Memory Lower Bounds and Impossibility Results for Anonymous Dynamic Broadcast

TL;DR

This work studies memory requirements for deterministic broadcast in anonymous, synchronous, 1-interval connected dynamic networks. It combines impossibility proofs with tight lower bounds and a practical logspace algorithm (Countdown) to achieve stabilizing termination without identifiers or knowledge of $n$, showing that nontrivial terminating tasks in such networks inherently require non-constant memory. The key contributions are a termination-detection impossibility for idle-start settings, a $\Omega(\log n)$ memory lower bound for non-idle-start, a $\omega(1)$ memory lower bound for idle-start stabilizing termination, and a $\mathcal{O}(\log n)$-space algorithm achieving stabilizing termination in $\mathcal{O}(n)$ rounds. These results illuminate the memory requirements for terminating computations in anonymous dynamic networks and have implications for privacy-preserving distributed systems and dynamic-overlay designs.

Abstract

Broadcast is a ubiquitous distributed computing problem that underpins many other system tasks. In static, connected networks, it was recently shown that broadcast is solvable without any node memory and only constant-size messages in worst-case asymptotically optimal time (Hussak and Trehan, PODC'19/STACS'20/DC'23). In the dynamic setting of adversarial topology changes, however, existing algorithms rely on identifiers, port labels, or polynomial memory to solve broadcast and compute functions over node inputs. We investigate space-efficient, terminating broadcast algorithms for anonymous, synchronous, 1-interval connected dynamic networks and introduce the first memory lower bounds in this setting. Specifically, we prove that broadcast with termination detection is impossible for idle-start algorithms (where only the broadcaster can initially send messages) and otherwise requires $Ω(\log n)$ memory per node, where $n$ is the number of nodes in the network. Even if the termination condition is relaxed to stabilizing termination (eventually no additional messages are sent), we show that any idle-start algorithm must use $ω(1)$ memory per node, separating the static and dynamic settings for anonymous broadcast. This lower bound is not far from optimal, as we present an algorithm that solves broadcast with stabilizing termination using $\mathcal{O}(\log n)$ memory per node in worst-case asymptotically optimal time. In sum, these results reveal the necessity of non-constant memory for nontrivial terminating computation in anonymous dynamic networks.

Figures (3)

• Figure 1: The time-varying graph $\mathcal{G}$ used in the proof of Theorem \ref{['thm:detection-impossibility']}. In each round $t$, the broadcaster $b$ is connected to node $v_t$ in the path $P$. Informed nodes are shown in green. When $b$ declares broadcast to be complete in round $k$, node $v_{k+1}$ is still uninformed.
• Figure 2: The inductive construction of configuration $C_i$ as described in the proof of Theorem \ref{['thm:detection-bound']}. The key idea is to find a new pair of states $(\beta_i, \alpha_i)$ that is not already in $(\beta_0, \alpha_0), \ldots, (\beta_{i-1}, \alpha_{i-1})$ by arranging a previously identified reachable configuration $C_k$ and extending the corresponding execution of $\mathcal{A}$ by one additional round.
• Figure 3: The time-varying graph structures used in the proof of Theorem \ref{['thm:stable-bound']}.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Lemma 5
• Lemma 6
• Theorem 7
• Lemma 8
• Lemma 9
• Theorem 10