We analyze greedy routing in a random graph G_n constructed on the vertex set V = {1, 2, ..., n} embedded in Z. Vertices are inserted according to a uniform random permutation pi, and each newly inserted vertex connects to its nearest already-inserted neighbors on the left and right (if they exist). This work addresses a conjecture originating from empirical studies (Ponomarenko et al., 2011; Malkov et al., 2012), which observed through simulations that greedy search in sequentially grown graphs exhibits logarithmic routing complexity across various dimensions. While the original claim was based on experiments and geometric intuition, a rigorous mathematical foundation remained open. Here, we formalize and resolve this conjecture for the one-dimensional case. For a greedy walk GW starting at vertex 1 targeting vertex n -- which at each step moves to the neighbor closest to n -- we prove that the number of steps S_n required to reach n satisfies S_n = Theta(log n) with high probability. Precisely, S_n = L_n + R_n - 2, where L_n and R_n are the numbers of left-to-right and right-to-left minima in the insertion-time permutation. Consequently, E[S_n] = 2H_n - 2 ~ 2 log n and P(S_n >= (2+c) log n) <= n^(-h(c/2) + o(1)) for any constant c > 0, with an analogous lower tail bound for 0 < c < 2, where h(u) = (1+u) ln(1+u) - u is the Bennett rate function. Furthermore, we establish that this logarithmic scaling is robust: for arbitrary or uniformly random start-target pairs, the expected routing complexity remains E[S_{s,t}] = 2 log n + O(1), closely mirroring decentralized routing scenarios in real-world networks where endpoints are chosen dynamically rather than fixed a priori.

翻译：我们分析了定义在整数集嵌入Z上的顶点集V={1,2,...,n}所构建的随机图G_n中的贪婪路由。顶点按照均匀随机排列π依次插入，每个新插入顶点连接其左侧和右侧最近的已插入邻居（若存在）。本研究针对一项源于实证观察的猜想（Ponomarenko等, 2011; Malkov等, 2012），这些模拟研究指出，在顺序生长图中，贪婪搜索在不同维度下均呈现对数级路由复杂度。尽管原始论断基于实验与几何直觉，但严格的数学基础一直缺失。本文对一维情形下的这一猜想进行了形式化证明与解决。对于从顶点1出发、以顶点n为目标的贪婪游走GW（每一步均移至最接近n的邻居），我们证明了到达n所需的步数S_n以高概率满足S_n = Θ(log n)。具体地，S_n = L_n + R_n - 2，其中L_n和R_n分别为插入时间排列中从左到右和从右到左的最小值数目。由此可得E[S_n] = 2H_n - 2 ≈ 2 log n，且对任意常数c > 0有P(S_n ≥ (2+c) log n) ≤ n^{-h(c/2)+o(1)}，对0 < c < 2有相应的尾部下界，其中h(u) = (1+u) ln(1+u) - u为Bennett速率函数。进一步，我们证明该对数标度具有鲁棒性：对于任意或均匀随机的起-终点对，期望路由复杂度保持为E[S_{s,t}] = 2 log n + O(1)，这与实际问题中端点动态选取而非先验固定的去中心化路由场景高度吻合。