History Trees and Their Applications

TL;DR

History trees are presented as a temporal, discrete data structure for analyzing anonymity in dynamic networks, capturing how indistinguishability evolves with message multisets. The note situates history trees among static notions such as Yamashita–Kameda views and Boldi–Vigna’s minimum base, and extends these ideas to dynamic settings to yield optimal linear-time solutions for core tasks like Average Consensus and Counting. It surveys a broad range of variations—Leader Election, Directed/Disconnected/Semi-Synchronous/Asynchronous, Port Awareness, Varying Inputs, Self-Stabilization, Finite-State Stabilization, Memoryless Computation, and Congested Networks—along with open problems and practical considerations, including simulation tooling. The results establish that, by building and merging agent views, a wide class of problems can be reduced to analyzing history-tree structures, enabling precise time bounds and robust handling of topology changes in anonymous networks.

Abstract

In the theoretical study of distributed communication networks, "history trees" are a discrete structure that naturally models the concept that anonymous agents become distinguishable upon receiving different sets of messages from neighboring agents. By conveniently organizing temporal information in a systematic manner, history trees have been instrumental in the development of optimal deterministic algorithms for networks that are both anonymous and dynamically evolving. This note provides an accessible introduction to history trees, drawing comparisons with more traditional structures found in existing literature and reviewing the latest advancements in the applications of history trees, especially within dynamic networks. Furthermore, it expands the theoretical framework of history trees in new directions, also highlighting several open problems for further investigation.

Figures (10)

• Figure 1: The first communication steps of a dynamic network and the corresponding levels of its history tree. Initial inputs are represented by agents' colors (cyan and yellow). Labels on agents and nodes have been added for the reader's convenience and indicate classes of indistinguishable agents. The portion of the history tree with a green background is the view of the two agents labeled $c_1$ after two communication steps.
• Figure 2: Updating the view of an agent represented by node $v$ after it receives the view of an agent represented by $u$ as a message. The two views are matched and merged starting from the roots, $v$ gets a child $v'$, and a new red edge from $u$ to $v'$ is added.
• Figure 3: A static network $G$ with a distinguished agent $p$ and its history tree $\mathcal{H}_G$ (truncated at level $L_4$). At stabilization, the directed graph of the red edges between non-branching levels (highlighted in green in $\mathcal{H}_G$) is isomorphic to the Boldi--Vigna minimum base $\widehat{G}$. The view $\mathcal{V}_G^3(p)$ after step $3$ is related to the Yamashita--Kameda view $\mathcal{T}_G^3(p)$ truncated at depth $3$, as well as to its Tani folded view $\widehat{\mathcal{T}}_G^3(p)$, as illustrated. The precise correspondence between these structures is described in \ref{['s:2.related']}.
• Figure 4: A static network with two connected components of sizes $n$ and $n+1$. The history tree shows that the two agents in cyan become distinguishable after $2n-1$ communication steps. Therefore, in static networks of $n$ agents with a unique leader, no Counting algorithm can terminate in $2n-2$ steps or stabilize in $2n-4$ steps.
• Figure 5: The numbers in the nodes are guesses made according to the algorithm in \ref{['s:3.term']}. Blue numbers indicate heavy nodes; their guesses are necessarily correct except for the one in (e), where guesses are not well spread.