A spectral approach to the shortest path problem

Let G = (V, E) be a simple, connected graph. One is often interested in a short path between two vertices u, v is an element of V. We propose a spectral algorithm: construct the function phi : V -> R->= 0 phi = arg min(f:V -> R f(u)=0, f not equivalent to 0) Sigma((w1,w2)is an element of E) (f(w(1)) - f(w(2)))(2)/w is an element of V f(w)(2). phi 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 vand construct a path from vto u: at each step, we move to the neighbor for which phi is the smallest. This algorithm provably terminates and results in a short path from vto 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. (C) 2021 Elsevier Inc. All rights reserved.