Constructing red-black spanners for mixed-charging vehicular networks

Motivated by the recent trend of increasing number of e-cars and hybrid cars, we investigate the problem of building a red-black spanner for a mixed-charging vehicular network. In such a network, we have two kinds of gas/charging stations: electric (black) and the traditional gas (red) stations. Our requirement is that one cannot connect two gas stations directly in the spanner (i.e., no red-red edge), and the goal is to build a linear-size spanner with a bounded stretch factor under this requirement. (In 2-d, it can be shown that a spanner with an optimal stretch factor could have a quadratic size and if one is constrained to build the spanner purely from a given road network then it is impossible to obtain a bounded stretch factor.) Our main results are summarized as follows.

• 1.
  In 1-d, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves the optimal stretch factor.
• 2.
  In 2-d and under the $L_2$ metric, we build a linear-size red-black spanner satisfying the ‘no red-red edge’ requirement which achieves a stretch factor of 1.998.
• 3.
  In 2-d and under the $L_1$ metric, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves a stretch factor of 3.613.