We present a method to approximate pairwise distance on a graph, having an amortized sub-linear complexity in its size. The proposed method follows the so called heat method due to Crane et al. The only additional input are the values of the eigenfunctions of the graph Laplacian at a subset of the vertices. Using these values we estimate a random walk from the source points, and normalize the result into a unit gradient function. The eigenfunctions are then used to synthesize distance values abiding by these constraints at desired locations. We show that this method works in practice on different types of inputs ranging from triangular meshes to general graphs. We also demonstrate that the resulting approximate distance is accurate enough to be used as the input to a recent method for intrinsic shape correspondence computation.