Deterministic Self-Stabilizing BFS Construction in Constant Space

TL;DR

This work resolves the long-standing question of whether a deterministic self-stabilizing spanning tree can be constructed in a semi-uniform network using constant per-node memory. The authors introduce TokBin, a synchronous algorithm that builds a BFS^{\equiv 3} structure by propagating a root-originated token stream and leveraging a binary bit mechanism to limit token lifetimes, thereby eliminating cycles and correcting misconfigurations. The protocol uses only 6 bits per node and stabilizes in $O(2^{\varepsilon})$ steps, where $\varepsilon$ is the eccentricity of the root, without requiring global knowledge such as diameter or node count. This constant-space approach opens new directions for self-stabilizing protocols under severe resource constraints and highlights a novel token-distribution technique that may inform other memory-restricted distributed tasks.

Abstract

In this paper, we resolve a long-standing question in self-stabilization by demonstrating that it is indeed possible to construct a spanning tree in a semi-uniform network using constant memory per node. We introduce a self-stabilizing synchronous algorithm that builds a breadth-first search (BFS) spanning tree with only $O(1)$ bits of memory per node, converging in $2^ε$ time units, where $ε$ denotes the eccentricity of the distinguish node. Crucially, our approach operates without any prior knowledge of global network parameters such as maximum degree, diameter, or total node count. In contrast to traditional self-stabilizing methods, such as pointer-to-neighbor communication or distance-to-root computation, that are unsuitable under strict memory constraints, our solution employs an innovative constant-space token dissemination mechanism. This mechanism effectively eliminates cycles and rectifies deviations in the BFS structure, ensuring both correctness and memory efficiency. The proposed algorithm not only meets the stringent requirements of memory-constrained distributed systems but also opens new avenues for research in self-stabilizing protocols under severe resource limitations.

Figures (4)

• Figure 1: The gray nodes in each subfigure represent activatable nodes. (a) Node $v$ possesses and offers a token. Node $u$ acquires the token by applying the rule $R_{tok}$, and node $v$ increments its bit using $R_{add}$. (b) Node $v$ applies rule $R_{ready}$ to indicate that it is ready to receive a new token. Node $u$ holds the token, it releases the token and increments its bit by applying $R_{add}$. Simultaneously, the node $w$ executes rule $R_{tok}$ to acquire the token. (d) Node $x$ holds the token; however, node $y$ does not accept it because node $x$ has $b_x = \text{\larger[0]{$\mathbf{0}$}}$.
• Figure 2: $f,\text{\larger[0]{$\mathbf{1}$}}$ means $t=\mathit{false}\wedge b=\text{\larger[0]{$\mathbf{1}$}}$, and $f,\text{\larger[0]{$\mathbf{0}$}}$ means $t=\mathit{false}\wedge b=\text{\larger[0]{$\mathbf{0}$}}$
• Figure 3: Dotted nodes denote nodes with a token. We have $d^r_v=9;\mathcal{R}(v)=(2^{9-1}-1)\times 3 +9-1=773;\texttt{B}^2(\gamma,v)=10011100;\texttt{B}^{10}(\gamma,v)=156;\mathcal{R}^c(\gamma,v)=773-156=617$ steps.
• Figure 4: The gray nodes are activatable. The information displayed on each node, respectively, corresponds to the rank, the value of the variable $t$, and the value of the variable $b$. (a) The node $v$ is not in error in this configuration, but it does not satisfy $Legal^{\pi}(\gamma,v)$. In fact, regarding the variable $b$, the path $u_1, w_1, x_1$ currently has the value $110$, while $u_2, w_2, x_2$ has $100$, so $Legal^{\pi}(\gamma,v)=\mathit{false}$. This discrepancy means that $v$ will receive the next token from node $x_1$, so $v$ will receive a token from only one of its parents rather than from both. Consequently, $TakeO(v)$ will be true, and node $v$ will execute $R_{erRank}$. (b) In this case as well, $x_1$ will obtain the token before $x_2$. Consequently, node $v$ executes $R_{er}$ due to $Er_\pi(v)$, and in the next step, both $x_1$ and $x_2$ enter an error state. Step by step, the two paths are deleted.

Theorems & Definitions (10)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Corollary 1
• Corollary 2
• Lemma 7