Computing a consensus trajectory in a vehicular network

Abstract

In this paper, we study the problem of computing a consensus trajectory of a vehicle given the history of Points of Interest (POIs) visited by the vehicle over a certain period of time. The problem arises when a system tries to establish the social connection between two vehicles in a vehicular network, where three versions of the problem are studied. Formally, given a set of m trajectories, the first version of the problem is to compute a target (median) sequence T over Σ 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 based on a greedy method. We implement the greedy algorithm and a variation of it which is based on a more natural greedy search on a new data structure called adjacency map. In the second version of the problem where the sequence T is restricted to be a permutation, we show that the problem remains NP-hard but the approximation factor can be improved to 1.5. In the third version where the sequence T has to contain all letters of Σ, we again prove that it is NP-hard. We implement a simple greedy algorithm and a variation of the 1.