It was shown previously that sortingnitems intonlocations with a polynomial number of processors requires Ω(log n/log log n) time. We sidestep this lower bound with the idea of padded sorting, or sortingnitems inton + o(n) locations. Because many problems do not rely on the exact rank of sorted items, a padded sort is often just as useful as an unpadded sort. Our algorithm for padded sort runs on the tolerant concurrent read, concurrent write parallel random access machine (CRCW PRAM) and takes Θ(log log n/log log log n) expected time usingn log log log n/log log nprocessors, assuming the items are taken from a uniform distribution. Using similar techniques we solve some computational geometry problems, including Voronoi diagram, with the same processor and time bounds, assuming points are taken from a uniform distribution in the unit square. Further, we present an arbitrary CRCW PRAM algorithm to solve the closest pair problem in constant expected time withnprocessors regardless of the distribution of points. All of these algorithms achieve linear speedup in expected time over their optimal serial counterparts.