Delay Tolerant Networking (DTN) aims to address a myriad of significant networking challenges that appear in time-varying settings, such as mobile and satellite networks, wherein changes in network topology are frequent and often subject to environmental constraints. Within this paradigm, routing problems are often solved by extending classical graph-theoretic path finding algorithms, such as the Bellman-Ford or Floyd-Warshall algorithms, to the time-varying setting; such extensions are simple to understand, but they have strict optimality criteria and can exhibit non-polynomial scaling. Acknowledging this, we study time-varying shortest path problems on metric graphs whose vertices are traced by semi-algebraic curves. As an exemplary application, we establish a polynomial upper bound on the number of topological critical events encountered by a set of $n$ satellites moving along elliptic curves in low Earth orbit (per orbital period). Experimental evaluations on networks derived from STARLINK satellite TLE's demonstrate that not only does this geometric framework allow for routing schemes between satellites requiring recomputation an order of magnitude less than graph-based methods, but it also demonstrates metric spanner properties exist in metric graphs derived from real-world data, opening the door for broader applications of geometric DTN routing.

翻译：延迟容忍网络（DTN）旨在解决时变环境（如移动网络和卫星网络）中出现的众多重大网络挑战，这些环境中网络拓扑变化频繁且常受环境约束。在此范式下，路由问题通常通过将经典图论路径查找算法（如Bellman-Ford或Floyd-Warshall算法）扩展至时变场景来解决；此类扩展易于理解，但其最优性标准严格且可能呈现非多项式复杂度。基于此，我们研究了度量图上顶点由半代数曲线轨迹描述的时变最短路径问题。作为示例应用，我们针对沿低地球轨道椭圆曲线运行的n颗卫星（每轨道周期内），建立了其遭遇拓扑关键事件数量的多项式上界。基于STARLINK卫星TLE数据构建的网络实验评估表明：该几何框架不仅使卫星间路由方案的重计算需求比基于图的方法降低一个数量级，还揭示了从实际数据导出的度量图中存在度量生成树性质，这为几何DTN路由的更广泛应用开启了新途径。