  We consider the planar dynamic convex hull problem. In the literature,
solutions exist supporting the insertion and deletion of points in
poly-logarithmic time and various queries on the convex hull of the current set
of points in logarithmic time. If arbitrary insertion and deletion of points
are allowed, constant time updates and fast queries are known to be impossible.
This paper considers two restricted cases where worst-case constant time
updates and logarithmic time queries are possible. We assume all updates are
performed on a deque (double-ended queue) of points. The first case considers
the monotonic path case, where all points are sorted in a given direction, say
horizontally left-to-right, and only the leftmost and rightmost points can be
inserted and deleted. The second case assumes that the points in the deque
constitute a simple path. Note that the monotone case is a special case of the
simple path case. For both cases, we present solutions supporting deque
insertions and deletions in worst-case constant time and standard queries on
the convex hull of the points in $O(\log n)$ time, where $n$ is the number of
points in the current point set. The convex hull of the current point set can
be reported in $O(h+\log n)$ time, where $h$ is the number of edges of the
convex hull. For the 1-sided monotone path case, where updates are only allowed
on one side, the reporting time can be reduced to $O(h)$, and queries on the
convex hull are supported in $O(\log h)$ time. All our time bounds are worst
case. In addition, we prove lower bounds that match these time bounds, and thus
our results are optimal. For a quick comparison, the previous best update
bounds for the simple path problem were amortized $O(\log n)$ time by Friedman,
Hershberger, and Snoeyink [SoCG 1989].
