Randomized Algorithms (Spring 2010)/Tail inequalities

Select the Median

The selection problem is the problem of finding the [math]\displaystyle{ k }[/math]th smallest element in a set [math]\displaystyle{ S }[/math]. A typical case of selection problem is finding the median, the [math]\displaystyle{ (\lceil n/2\rceil) }[/math]th element in the sorted order of [math]\displaystyle{ S }[/math].

The median can be found in [math]\displaystyle{ O(n\log n) }[/math] time by sorting. There is a sophisticated deterministic algorithm, "median of medians" algorithm, which finds the median in [math]\displaystyle{ O(n) }[/math] time. Here we introduce a much simpler randomized algorithm which also runs in linear time. The idea of this randomized algorithm is based on sampling.