Before revision:  We successfully design linear-size shallow cuttings for general distance functions, indicating that for general distance functions it could still be possible to achieve the same complexities as point sites in the Euclidean metric. Our breakthrough to achieve the linear size is a new random sampling technique (for the configuration space) that employs relatively many local conflicts to prevent relatively few global conflicts.
After revision:  Our innovation is a new random sampling technique (for the configuration space) that employs relatively many local conflicts to prevent relatively few global conflicts.