On the discrete frchet distance in a graph

The Fr & eacute;chet distance is a well-studied similarity measure between curves thatis widely used throughout computer science. Motivated by applications where curves stemfrom paths and walks on an underlying graph (such as a road network), we define and studythe Fr & eacute;chet distance for paths and walks on graphs. When provided with a distance oracleofGwithO(1)query time, the classical quadratic-time dynamic program can compute theFr & eacute;chet distance between two walksPandQin a graphGinO(|P||Q|)time. We showthat there are situations where the graph structure helps with computing Fr & eacute;chet distance:when the graphGis planar, we apply existing (approximate) distance oracles to computea(1 +epsilon)-approximation of the Fr & eacute;chet distance between any shortest pathPand any walkQinO(|G|log|G|/epsilon+|P|+|Q|epsilon 2)time. We generalise this result to near-shortest paths,i.e.kappa-straight paths, as we show how to compute a(1 +epsilon)-approximation between a kappa-straight pathPand any walkQinO(|G|log|G|/epsilon+|P|+kappa|Q|epsilon 2)time. Specifically, for the(strong) Fr & eacute;chet distance, we show how to preprocess the graph and the trajectoryPinO(|G|log|G|/epsilon+|P|)time such that, for anyQ, we can compute a(1 +epsilon)approximationof the Fr & eacute;chet distance betweenPandQinO(kappa|Q|epsilon 2)time. Finally, we show that additional assumptions on the input, such as our assumptionon path straightness, are indeed necessary to obtain truly subquadratic running time. Weprovide a conditional lower bound showing that the Fr & eacute;chet distance, or even its1.01-approximation, between arbitrarypathsin a weighted planar graph cannot be computed inO((|P||Q|)1-delta)time for any delta >0unless the Orthogonal Vector Hypothesis fails. Forwalks, this lower bound holds even whenGis planar, unit-weight and hasO(1)vertices