A Spectral Approach to the Shortest Path Problem

04/02/2020

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by   Stefan Steinerberger, et al.

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Let G=(V,E) be a simple, connected graph. One is often interested in a short path between two vertices u,v. We propose a spectral algorithm: construct the function ϕ:V →ℝ_≥ 0 ϕ = min_f:V →ℝ f(u) = 0, f ≢0∑_(w_1, w_2) ∈ E(f(w_1)-f(w_2))^2/∑_w ∈ Vf(w)^2. ϕ can also be understood as the smallest eigenvector of the Laplacian Matrix L=D-A after the u-th row and column have been removed. We start in the point v and construct a path from v to u: at each step, we move to the neighbor for which ϕ is the smallest. This algorithm provably terminates and results in a short path from v to u, often the shortest. The efficiency of this method is due to a discrete analogue of a phenomenon in Partial Differential Equations that is not well understood. We prove optimality for trees and discuss a number of open questions.