Tight Bounds on Window Size and Time for Single-Agent Graph Exploration under T-Interval Connectivity

2026-04-06

AI Summary

This study investigates the minimal window size and time required for a single agent to deterministically explore an unknown T-interval-connected dynamic graph, without prior knowledge of T, the number of nodes n, or edges m. Under two local perception models—KT₀ (neighbor IDs invisible) and KT₁ (neighbor IDs visible)—the work establishes, for the first time, a necessary window size of Ω(m). It also proves lower bounds on exploration time: Ω((m−n+1)n) under KT₀ and Ω(m) under KT₁. The proposed deterministic algorithms achieve tight bounds when m = n^{1+Θ(1)}, attaining Θ(n³) time complexity under KT₀ and Θ(n²) under KT₁, with both window size and exploration time matching the respective theoretical lower bounds.

Abstract

We study deterministic exploration by a single agent in $T$-interval-connected graphs, a standard model of dynamic networks in which, for every time window of length $T$, the intersection of the graphs within the window is connected. The agent does not know the window size $T$, nor the number of nodes $n$ or edges $m$, and must visit all nodes of the graph. We consider two visibility models, $KT_0$ and $KT_1$, depending on whether the agent can observe the identifiers of neighboring nodes. We investigate two fundamental questions: the minimum window size that guarantees exploration, and the optimal exploration time under sufficiently large window size. For both models, we show that a window size $T = Ω(m)$ is necessary. We also present deterministic algorithms whose required window size is $O(ε(n,m)\cdot m + n \log^2 n)$, where $ε(n,m) = \frac{\ln n}{1 + \ln m - \ln n}$. These bounds are tight for a wide range of $m$, in particular when $m = n^{1+Θ(1)}$. The same algorithms also yield optimal or near-optimal exploration time: we prove lower bounds of $Ω((m - n + 1)n)$ in the $KT_0$ model and $Ω(m)$ in the $KT_1$ model, and show that our algorithms match these bounds up to a polylogarithmic factor, while being fully time-optimal when $m = n^{1+Θ(1)}$. This yields tight bounds when parameterized solely by $n$: $Θ(n^3)$ for $KT_0$ and $Θ(n^2)$ for $KT_1$.

Problem

• graph exploration
• T-interval connectivity
• window size
• exploration time
• dynamic networks

Innovation

• T-interval connectivity
• graph exploration
• deterministic algorithm
• tight bounds
• dynamic networks