arXiv:2604.04619 · v2

Online Graph Exploration under T-Interval Connectivity

首次提交 2026-04-06 20:10

版本更新 2026-11-06 10:33

雷达观测 2026-11-09 10:16

摘要

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: in $KT_0$, the agent observes only its current node's identifier and port information; in $KT_1$, it additionally observes the identifiers of its currently available neighbors. 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, and we present deterministic algorithms whose required window size is $O(ξ(n,m)\cdot m + n \log^2 n)$, where $ξ(n,m) = \frac{\ln n}{1 + \ln m - \ln n}$. This window-size bound is tight for a wide range of $m$, in particular when $m = n^{1+Θ(1)}$. Separately, for 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, even when the dynamic graph is $\infty$-interval-connected (i.e., even when all snapshots share a common spanning tree), and our algorithms match these time bounds up to a polylogarithmic factor. These bounds are also tight when parameterized solely by $n$: $Θ(n^3)$ for $KT_0$ and $Θ(n^2)$ for $KT_1$.