Let $$G$$ be a random geometric graph formed by $$n$$ nodes with adjacency distance $$r_n$$ and let each edge of $$G$$ be assigned an independent exponential passage time with mean that depends on the graph size $$n.$$ We connect $$G$$ to two nodes source $$s_A$$ and destination $$s_B$$ at deterministic locations spaced $$d_n$$ apart in the unit square and find upper and lower bounds on the minimum passage time between $$s_A$$ and $$s_B$$ through paths in $$G$$ having constant stretch, i.e., whose length is constrained to be proportional to the Euclidean distance between $$s_A$$ and $$s_B.$$

中文翻译：

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随机几何图中的受限最小通过时间

令 $$G$$ 是由 $$n$$ 个节点形成的随机几何图，相邻距离为 $$r_n$$，并让 $$G$$ 的每条边被分配一个独立的指数通过时间，其均值取决于图大小 $$n.$$ 我们将 $$G$$ 连接到两个节点，源 $$s_A$$ 和目的地 $$s_B$$ 在单位正方形中间隔 $$d_n$$ 的确定性位置，并找到上和下$$s_A$$ 和 $$s_B$$ 之间通过具有恒定拉伸的 $$G$$ 路径的最短通过时间的界限，即，其长度被约束为与 $$s_A$$ 和 $$s_A$$ 之间的欧几里得距离成比例$$s_B.$$