4.3 Four levels of stableness
Stableness is the quality or state of being stable, or firmly established. In distributed computing, stableness informally requires that “good configurations eventually stop” in a distributed system or algorithm (i.e., the system or the algorithm “resists change”).
We have defined self-stabilizing MDS-covered, MDS-silent, and MDS-stable algorithms.
We now generalize these definitions. A configuration is silent if no rules are enabled at any node. In the literature, self-stabilizing graph algorithms are usually silent, that is, the algorithms eventually reach a silent configuration in a finite time. Let P denote a desired property. Let Λ′ be the set of all decisions (d1, d2, . . . , dn) satisfy the desired property P , where dv is the decision of node v. Self-stabilizing algorithms can be described as having different degrees of stableness with respect to property P .
Definition 4.3.1. An algorithm is P if after stabilizing, the configuration conforms the desired property P . A P algorithm is:
P -covered if after stabilizing, the configuration potentially can be decoded to any possible element in Λ′.
P -stable if the distributed system starts with an element in Λ′, the algorithm makes no membership moves.
P -silent if the distributed system starts with an element in Λ′, the algorithm makes no moves.
Let ΣAbe the set of all silent configurations in a silent self-stabilizing algorithm A. If ΣA is a proper subset of Λ, i.e., all silent configurations are legitimate, then it is easy to design a self-stabilizing P algorithm by shrinking the definition of legitimate configurations ΛA of an algorithm A from Λ to ΣA. One advantage of doing this is that we do not need to prove the closure property. Thus, by setting ΛA := ΣA, a silent self-stabilizing graph algorithm
CHAPTER 4. THE STABLENESS OF MDS ALGORITHMS 4.3. FOUR LEVELS OF STABLENESS
only needs to prove the convergence property and the correctness, where correctness is that all silent configurations are legitimate, that is, ΣA⊆ Λ
Now consider a silent self-stabilizing P algorithm and let Λ⊆ Γ be the set of all legitimate configurations in a input graph G, which conform to P . By the arguments in previous section, there exists a variable called membership and a variable called inf ormation which keeps the neighborhood information. Let Γ′ be the vector space expanded by decisions of all nodes, and let P roj : Γ→ Γ′ be the projection function defined by
P roj((q1, q2, . . . , qn)) = (d1, d2, . . . , dn),
where dv = Dec(qv) for all v ∈ V . An equivalent definition of stableness can be given by setting the legitimate configurations of algorithm ΛA equal to the silent configurations ΣA, which conform to the desired property P as follows.
Definition 4.3.2. A silent self-stabilizing P algorithm A is
• S1-stable : if P roj(ΛA)⊆ Λ′.
• S2-stable : if P roj(ΛA) = Λ′.
• S3-stable : if P roj(ΛA) = Λ′ and
for all γ ∈ P roj−1(Λ′) and γ → γ′, P roj(γ′) = P roj(γ).
• S4-stable : if P roj−1(Λ′) = ΛA.
The level of stableness is shown by the index i of Si. The larger index will be seen as being more stable. Note that these are increasingly stringent requirements; see the following theorem.
Theorem 4.3.3. Si+1-stableness implies Si-stableness, for i∈ {1, 2, 3}.
Proof. We give one line proof for each index i:
CHAPTER 4. THE STABLENESS OF MDS ALGORITHMS 4.3. FOUR LEVELS OF STABLENESS
(S4 ⇒ S3) : P roj−1(Λ′) = ΛA implies ∀γ ∈ P roj−1(Λ′), γ′ = γ; hence P roj(γ′) = P roj(γ).
(S3 ⇒ S2) : by definition, S3 satisfies P roj(ΛA) = Λ′. (S2 ⇒ S1) : Λ′ ⊆ Λ′.
Another way to classify the degrees of stableness is to describe the convergence of an algorithm A.
Definition 4.3.4. If the desired property is P , we say an algorithm A is
• single-point-stable: if A converges to one specific point in Λ′.
• subset-stable: if A converges to a subset of Λ′.
• whole-set-stable: if A potentially can converge to every point in Λ′.
• no-membership-move: if A makes no membership move when the initial configuration is in Λ′.
• no-move: if A makes no move when the initial configuration is in Λ′.
For example, MDS algorithms Turau9n, Goddard5n, and Well4n are whole-set-stable, and MIS algorithm Turau3n is subset-stable with respect to MDS. The containment relations and the corresponding proofs are listed below.
• whole-set-stable ⇒ subset-stable: Λ′ ⊆ Λ′.
• single-point-stable ⇒ subset-stable: For all λ′ ∈ Λ′, {λ′} ⊆ Λ′.
• no-membership-move ⇒ whole-set-stable: For all λ′ ∈ Λ′, if A is initialized with λ′, then the projection of the configuration after stabilizing is λ′.
• no-move ⇒ no-membership-move: membership moves are moves.
CHAPTER 4. THE STABLENESS OF MDS ALGORITHMS 4.3. FOUR LEVELS OF STABLENESS
Table 4.1: The levels of stableness and convergence of self-stabilizing algorithms.
Stableness defined by
Level Degree Convergence Algorithms w.r.t. MDS
S1-stable P subset-stable MIS algorithms (e.g. Turau3n) S2-stable P -covered whole-set-stable Turau9n, Goddard5n, Well4n S3-stable P -stable no-membership-move ?
S4-stable P -silent no-move none
We now summarize the levels of stableness and convergence in Table 4.1.
In the remaining tables, Dist-2 means Distance-2. Table 4.2 summarizes the levels of stableness of self-stabilizing MDS algorithms. As can be observed from Table 4.2, Algorithm Well4n has the best performance among known S2-stable MDS algorithms.
Table 4.2: Self-stabilizing MDS algorithms on general graphs with different levels of stableness.
Reference Degree Model Daemon Complexity
Hedetniemi et al. [36] MDS-covered Normal Central (2n + 1)n moves Xu et al. [83] MDS-covered Normal Synchronous 4n rounds Turau [75] MDS-covered Normal Distributed 9n moves Goddard et al. [30] MDS-covered Normal Distributed 5n moves Sec. 3.2 in this thesis MDS-covered Normal Distributed 4n− 2 moves Sec. 5.3 in this thesis MDS-silent Dist-2 Distributed 2n− 1 moves
Similar to the tables in [34], we list the stableness of each self-stabilizing independent set and dominating set algorithms in Table 4.3 and Table 4.4.
Recall that for a graph G = (V, E), a vertex subset S is independent if no two nodes in S are adjacent. A maximal independent set (MIS) is an independent set that is not properly contained in any independent set. Note that an MIS is also an MDS. A 1-maximal independent set (1-MIS) is an MIS, with the additional property that one cannot increase the cardinality of the independent set by removing one node and adding more nodes. A vertex subset of S is a k-packing if d(u, v) > k for all pairs of distinct vertices u and v of S [61]. A maximal 2-packing (M2P) is a 2-packing that is not properly contained in any
CHAPTER 4. THE STABLENESS OF MDS ALGORITHMS 4.3. FOUR LEVELS OF STABLENESS
2-packing.
In a graph G = (V, E), a vertex subset S is called a dominating set (DS) if every vertex is either a member of S or is adjacent to a member of S. A minimal dominating set (MDS) is a dominating set such that no proper subset of it is a dominating set. A vertex subset S is a total dominating set (TDS) if each vertex of the graph is adjacent to a member in S. A vertex subset S is a k-dominating set (kDS) if each nonmember of S is adjacent to at least k members in S. When the positive integer k = 1, it is a question of (single) domination.
The minimal total dominating sets (MTDS) and minimal k-dominating sets (MkDS) can be similarly defined. A function f : V → N is {k}-dominating if for every v ∈ V we have
∑
u∈N[v]f (u)≥ k. The case k = 1 is a normal dominating set [21].
In summery, a subset S of V is:
independent if ∀i, j ∈ S, ij ̸∈ E.
maximal independent if S is independent and any subset of V properly containing S is not independent.
k-packing if ∀i ∈ V , either i ∈ S or i ∈ Nk(S).
dominating if ∀i ∈ V , either i ∈ S or i ∈ N(S).
minimal dominating if S is dominating and no proper subset of S is dominating.
total dominating if ∀i ∈ V , i ∈ N(S).
k-dominating if ∀i ∈ V , either i ∈ S or |N(i) ∩ S| ≥ k.
The stableness of each algorithm in Table 4.3 is S1 if the output is replaced by MDS.