Trajectory Comparison in a Vehicular Network I: Computing a Consensus Trajectory

Abstract

In this paper, we investigate the problem of computing a consensus trajectory of a vehicle giving the history of Points of Interest (POIs) visited by the vehicle over certain period of time. The problem originates from building the social connection between two vehicles in a vehicular network. Formally, given a set of m trajectories (sequences $S_i$’s over a given alphabet $\varSigma$, each with length at most O(n), with $n=|\varSigma |$), the problem is to compute a target (median) sequence T over $\varSigma$ such that the sum of similarity measure (i.e., number of adjacencies) between T and all $S_i$’s is maximized. For this version, we show that the problem is NP-hard and we present a simple factor-2 approximation. If T has to be a permutation, then we show that the problem is still NP-hard but the approximation factor can be improved to 1.5. We implement the greedy algorithm and a variation of it which is based on a more natural greedy search. Using simulated data over two months (e.g., $m=60$) and variants of $|S_i|$ and $\varSigma$ (e.g., $30\le |S_i|\le 100$ and $30\le |\varSigma | \le 60$), the empirical results are very promising and with the local adjustment algorithm the actual approximation factor is between 1.5 and 1.6 for all the cases.