AI 中文总结
该研究基于之前对贝尔曼森林迷路问题的计算解，提出新方法和公式解决无限单位带形森林和单位宽虫逃逸路径问题，将其转化为区间覆盖及约束泛函最小化问题并离散化求解，还扩展线段分析，指出单位带形封闭路径最优解是恒定单位宽度曲线。
AI 中文摘要
基于我们之前对贝尔曼森林迷路问题的一般计算解决方案，我们提出了一种新方法和解析公式，用于解决扎尔加勒尔之前著名的无限单位带形森林和单位宽虫的逃逸路径问题。早期研究仅通过几何方法解决这些问题。我们将问题重新表述为区间覆盖问题，然后将其表述为约束泛函最小化问题。此约束泛函最小化可直接离散化并作为凸优化求解。此外，我们扩展了对各种线段的分析。最后，我们表明，在单位带形的封闭逃逸路径情况下，最优解是恒定单位宽度的曲线。
英文摘要
Building on our previous general computational solution to Bellman's Lost-in-a-Forest Problem, we present a new approach and analytical formulas for the previously well-known escape path for the infinite unit-strip forest and unit broadworm by Zalgaller. Earlier studies addressed these problems exclusively through geometric methods. We reformulated the problem as an interval-cover problem and then formulated it as a constrained functional minimization problem. This constrained functional minimization can be directly discretized and subsequently solved as a convex optimization. Furthermore, we extend the analysis of various line segment. Finally, we show that, in the case of a closed escape path for the unit strip, the optimal solution is a curve of constant unit width.
Comments12 pages, 5 figures
