Benchmarking of Pivot Selection Strategies
Timothy Rolfe
The detailed behavior of QuickSort depends on the strategy used for the selection of the pivot
value within the Partition algorithm.
The fastest method is simply to accept the value at the beginning of the region being sorted. As
Tony Hoare, the originator of QuickSort, noted in his original discussion of the algorithm1, this is
known to have disastrous results if the data being sorted are already ordered, either in a forward
direction or reversed: QuickSort becomes an O(n2) algorithm rather than the average case as an
O(n log n) algorithm.
One can avoid this worst-case behavior by choosing a random location within the region being
sorted. This has the overhead of computing the random location — the method invocation of
calling the random number generator, the integer arithmetic of computing the complete subscript,
and the three assignment statements to swap the front element with that random element. This is
actually in the original publication of the algorithm.2
Finally, Sedgewick suggests a median-of-three algorithm.3 Assume that the region being sorted
is defined by subscripts “lo” and “hi”. Compute the midpoint subscript as “(lo+hi)/2”. Perform
the necessary comparisons and swaps to guarantee that the median of the three values is at x[lo].
Below is my implementation in Java, which sorts [lo], [mid], and [hi]:
mid = (lo + hi) / 2;
// We wish to move the median -- b in "abc" -- to position lo
if (x[lo].compareTo(x[hi]) < 0) // abc acb bac
if (x[lo].compareTo(x[mid]) < 0) // abc acb
if (x[mid].compareTo(x[hi]) < 0) // abc
swap(lo, mid, x);
else // acb --- place sentinels
{ swap(lo, hi, x); swap (mid, hi, x); }
else ; // bac --- null statement
else // bca cab cba
if (x[lo].compareTo(x[mid]) > 0) // cab cba
if (x[mid].compareTo(x[hi]) > 0) // cba --- place sentinels
{ swap(lo, mid, x); swap (mid, hi, x); }
else // cab
swap(lo, hi, x);
else
swap(mid, hi, x); // bca: move c into place
On average, there are 8/3 invocations of compareTo: four permutations require three, while two
require two. Seven swaps are performed across the six possible permutations. Thus we probably
have a somewhat larger overhead than the random pivot. Given the complexity of the above, it
might be preferable to use a pure three-comparison method with potentially three swaps, as given
in Koffman's implementation.4
1 Charles Anthony Richard Hoare, Computer Journal, Vol 2 (1962), pp. 10-15.
2 Charles Anthony Richard Hoare, CACM, Vol 4, No. 7 (July 1961), pp. 321-22.
3 Robert Sedgewick, Algorithms in C; Parts 1-4 (3rd edition; Addison, Wesley, 1998), p. 319-24.
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