A 4n-move self-stabilizing algorithm for the minimal dominating set problem using an unfair distributed daemon

Information Processing Letters 114 (2014) 515–518

Contents lists available atScienceDirect

Information Processing Letters

www.elsevier.com/locate/ipl

A 4n-move self-stabilizing algorithm for the minimal

dominating set problem using an unfair distributed daemon

✩

Well Y. Chiu, Chiuyuan Chen

∗

, Shih-Yu Tsai

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 29 April 2013

Received in revised form 19 March 2014 Accepted 10 April 2014

Available online 23 April 2014 Communicated by Tsan-sheng Hsu Keywords: Self-stabilizing algorithm Fault tolerance Distributed computing Graph algorithm Domination

A distributed system is self-stabilizing if, regardless of its initial state, the system is guaranteed to reach a legitimate (i.e., correct) state in finite time. In 2007, Turau proposed the first linear-time self-stabilizing algorithm for the minimal dominating set (MDS) problem under an unfair distributed daemon [9]; this algorithm stabilizes in at most 9n moves, where n is the number of nodes in the system. In 2008, Goddard et al.[4]proposed a 5n-move algorithm. In this paper, we present a 4n-move algorithm.

©2014 Elsevier B.V. All rights reserved.

1. Introduction

Self-stabilization is a concept of designing a distributed system for transient fault toleration and was introduced by Dijkstra in [2]. A fundamental idea of self-stabilizing algorithms is that the distributed system may be started from an arbitrary state, and after finite time, the system will reach a legitimate (i.e., correct) state. Notice that the (global) state of a distributed system consists of the (local) state of every node (also called process) and the content of every communication channel (or communication register). A distributed algorithm is self-stabilizing if the following two properties hold: convergence and closure. The conver-gence property ensures that, starting from any illegitimate state, the distributed system reaches a legitimate state in finite time without any external intervention. The closure

✩ _{This research was partially supported by the National Science Council}

of the Republic of China under the grant NSC100-2115-M-009-004-MY2.

*

Corresponding author.

E-mail addresses:weeeeeeeeell@gmail.com(W.Y. Chiu), cychen@mail.nctu.edu.tw(C. Chen).

property ensures that, after convergence, the system re-mains in the set of legitimate states.

This paper follows the conventions used in[9]. In par-ticular, every node executes the same program, and main-tains and changes its own state based on its current state and the states of its neighbors. A node can change its state by making a move, that is, by changing the value of at least one of its local variables. Our algorithm requires locally unique identifiers (i.e., no closed neighborhood contains two identical identifiers) and assumes the shared mem-ory model of communication, where neighboring nodes can communicate via common variables or registers. No-tice that the shared memory models have two variants: state reading model and link-register model; we assume the former, in which each node can directly read the inter-nal state of its neighbors.

The program of every node comprises a collection of rules of the form:



precondition

 → 

statement

.

The precondition is a Boolean expression involving the states of the node and its neighbors. The statement up-dates the state of the node. The execution of a statement is http://dx.doi.org/10.1016/j.ipl.2014.04.011

516 W.Y. Chiu et al. / Information Processing Letters 114 (2014) 515–518

called a move. A rule is enabled (or privileged) if its precon-dition evaluates to be true. A node is enabled (or privileged) if at least one of its rules is enabled. It is assumed that rules are atomically executed, that is, the evaluation of a precondition and the move are performed in one atomic step. More precisely, we use composite atomicity, meaning that a node may read all its input variables, perform a state transition, and write all its output variables in a sin-gle atomic step.

Various execution models have been used in self-stabilizing algorithms and these models are encapsulated within the notion of a daemon (or scheduler): the central daemon, the synchronous daemon, and the distributed dae-mon. A daemon can be fair or unfair. Refer to[6]for these definitions. It is well-known that an unfair distributed dae-mon is more practical for implementations than the other types of daemons, and it is the daemon used in this paper and[4,9].

In this paper, we consider a distributed system whose topology is represented by an undirected, simple graph G

= (

V

,

E

)

, whose V represents the set of nodes and E represents the set of edges (i.e., interconnections between nodes). Let n

= |

V

|

. If two nodes are connected by an edge, they are called neighbors. A subset S

⊆

V is a dominating set of G if every node of G is either a member of S or adja-cent to a member of S. A dominating set of G is a minimal dominating set (MDS) of G if none of its proper subsets is a dominating set of G. An MDS has an application of clustering in wireless networks and is maintained for min-imizing the number of required resource centers [7]. The MDS problem is that of finding an MDS of a given graph and this is the problem concerned in this paper.

We now briefly review previous results. In [8], Hedet-niemi et al. proposed the first self-stabilizing algorithm for the MDS problem; their algorithm assumes the cen-tral daemon. In [11], Xu et al. presented an algorithm under the synchronous daemon. In [9], Turau proposed a 9n-move algorithm under an unfair distributed dae-mon; this algorithm is the first linear-time self-stabilizing algorithm for the MDS problem. In [4], Goddard et al. presented a 5n-move algorithm. A good survey for self-stabilizing algorithms for the MDS problem can be found in[6].

The time complexity of a self-stabilizing algorithm is estimated in terms of moves or in terms of rounds. As was mentioned in [9], for a wireless system with bounded re-sources, the number of moves is at least as important as the number of rounds. The reason is that a node has to broadcast the state to its neighbors after making a move and therefore a reduction of the number of moves pro-longs the lifetime of the network. The paper[11]uses the number of rounds to estimate the time complexity; but [4,8,9]and this paper uses the number of moves. The main contribution of this paper is to propose a 4n-move self-stabilizing algorithm for the MDS problem under an unfair distributed daemon. We now summarize all the known re-sults inTable 1.

This paper is organized as follow. In Section 2, we present a 4n-move self-stabilizing algorithm for the MDS problem. Concluding remarks are given in Section3. A pre-liminary version of this paper appeared as[1].

Table 1

Self-stabilizing algorithms for the MDS problem.

stabilization time daemon type Hedetniemi et al.[8] (2n+1)n moves central

Xu et al.[11] 4n rounds synchronous

Turau[9] 9n moves distributed

Goddard et al.[4] 5n moves distributed

this paper 4n moves distributed

2. Main result: a 4n-move algorithm

The purpose of this section is to present our main re-sult:

A

MDS, a 4n-move self-stabilizing algorithm for the

MDS problem under an unfair distributed daemon.

A

MDS

uses four (local) states, which are defined by the four-valued variable state. The range of values of state is:

IN

,

OUT1

,

OUT2

, and

WAIT

. A node with state

= IN

will be referred to as an

IN

node. A neighbor is an

IN

neighbor if it is an

IN

node. Let S

= {

v

:

v

.

state

= IN}

. Nodes with state

= OUT1

or

OUT2

or

WAIT

will be referred to as

OUT

nodes.

The values of state have the following meaning. The value

IN

indicates that the node is in the MDS. The value

OUT1

means that the node is not in the MDS and it has a unique

IN

neighbor. The value

OUT2

indicates that the node is not in the MDS and it has at least one

IN

neighbor. The value

WAIT

means that the node is not in the MDS and it does not have any

IN

neighbor. To make it precise, in our self-stabilizing MDS algorithm, a legitimate state is: the set of

IN

nodes form an MDS, every

OUT

node with state

= OUT1

has a unique

IN

neighbor, every

OUT

node with state

= OUT2

has at least one

IN

neighbor, and there is no node with state

= WAIT

.

Let N

(

v

)

denote the set of neighbors of node v and v

.

id denote the identifier of v. To formally define the rules of

A

MDS, the following predicates defined for each node v are

needed:

•

noBtNbr

≡ 

u

∈

N

(

v

)

:

u

.

state

= WAIT ∧

u

.

id

<

v

.

id.

•

noDpNbr

≡ 

u

∈

N

(

v

)

:

u

.

state

= OUT1

.

When two or more neighboring nodes want to enter the MDS simultaneously, our algorithm chooses the one with the smaller (smallest) id. According to this, if v is an

OUT

node and has a neighbor u such that u

.

state

= WAIT

and u

.

id

<

v

.

id, then u is called a better neighbor. The predicate noBtNbr indicates that v has no better neigh-bor. Also, if v is an

IN

node and has a neighbor u with u

.

state

= OUT1

, then u is called a dependent neighbor since u depends on its unique neighbor in the MDS (the neigh-bor is v). The predicate noDpNbr indicates that v has no dependent neighbor.

For convenience, we introduce InNbr

= |{

u

∈

N

(

v

)

|

u

.

state

= IN}|

. The algorithm

A

MDS uses the following six

rules and its state diagram is given inFig. 1.

R1. state

= WAIT ∧

InNbr

=

0

∧

noBtNbr

→

state

:= IN

. R2. state

= IN ∧

InNbr

=

1

∧

noDpNbr

→

state

:= OUT1

. R3. state

= IN ∧

InNbr

>

1

∧

noDpNbr

→

state

:= OUT2

. R4. state

= WAIT ∧

InNbr

=

1

→

state

:= OUT1

.

W.Y. Chiu et al. / Information Processing Letters 114 (2014) 515–518 517

Fig. 1. The state diagram ofAMDS.

R5.

(

state

= OUT1 ∨

state

= WAIT) ∧

InNbr

>

1

→

state

:=

OUT2

.

R6.

(

state

= OUT1 ∨

state

= OUT2) ∧

InNbr

=

0

→

state

:=

WAIT

.

We now prove the correctness of

A

MDS.

Lemma 2.1. In any state in which no node is enabled the set S

is a minimal dominating set of G.

Proof. Suppose to the contrary that S is not a minimal

dominating set of G. Then either (i) S is not a dominat-ing set or (ii) S is a dominatdominat-ing set but not minimal. First consider (i). Since S is not a dominating set, there exists at least one node u

∈

/

S which has no

IN

neighbor; let Sbe the set of all such nodes. Since rule 6 is not enabled, every node in Shas state

= WAIT

. Let u0be the node in Swith

minimum id. Then u0satisfies all the constraints of rule 1.

Hence, rule 1 is enabled and this contradicts to the as-sumption that no node is enabled. Now consider (ii). Since S is a dominating set but not minimal, there must exist at least one node u

∈

S such that S

\{

u

}

is also a dominat-ing set of G. Then

|

N

(

u

)

∩

S

| ≥

1 and for all u in N

(

u

)

\

S, we have

|

N

(

u

)

∩

S

| ≥

2. Thus, every node u in N

(

u

)

\

S has InNbr

>

1. Hence, every node uin N

(

u

)

\

S must have u

.

state

= OUT2

; otherwise rule 5 is enabled on u. Conse-quently, node u has noDpNbr

=

true and either InNbr

=

1 (if

|

N

(

u

)

∩

S

| =

1) or InNbr

>

1 (if

|

N

(

u

)

∩

S

| >

1). Hence, either rule 2 or rule 3 is enabled on node u, which is a contradiction.

2

We now show that

A

MDS converges in finite time. In

particular, we show that the number of moves of

A

MDS

is at most 4n

−

2. Let k be a nonnegative integer and



r1

,

r2

, . . . ,

rk



be a sequence of rules (ri’s are not necessar-ily distinct). The sequence



r1

,

r2

, . . . ,

rk



is called a move sequence if a node can execute r1, then r2

, . . .

, then rk. The following two lemmas show that in any possible move se-quence of a specific node, rule 1 and rule 6 will appear at most once.

Lemma 2.2. If a node executes rule 1, then it will not execute

any other rule. Consequently, if a node enters the set S, then it will never leave S.

Proof. Let v be a node which executes rule 1. Then v.state is set to

IN

and v enters S. By the precondition of rule 1,

v has no

IN

neighbor and no better neighbor; therefore no neighbor of v enters S at the same time. Thus, no node in N

(

v

)

enters S and therefore InNbr

=

0. After executing rule 1, v

.

state is

IN

and the possible rule that v can ex-ecute is either rule 2 or rule 3. Rule 2 is impossible since it requires InNbr

=

1; rule 3 is also impossible since it re-quires InNbr

>

1. Therefore, v will not execute any other rule. The second statement of this lemma now follows.

2

Lemma 2.3. A node can execute rule 6 at most once, or

equiva-lently, a node can set its state to

WAIT

at most once.

Proof. Let v be a node which executes rule 6. By the

pre-condition of rule 6, v has no

IN

neighbor. After executing rule 6, v

.

state is set to

WAIT

and the possible rule that v can execute is rule 1 or rule 4 or rule 5. If v executes rule 1, then byLemma 2.2, v will not execute any other rule and we have this lemma. If v executes rule 4, then InNbr

=

1 must be true before rule 4 is enabled, meaning that v has a neighbor (say, u) which has executed rule 1; byLemma 2.2, u will never leave S and therefore it is im-possible to have InNbr

=

0, which means that v cannot execute rule 6 again. Similarly, if v executes rule 5, then InNbr

>

1 must be true before rule 5 is enabled, meaning that v has two neighbors (say, u and w) which have exe-cuted rule 1; byLemma 2.2, both u and w will never leave S and therefore it is impossible to have InNbr

=

0, which means that v cannot execute rule 6 again.

2

Theorem 2.4. The proposed algorithm

A

MDSis self-stabilizing

under an unfair distributed daemon and it stabilizes after at most 4n

−

2 moves with a minimal dominating set, where n is the number of nodes. Moreover, the bound 4n

−

2 is tight.

Proof. ByLemma 2.1,

A

MDSis correct. To prove that

A

MDS

stabilizes after at most 4n

−

2 moves, we first prove that it stabilizes after at most 4n moves. To do this, it suffices to show that any move sequence of a node is of length at most 4 under an unfair distributed daemon. Let v be an arbitrary node in G. ByLemma 2.3, v can execute rule 6 at most once. Thus, there are two cases: v never executes rule 6 and v executes rule 6 once.

First consider the case that v never executes rule 6. Then v

.

state never changes to

WAIT

. Thus, the move se-quence of v is either



1



or



2

,

5



or



4

,

5



. It follows that any move sequence of v is of length at most 2.

Now consider the case that v executes rule 6 once. In this case, regard a move sequence of v as the concate-nation of a prefix and a suffix. ByLemma 2.2, the prefix of any move sequence of v cannot contain 1 since if v executes rule 1 then v will not execute any other rule, including rule 6. Hence, the possible prefix of any move sequence of v is either



2

,

6



or



3

,

6



or



4

,

6



or



5

,

6



. After v executes rule 6, v

.

state changes to

WAIT

. Thus, the possible suffix of any move sequence of v is either



6

,

1



or



6

,

4

,

5



or



6

,

5



. Concatenating the prefix and suffix, we conclude that any move sequence of v is of length at most 4.

From the above,

A

MDSstabilizes after at most 4n moves

518 W.Y. Chiu et al. / Information Processing Letters 114 (2014) 515–518

bound can be strengthened to 4n

−

2. The cases of n

=

1 and n

=

2 are trivial. Suppose n

≥

3 and one of the nodes makes 4 moves; by the above argument, this node has two neighbors executing rule 1. Thus, at least two nodes in G make less than 4 moves and the upper bound can be strengthened to 4n

−

2. We now prove that this bound is tight. Consider the complete bipartite graph K2,n−2, where

n

≥

3. Let the two nodes in the partite of cardinality two have the maximum and the minimum identifiers among the n nodes. If initially all nodes are in state

IN

, then there is a way that all the rest of the nodes executes



3

,

6

,

4

,

5



but nodes with the maximum and minimum identifiers ex-ecute



3

,

6

,

1



. All together 4n

−

2 moves are made.

2

Before ending this section, we would like to point out an error in the modified MDS algorithm in [9](denote it as

A

_MDShere). The author claimed (in page 93) that rule 4 can be changed by replacing the predicate inNeighbor with inNeighborWithLowerId so that the total number of moves can be further reduced. We now show that the author’s claim is incorrect. Suppose the replacement is done and the resultant rule is called rule 4. Let G be a path of three nodes v1

−

v2

−

v3. Suppose vi

.

id is i and suppose the initial states are: v1

,

v2 are

IN

nodes with dependent

= Λ

and v3is an

OUT

node with dependent

=

v2. Since

{

v1

,

v2

}

is not an MDS, algorithm

A

_MDS must make a move. It is easy to verify that no rule can be enabled by

A

_MDS.

3. Concluding remarks

The main result of this paper is a

(

4n

−

2

)

-move self-stabilizing algorithm for the MDS problem using an unfair distributed daemon and the bound 4n

−

2 is tight. The pre-vious best algorithm is a 5n-move algorithm. The model that we use is the normal model, also called the distance-1 model. Notice that in [3] and [10], the authors consid-ered the distance-2 model, in which every node can read

the states of nodes up to distance 2; see also [5]for the distance-k model.

Acknowledgements

The authors would like to greatly thank the referees for their constructive comments and suggestions that greatly improve the presentation of this paper.

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