  We present a number of new results about range searching for colored (or
"categorical") data:
  1. For a set of $n$ colored points in three dimensions, we describe
randomized data structures with $O(n\mathop{\rm polylog}n)$ space that can
report the distinct colors in any query orthogonal range (axis-aligned box) in
$O(k\mathop{\rm polyloglog} n)$ expected time, where $k$ is the number of
distinct colors in the range, assuming that coordinates are in
$\{1,\ldots,n\}$. Previous data structures require $O(\frac{\log n}{\log\log n}
+ k)$ query time. Our result also implies improvements in higher constant
dimensions.
  2. Our data structures can be adapted to halfspace ranges in three dimensions
(or circular ranges in two dimensions), achieving $O(k\log n)$ expected query
time. Previous data structures require $O(k\log^2n)$ query time.
  3. For a set of $n$ colored points in two dimensions, we describe a data
structure with $O(n\mathop{\rm polylog}n)$ space that can answer colored
"type-2" range counting queries: report the number of occurrences of every
distinct color in a query orthogonal range. The query time is $O(\frac{\log
n}{\log\log n} + k\log\log n)$, where $k$ is the number of distinct colors in
the range. Naively performing $k$ uncolored range counting queries would
require $O(k\frac{\log n}{\log\log n})$ time.
  Our data structures are designed using a variety of techniques, including
colored variants of randomized incremental construction (which may be of
independent interest), colored variants of shallow cuttings, and bit-packing
tricks.
