  We study a fundamental problem in Computational Geometry, the planar
two-center problem. In this problem, the input is a set $S$ of $n$ points in
the plane and the goal is to find two smallest congruent disks whose union
contains all points of $S$. A longstanding open problem has been to obtain an
$O(n\log n)$-time algorithm for planar two-center, matching the $\Omega(n\log
n)$ lower bound given by Eppstein [SODA'97]. Towards this, researchers have
made a lot of efforts over decades. The previous best algorithm, given by Wang
[SoCG'20], solves the problem in $O(n\log^2 n)$ time. In this paper, we present
an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which
completely resolves this open problem.
