. This set isexactly the set of compositions of n into V .

It easy to check that the configuration space of CFG is asubset of the configuration space of CCFG on the same supportgraph and we are going to see that the state space of rotor-router model is too. In the rotor-route model, at each time achip at a vertex which is not the sink just can walk to only oneof its neighbor, while a chip in the CCFG model can walk tomany other neighbor depending on the out-degree of the vertexwhich it stores. The structure of configuration space of CCFGis not a lattice, but we has shown the order structure of thismodel on directed acyclic graph by using energy functions [12]and we has constructed the algorithm to determine this orderin [8]. We has also studied the reachability of this model in therelation with the flow network problem and we has constructedthe polynomial algorithm to determine the reachability of thismodel on general digraph in [9].

In Section 2, we first represent the definition of rotor-routermodel in the relation with the discrete dynamical systemsChip Firing Games. Then some important results of the orderstructure, the lattice structure of CFGs are also represented inthis section.

In section 3, we investigate the rotor-router models withno closed component and give the strong structural propertiesof this model. We show the state space of this model haslattice structure by using the notation of shot-vector and sometechniques in [17].

In Section 4, we attempt to characterize the class L(R) oflattices that are state space of a rotor-router model.

978-1-4244-8075-3/10/$26.00 ©2010 IEEE

II. CHIP FIRING GAMES AND ROTOR - ROUTER MODEL

The goal of this section is to introduce some results of ChipFiring Games in the relation with rotor-router models.

First of all, we recall here some basic notions in the Orderand Lattice Theory. After that, we represent some results oflattice structure of CFG on digraph with no closed componentand order structure of CCFG on directed acyclic graph (DAG).

An order relation or partial order relation is a binaryrelation ≤ over a set, such that for all x, y and z in thisset, x ≤ x (reflexivity), x ≤ y and y ≤ z implies x ≤ z(transitivity), x ≤ y and y ≤ x implies x = y (antisymmetry).The set is then called a partially ordered set or, for short, aposet.

A lattice L is an poset such that for any two elements a andb of L, there exists a unique smallest element which is greaterthan a and b (the supremum of a and b, denoted by a∨ b) andthere exists a greatest element which is smaller than a and b(the infimum of a and b, denoted by a ∧ b).

A lattice L is distributive if it satisfies one of the twofollowing laws of distributivity (which are equivalent):

∀x, y, z ∈ L, x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

∀x, y, z ∈ L, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

A lattice is a hypercube of dimension n if it is isomorphicto the set of all subsets of a set of n elements, ordered byinclusion. It is also called a boolean lattice.

A lattice is upper locally distributive (denoted by ULD[21]) if the interval between an element and the supremumof all its upper covers is a hypercube. The lower locallydistributive lattices (LLD) is dual.

Let us present here one of the most important results aboutthe lattice structure of CFG.

Theorem 1. [17] The configuration space of a CFG ona directed graph with no closed component and with anarbitrary initial configuration O ordered with the reachabilityrelation is a lower locally distributive lattice.

1 0

v1

3

e3

e2 e1 4 v2

e4

v3 2

5

3

2

4

3

2

43

4

5

0

4

5

3

1

1 2 0

2 0 1

6 7 8

5

Fig. 1. The configuration space of a CFG with 9 chips.

This Theorem derives many consequences: first, it provesthe convergent property of CFG, then it shows that whatever

the way of firings is, after exactly the same time, the systemreaches the same fixed point. Moreover, this convergence isvery strong in the following sense: for two configurations ofa system, there exists a unique first configuration obtain forthem, and every configuration which can be obtained fromboth of them can be obtained from this first one.

In [20], the authors proved that the set of configurationspaces of all CFG is strictly included in the set of ULDlattices and they introduced the coloured Chip Firing Game,that generates exactly the class of ULD lattices.

Next, we recall one of the most important result about thecharacterization of the order structure of CCFG on a specialclass support graph.

Theorem 2. [12] The configuration space CCFG(G,n) ofconflicting Chip firing game on DAG G is ordered with thereflexive and transitive closure of the successor relation.

2

1

11

1 1

22 1 1

11

1 1 1 1

1 11 1

1 212

Fig. 2. The configuration space of a CCFG with 2 chips.

Now, we define the rotor-router model and we will see thaton the support digraph G in which every vertex has out-degreeat most one, the rotor-router model is nothing but the CCFGmodel on the same support graph.

Given a directed graph G, fix for each vertex v a cyclicordering of the edges emanating from v. For an edge e withtail v we denote by e+ the next edge after e in the prescribedcyclic ordering of the edges emanating from v.

Definition 1. A chip configuration σ on G, is a vector ofnonnegative integers indexed by the non-sink vertices of G,where σ(v) represents the number of chips at vertex v.

Definition 2. [22] A rotor configuration is a function ρ thatassigns to each non-sink vertex v of G an edge ρ(v) emanatingfrom v. If there is a chip at a non-sink vertex v of G, routingthe chip at v (for one step) consists of updating the rotorconfiguration so that ρ(v) is replaced with ρ(v)+, and then

moving the chip to the head of ρ(v)+. A single-chip-and-rotorstate is a pair consisting of a vertex w (which represents thelocation of the chip) and a rotor configuration ρ. The rotorrouter operation is the map that sends a single-chip-and-rotorstate (w, ρ) (where w is not a sink) to the state (w+, ρ+)obtained by routing the chip at w for one step. A chip-and-rotor state (for short, state) is a pair τ = (σ, ρ) consisting ofa chip configuration σ and rotor configuration ρ on G.

By using the definition of rotor-router in [22], we considerrotor-router as a discrete dynamical system on digraph andredefine it as follows:

Definition 3. The rotor-router on digraph G, denoted byR(G), is a dynamical model defined as follows: each stateis a chip-and-rotor-state on G; a non-sink vertex is active ifit has at least one chip; the evolution rule (firing rule) of thissystem is the firing of one active vertex v and firing v resultsin a new state given by replacing the rotor ρ(v) with ρ(v)+

and moving a chip from v to the head of ρ(v)+ (and removingthe chip if the head of ρ(v)+ is a sink).

We call state space, and denote by R(G), the set of all stateon G.

Definition 4. Given two state τ and τ ′ of a R(G), we saythat τ ′ is reachable from τ , denoted by τ ′ ≤ τ , if τ ′ can beobtained from τ by a firing sequence (in the case the firingsequence is empty, τ = τ ′). In particular, we write τ ≺ τ ′ ifτ ′ is obtained from τ by applying once firing rule and we callτ ′ a successor of τ . We say that τ is stable if no vertex canfire, i.e., all chips have moved to sinks.

Definition 5. Given R(G) and let τ0 be a state of R(G). Wedenote by R(G, τ0) the state space of all reachable states fromτ0.

21,13

12,23 20,14

11,2402,13

01,14

10,13

01,23

00,24

20,231

2

132

4

Fig. 3. The state τ0 = ((2, 1); (1, 3)) (left) and its state space R(G, τ0)(right).

III. ROTOR-ROUTER WITH NO CLOSED COMPONENT

As we will see, there is an important link between chipfiring game and rotor-router. A hint at this link comes from astraightforward count of configurations. Recall that a stablechip configuration of classical CFG is a way of assigningsome number of chips between 0 and dv − 1 to each non-sink vertex v of G. Thus, the number of stable configurationsis exactly

∏v dv , where the product runs over all non-sink

vertices. This is also the number of rotor configurations onG. Other connections become apparent when one exploresthe appropriate notion of recurrent states for the rotor-routermodel. We will treat the cases of digraphs with no closedcomponent.

Definition 6. Let G be a digraph. A closed component of Gis a nontrivial (more than one element) proper subset S of theset of the vertices of G such that

• there exists a path from any element of S to any otherelement of S (S is a nontrivial strongly connected com-ponent)

• there is no outgoing edge from any element of S to anyvertex of G \ S (S is closed)

It is clear that the (support of the) CFG in Figure 1 hasno closed component, since its unique nontrivial stronglyconnected component is composed of the two topmost vertices,and there is an edge from this component to the third vertex,which is a sink. Let us recall that when a chip arrive to sucha sink, it can never go out. Notice that a closed componentbehaves as a sink, since it also has this property.

Now, we will show an important lemma which allows thestudy of the special case where all the nontrivial stronglyconnected components of the support of the game have atleast one outgoing edge, i.e, an edge from one vertex in thecomponent to a vertex outside the component. This means thatthe support graph of the game has no closed component.

By using the same technique in [17], we can show the orderstructure of state space of a rotor-router model with no closedcomponent.

Lemma 1. Let us consider a non closed strongly connectedcomponent C. Starting from a state τ = (σ, ρ) there is nononempty sequence of firings of vertices in C such that thestate τ is reached again.

This lemma implies that if we only fire vertices which arenot in a closed component, then we can not have a cycle ofstate. Therefore, we deduce that if the support graph of a rotor-router model has no closed component then its state spacecontains no cycle, and so it is a poset.

Theorem 3. The state space of a rotor-router with no closedcomponent is partially ordered by the reflexive and transitiveclosure of the successor relation.

Recall that a vertex is a sink if its out-degree is zero. Aglobal sink is a sink s such that from every other vertex there isa directed path leading to s. Note that if there is a global sink,

then it is the unique sink. It easy to see that if G is a digraphwith a global sink then G has no closed component. So, byusing the similar reasoning in [22], we have the followinglemma.

Lemma 2. [22][Lemma 3.9] Let G be a digraph with noclosed component. Let τ0, τ1, . . . , τn be a sequence of chip-and-rotor states of R(G), each of which is a successor of theone before. If τ0, τ

′1, . . . , τ

′m is another such sequence, and τn

is stable, then m ≤ n. If in addition τm is stable, then m = nand τn = τ ′

n, and for each vertex w, the number of times wfires is the same for both sequences.

This lemma shows that starting from a state τ there is atmost one stable state which can be reached by a finite sequenceof firings. Moreover, the Lemma 1 shows that the state spaceof a rotor-router with no closed component has no cycle andthe number of state is finite, so starting from a state τ thereis exactly one stable state. We denote it τ0 and call it thestabilization of τ . Now, we prove that every path betweenany two states has the same length, that means the number ofapplications of firing rule to the vertices v ∈ V during everyfiring sequence to obtain state τ ′ from state τ are the same.Given a firing sequence p, we denote by |p|i the number ofapplications of firing rule to the ith vertex during the sequencep and denote by |p| the number of vertices (may be repeated)in sequence p. By using the Lemma 2, we have the followingresult:

Lemma 3. Given a rotor-router R(G) with no closed compo-nent, if starting from the same state, two sequences of firings and t lead to the same final state, then |s|i = |t|i for eachvertex i.

This lemma allows us to define the shot vector k(τ, τ ′) oftwo states τ and τ ′ if b′ can be obtained from τ in a rotor-router R(G) : k(τ, τ ′) = (k1(τ, τ ′), k2(τ, τ ′), . . . , kn(τ, τ ′)),where ki(τ, τ ′) is the number of firings of vertex i to obtainτ ′ from τ . If τ and τ ′ are two states obtained from the samestate τ0, we order k(τ0, τ) ≤ k(τ0, τ

′) if for all i, ki(τ0, τ) ≤ki(τ0, τ

′). Moreover, if τ ′ ≥ τ then k(τ0, τ′) = k(τ0, τ) +

k(τ, τ ′).By using the proof technics in [17], we can characterize the

order between all the state obtained from the initial one τ0 ina rotor-router by comparing their shot-vectors as follows.

Theorem 4. Let τ and ς be two states of R(G, τ0). Thenτ ≥ ς in R(G, τ0) if and only if k(τ0, τ) ≤ k(τ0, ς).

By using the same technique in [17], we can show thestrong structure, that is the lattice structure of the state spaceof R(G, τ0).

Theorem 5. Let R(G) be a rotor-router with no closedcomponent and τ0 be a state of R(G). Then R(G, τ0), orderedwith the reflexive and transitive closure of the successorrelation, is a lattice. Moreover, the infimum of two elements τand ς is defined as follows: let k be a vector such that for allvertex i, ki = max(ki(τ0, τ), ki(τ0, ς)) then the state κ such

that k(τ0, κ) = k is the infimum of τ and ς .

IV. STUDY OF THE CLASS L(R)

The study of the configuration spaces structure of somediscrete dynamical systems has been presented in [20] and[11]. We have seen above that the proof of lattice structureof rotor-router model is similar to the lattice structure of ChipFiring Game. Denote by L(CFG) the class of lattices inducedby CFG. In [20] it has been proved that L(CFG) is not thewhole ULD class (i.e, there exists a ULD lattice which is theconfiguration space of no CFG), but contains the class D ofdistributive lattices. This is an interesting result from the latticetheory point of view, since the distributive and ULD latticesclasses are very closed one to another. We have the followingrelations:

D � L(CFG) � ULD.

The classify of lattices which induced by some modelsemphasizes the complexity of the characterization problems inlattice theory and plays an important role in the computationcomplexity of systems. Given a rotor-router model with noclosed component R = R(G, τ0), we denote by L(R) itsstate space which is a lattice. We denote by L(R) the classof lattices that are the state space of a rotor-router model. Fora lattice L ∈ L(R), we say a corresponding rotor-router anyrotor-router model R = R(G, τ0) such that L(R) = L. In thissection, we are going to study the class L(R) in the relationwith the class D and class ULD. The rotor-router model arequite similar to the CFG, not only at the behavior but also atgroup structure (sandpile group) (see example [6], [22]) andlattice structure. However, the classes L(R) and L(CFG) arenot the same. The class D is not included the class L(R).First of all, we can give an example to show that L(R) isnot a subset of D. In Figure 4, we show the state space of arotor-router model which is not in D.

Next, in the following theorem, we show that there existsdistributive lattices that is not the state space of any rotor-router model.

Theorem 6. The lattice D5 = 1 ⊕ 22 is not in L(R).

Proof: Let us suppose that D5 is in L(R). Then thereexists rotor-router model R = R(G, τ0) such that L(R) = D5

(see Figure 5), where τf is the stable state. Let a and b be thevertices that are fired in τ2 and τ1 to obtain τ3. Clearly, a = bthen by Lemma 3, we have a and b are vertices that fired atthe state τ0 to obtain τ1 and τ2, respectively.

At the state τ0 = (σ0, ρ0), there are only vertices a and bactive, so σ0(a) = 0, σ0(b) = 0 and σ0(v) = 0 for all v = a, b.It easy to see that σ0(a) = σ0(b) = 1. Similarly, at the stateτ3 = (σ3, ρ3), there is only the vertex c active and it is firedexactly once, so σ3(c) = 1 and σ3(v) = 0 for all v = c. Wecan describe the states τ0 and τ3 in the Figure 7.

After firing the vertex a at the state τ0 = (σ0, ρ0) we obtainthe state τ1 = (σ1, ρ1). So we have ρ0(b) = ρ1(b). At the stateτ1 = (σ1, ρ1), there is only the vertex b active, so σ1(b) =0 and σ1(v) = 0 for all v = b. Moreover, after firing the

11

e1

e3

e2

1

e1

e3

e2

1

e1

e3

e2

11

e1

e3

e2

21

e1

e3

e2

1

e1

e3

e2

e1

e3

e2

1

e1

e3

e2

Fig. 4. The state space lattice of a rotor-router is not distributive.

c

0

21

3

f

ab

a b

Fig. 5. A distributive lattice that is not the state space of any rotor-routermodel.

vertex b at the state τ1, we obtain the state τ3, this impliesthat σ1(b) = 1 and σ1(v) = 0 for all v = b. Thus, after firingthe vertex b at the state τ1 chip must be transferred from bto c, that mean ρ+

1 (b) = (b, c) and therefore ρ+0 (b) = (b, c)

(here we denote by (u, v) for the arc in digraph which u isthe head and v is the tail). By using similar argument we haveρ+0 (a) = ρ+

2 (a) = (a, c). Now, at the state τ0, consider thefiring sequence

τ0a−→ τ1

b−→ τ3.

Since ρ+0 (a) = (a, c) and ρ+

1 (b) = (b, c) then σ3(c) = 2.This contraction implies that there is no rotor-router modelsthat its state space is the lattice D5 = 1 ⊕ 22.

Now, we will show that the class L(R) is included in theclass of ULD by using the notation of shot-set of a statepresented in [20]. By Lemma 3, we can define the shot-sets(τ) of a state τ the set of rules fired to reach τ from the

11

sink

b a

c 1

sink

b a

c

Fig. 6. Left: the state τ0 and right: the state τ3.

DL(R)

L(CFG)

ULD

Fig. 7. The classes of lattices induced by various systems.

initial state. A subset X ⊆ V is a valid shot-set if there existsa state τ reachable from the initial state such that s(τ) = X.It is easy to check that the lattice of the state space of aCFG is isomorphic to the lattice of the shot-sets of its statespace ordered by inclusion. The join is given by the followingformula:

Lemma 4. Let R = R(G, τ0) be a rotor-router model, L(R)be its state space and τ, τ ′ be two states. The join of τ andτ ′ in L(R) is determined by:

s(τ ∨ τ ′) = s(τ) ∪ s(τ ′).

As we see, the behavior of rotor-router models is similar tothe behavior of CFG in the sense the firing of a vertex v doesnot prevent the firing of any vertex v′, v′ = v because when vis fired, the number of chips in v′ stays the same or increases.So the following result is immediate:

Theorem 7. The lattice of the state space of a rotor-routermodel with no closed component is ULD.

From this Theorem and Theorem 6, we conclude that theclass L(R) of lattices induced by rotor-router models is strictlyincluded in the class of ULD.

Now, fix a chip configuration σ on digraph G by consideringdifferent rotor configurations, we obtain different states. Thecorresponding lattices induced from these states are alsodifferent. They are eventually very different from one anotherat the time to reach fixed points and their complexities.These issues are particular of interest in learning about therelationship between the dynamical systems. Firstly, in thispaper, we see that if two rotor configuration are acyclic thentwo corresponding induced lattices are disjoint (see Figure 3,

3

21,24

11,14 30,23

20,1302,24

11,23

10,24

1

2

1

2

4

01,13

00,14

Fig. 8. The state τ0 = ((2, 1); (2, 4)) (left) and its state space R(G, τ0)(right).

21,14

12,24 30,13

21,2302,14

20,2411,13

10,14

10,23

00,13

02,23

01,24

1

2

132

4

Fig. 9. The state τ0 = ((2, 1); (1, 4)) (left) and its state space R(G, τ0)(right).

Figure 9 and Figure 8, lattices are pairwise disjoint) otherwisethey contain each other (see Figure 9 lattice L(R(G, (21, 23)))is included in lattice L(R(G, (21, 14)))). Let ρi, i = 1,2, betwo rotor configurations on G and let τi = (σ, ρi) be twostates. Let Ri = R(G, τi) and let Li = L(Ri), for i = 1, 2.Then we have the following result:

Theorem 8. If ρ1 and ρ2 are acyclic then L1 ∩ L2 = ∅otherwise L1 ⊆ L2 or L2 ⊆ L1.

This theorem implies that if there is a state τ obtained fromboth τ1 and τ2 then there exists the path between τ1 and τ2.

Proof: Suppose that L1 ∩ L2 = ∅, that means thereexists state τ such that τ ≤ τ1 and τ ≤ τ2. Without lossof generality we may assume that τ is a fixed point and therotor configuration of state τ is acyclic. Notice that the chipaddition operator is a permutation on the set of acyclic rotorconfiguration (for more detail see [22]), so either ρ1 or ρ2 isnot acyclic. Let ci be the number of directed cycles in rotorconfiguration ρi, (i = 1, 2) and assume that c1 ≤ c2. Let s(τi)be the set of rules fired to reach τ from τi. Because of σ1 = σ2

and τ1 differ from τ2 at some cycles of walking of chips andc1 ≤ c2 implies that s(τ2) ⊆ s(τ1). Therefore, at the state τ1,we may apply the firing rules at vertices to routing chip to thecycles of ρ2 and we have τ2 ≤ τ1, that mean L2 ⊆ L1.

V. CONCLUSION

We study the rotor-router model as a discrete dynamicalsystem. We show the lattice structure of this model. We havealso compared the class L(R) of lattices that are the statespace of rotor-router models with class D and class ULD. Wepredict that the class L(R) is included in the class L(CFG).In future works, we will study the relation between latticeswhich have the common initial chip configurations but differ atrotor configurations and the problem determining the shortestpath and study the complexity of lattices.

Acknowledgements: The author would like to thankPHAN Thi Ha Duong for her invaluable advising, and forproviding the initial motivation of this work.

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