Greedy Routing in a Sequentially Grown One-Dimensional Random Graph

2026-04-21

AI Summary

This work provides the first rigorous proof of the long-standing conjecture that greedy routing in a one-dimensional stochastic graph exhibits logarithmic complexity. The graph is constructed by inserting integer lattice points in a uniformly random order and connecting each new node to its nearest left and right neighbors already present. Through a combination of probabilistic analysis, permutation statistics—specifically the count of left-to-right and right-to-left minima—large deviation theory, and asymptotic estimates of harmonic numbers, we establish that the routing length $S_n$ satisfies $S_n = \Theta(\log n)$ with high probability and that $\mathbb{E}[S_n] = 2\log n + o(1)$. Our results precisely characterize the relationship between routing steps and the number of directional minima in the insertion sequence, demonstrate robustness for both arbitrary and random source–destination pairs, and show exponentially decaying tail probabilities for both upper and lower deviations.

Technology Category

• Planning, Routing, and Scheduling: Routing
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

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.

Problem

• greedy routing
• random graph
• logarithmic complexity
• sequential growth
• routing conjecture

Innovation

• greedy routing
• random graph
• logarithmic complexity
• sequential growth
• probabilistic analysis