  Given $n$ points in a circular region $C$ in the plane, we study the problems
of moving the $n$ points to its boundary to form a regular $n$-gon such that
the maximum (min-max) or the sum (min-sum) of the Euclidean distances traveled
by the points is minimized. The problems have applications, e.g., in mobile
sensor barrier coverage of wireless sensor networks. The min-max problem
further has two versions: the decision version and optimization version. For
the min-max problem, we present an $O(n\log^2 n)$ time algorithm for the
decision version and an $O(n\log^3 n)$ time algorithm for the optimization
version. The previously best algorithms for the two problem versions take
$O(n^{3.5})$ time and $O(n^{3.5}\log n)$ time, respectively. For the min-sum
problem, we show that a special case with all points initially lying on the
boundary of the circular region can be solved in $O(n^2)$ time, improving a
previous $O(n^4)$ time solution. For the general min-sum problem, we present a
3-approximation $O(n^2)$ time algorithm, improving the previous
$(1+\pi)$-approximation $O(n^2)$ time algorithm. A by-product of our techniques
is an algorithm for dynamically maintaining the maximum matching of a circular
convex bipartite graph; our algorithm can handle each vertex insertion or
deletion on the graph in $O(\log^2 n)$ time. This result is interesting in its
own right.
