Refined Buneman Trees

is O(n 2 ). The algorithm uses a divide-and-conquer strategy, where the idea
is to cut away search space to quickly find the element we are looking for. If
we were to divide in half every time we would expect to spend linear time
dividing and searching recursively (n + n/2 +n/4 +···+1). However, since
we are partitioning around some element in an array, we are not guaranteed to
partition in the middle each time. The randomized part of the algorithm tries
to remedy this, and the expectation is that selecting an element at random will
on average partition the search space in half. Of course, if we are very unlucky
we will partition such that we only cut away one element each time, for a worst
case performance of O(n 2 ).
10.2 Performance of the selection algorithm
The selection algorithm runs in expected linear time, but worst case quadratic
time. The following test shows that the performance is indeed linear, as expected.
The test was done by running the randomized selection algorithm 50
times on input sizes starting at 10000 and increasing the size by 1000 for each
new run. Each runs consists of 100 repetitions. A plot is made of input size in
number of elements, against running time in milliseconds. This plot is shown in
Figure 10.1. The plot clearly shows that the selection algorithm does indeed run
in linear time, as expected. If one is not satisfied with this algorithm, [CLR90]
has a more complicated selection algorithm which runs in guaranteed worst case
linear time.
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