Often when analysing randomized algorithms, especially
parallel or distributed algorithms,
one is called upon to show that some function of many
independent choices is tightly
concentrated about its expected value. For example, the
algorithm might colour the
vertices of a given graph with two colours and one would
wish to show that, with high
probability, very nearly half of all edges are monochromatic.
The classic result of Chernoff [3] gives such a
large deviation result when the function
is a sum of independent indicator random variables. The
results of Hoeffding [5] and
Azuma [2] give similar results for functions which
can be expressed as martingales with
a bounded difference property. Roughly speaking, this means
that each individual choice
has a bounded effect on the value of the function. McDiarmid
[9] nicely summarized
these results and gave a host of applications. Expressed a
little differently, his main result is as follows.