Before revision:  Related work only tackles the case where E consists of axis-parallel segments, resulting in an O( |E|\log |E| )-time and O( |E|\log |E| )-space 8-approximation.
After revision:  Related work only tackles the case where E consists of axis-parallel segments, resulting in an O( |E|\log |E| )-time and O( n\log n )-space 8-approximation.