Segment Not A Line

In a Li Chao tree over a coordinate range of size $C$ , what does inserting a full line cost against inserting a line restricted to $[l, r]$ ?

A full line is $O(\log C)$ , one root-to-leaf descent. A segment first decomposes $[l, r]$ into $O(\log C)$ canonical nodes and inserts into each, so it is $O(\log^2 C)$ . Queries become $\Theta(\log^2 C)$ once segments are present.

A full line is $O(\log C)$ , one root-to-leaf descent. A segment first decomposes $[l, r]$ into $O(\log C)$ canonical nodes and inserts into each, so it is $O(\log^2 C)$ . Queries stay $O(\log C)$ either way.

A full line is $O(\log C)$ , one root-to-leaf descent. A segment prunes at the first node outside $[l, r]$ and inserts only below it, so it is $O(\log C)$ as well. Queries stay $O(\log C)$ either way.

A full line is $O(\log^2 C)$ , since the descent branches into both halves. A segment first decomposes $[l, r]$ into $O(\log C)$ canonical nodes and inserts into each, so it is $O(\log C)$ . Queries stay $O(\log C)$ either way.