Building Fastest Broadcast Trees in Periodically-Varying Graphs

Delay-tolerant networks (DTNs) are characterized by a possible absence of end-to-end communication routes at any instant. Still, connectivity can generally be established over time and space. The optimality of a temporal path (journey) in this context can be defined in several terms, including topological (e.g. {\em shortest} in hops) and temporal (e.g. {\em fastest, foremost}). The combinatorial problem of computing shortest, foremost, and fastest journeys {\em given full knowledge} of the network schedule was addressed a decade ago (Bui-Xuan {\it et al.}, 2003). A recent line of research has focused on the distributed version of this problem, where foremost, shortest or fastest {\em broadcast} are performed without knowing the schedule beforehand. In this paper we show how to build {\em fastest} broadcast trees (i.e., trees that minimize the global duration of the broadcast, however late the departure is) in Time-Varying Graphs where intermittent edges are available periodically (it is known that the problem is infeasible in the general case even if various parameters of the graph are know). We address the general case where contacts between nodes can have arbitrary durations and thus fastest routes may consist of a mixture of {\em continuous} and {\em discontinuous} segments (a more complex scenario than when contacts are {\em punctual} and thus routes are only discontinuous). Using the abstraction of \tclocks to compute the temporal distances, we solve the fastest broadcast problem by first learning, at the emitter, what is its time of {\em minimum temporal eccentricity} (i.e. the fastest time to reach all the other nodes), and second by building a {\em foremost} broadcast tree relative to this particular emission date.

💡 Research Summary

The paper tackles the problem of constructing a fastest broadcast tree in time‑varying graphs (TVGs) whose edges appear intermittently but follow a known periodic schedule. In delay‑tolerant networks (DTNs), end‑to‑end connectivity may be absent at any instant, yet communication becomes possible over time through temporal paths (journeys). While earlier work (Bui‑Xuan et al., 2003) solved shortest, foremost, and fastest journeys under the assumption of complete knowledge of the network schedule, the distributed version—where nodes must operate without prior schedule information—remains largely unsolved. In particular, the fastest broadcast problem is provably infeasible in arbitrary TVGs, even when some graph parameters are known.

Model and Assumptions
The authors consider a periodic TVG: every edge e ∈ E repeats its availability every period p. Each occurrence of e is defined by a start offset s(e) (0 ≤ s(e) < p) and a duration δ(e) ≥ 0. The duration may be zero (punctual contact) or positive (continuous contact), allowing the model to capture realistic scenarios where links persist for non‑trivial intervals. Nodes are unaware of the schedule a priori; they can only observe edge activations and deactivations as they happen.

Temporal Eccentricity and the Fastest Broadcast Goal
For a node u and a reference time t, the temporal distance d_t(u, v) is the earliest arrival time at v when a message departs u at t, respecting causality and edge availability. The temporal eccentricity ε_t(u) = max_v d_t(u, v) measures how long it takes u to reach the farthest node if it starts broadcasting at time t. The fastest broadcast problem asks for a departure time t* that minimizes ε_t(u); the resulting broadcast duration ε_min = ε_{t*}(u) is the global optimum, irrespective of when the broadcast actually starts.

t‑Clocks: A Distributed Temporal‑Distance Tracker
The central technical tool is the t‑clock, a lightweight local data structure that each node maintains to estimate its temporal distances to all other nodes in real time. Whenever an edge becomes active, its incident nodes exchange their current distance vectors, compute potential new arrival times through the edge, and update entries if a strictly earlier arrival is discovered. When an edge deactivates, nodes invalidate any paths that relied exclusively on that edge and recompute alternatives. This event‑driven propagation ensures that, after any finite observation window, each node’s t‑clock contains the exact temporal distances for the observed interval.

Algorithm Overview
The solution proceeds in two phases:

1. Learning Phase – The source node s monitors the network for one full period p. Using its t‑clock, s records ε_t(s) for every discrete time instant t ∈