**Abstract:**
This paper gives hypercube algorithms for some simple problems
involving geometric properties of sets of points. A hypercube
is a type of parallel computer, with the processors linked together
as a hypercube graph.
The properties considered emphasize aspects of convexity and
domination.
Efficient algorithms are given for both fine-grain
and medium-grain hypercube computers. For both serial and
parallel computers, sorting plays an important role in geometric
algorithms for determining simple properties, often
being the dominant component of the time.
On a hypercube computer the time required to sort is still not fully
understood, so the times of some of our algorithms for unsorted data are
not completely determined.
For the fine-grain model using
worst-case timing we show that if the data are presorted then
faster algorithms are possible, provided that sorting one item per processor
requires time growing faster than the dimension of the
hypercube.
For both models we show that faster expected-case running time algorithms
are possible for point sets generated randomly.
Our algorithms are developed
for sets of planar points, with several of them extending to
sets of points in spaces of higher dimension.
**Keywords:** convex hull, domination, sorting, hypercube algorithms,
computational geometry, worst-case, expected case, parallel algorithm,
parallel computer, supercomputing, computer science

Russ Miller (miller@cse.buffalo.edu)