Merge sort - Definition and Overview

In computer science, merge sort or mergesort is a sort algorithm for rearranging lists (or any other data structure that can only be accessed sequentially, e.g. file streams) into a specified order. It is a particularly good example of the divide and conquer algorithmic paradigm.

Conceptually, merge sort works as follows :

1. Divide the unsorted list into two sublists of about half the size
2. Sort each of the two sublists
3. Merge the two sorted sublists back into one sorted list.

The algorithm was invented by John von Neumann in 1945.

Analysis

Mergesort has an average and worst-case performance of O(n log n). In the worst case, mergesort does about 30% fewer comparisons than quicksort does in the average case; thus mergesort very often needs to make fewer comparisons than quicksort. In terms of moves, mergesort's worst case complexity is O(n log n)--the same complexity as quicksorts best case, and mergesort's best case is <math>\Theta(\frac{1}{2}nlogn)<math>.

However, mergesort performs O(2n -1) method calls in the worst case, compared to quicksort's O(n), thus has roughly twice as much recursive overhead as quicksort. Mergesort's most common implementation does not sort in place, meaning memory the size of the input must be allocated for the sorted output to be stored in. Sorting in-place is possible but requires an extremely complicated implementation.

Although mergesort can sort linked lists, it is also much more efficient than quicksort if the data to be sorted can only be efficiently accessed sequentially, and is thus popular in languages, such as Lisp, where sequentially accessed data structures are very common. Unlike some optimized implementations of quicksort, mergesort is a stable sort, as long as the merge operation is implemented properly.

More precisely, mergesort does between ⌈ n log n - n + 1 ⌉ and ⌈ n log n - 0.914·n ⌉ comparisons in the worst case.

Merge sorting tape drives

Merge sort is so inherently sequential that it's practical to run it using slow tape drives as input and output devices. It requires very little memory, and the memory required does not change with the number of data elements. If you have four tape drives, it works as follows:

1. divide the data to be sorted in half and put half on each of two tapes
2. merge individual pairs of records from the two tapes; write two-record chunks alternately to each of the two output tapes
3. merge the two-record chunks from the two output tapes into four-record chunks; write these alternately to the original two input tapes
4. merge the four-record chunks into eight-record chunks; write these alternately to the original two output tapes
5. repeat until you have one chunk containing all the data, sorted --- that is, for lg n passes, where n is the number of records.

On tape drives that can run both backwards and forwards, you can run merge passes in both directions, avoiding rewind time. For the same reason it is also very useful for sorting data on disk that is too large to fit entirely into primary memory.

Optimizing merge sort

This might seem to be of historical interest only, but on modern computers, locality of reference is of paramount importance in software optimization, because multi-level memory hierarchies are used. In some sense, main RAM can be seen as a slow tape drive, level 3 cache memory as a slightly faster one, level 2 cache memory as faster still, and so on. In some circumstances, cache reloading might impose unacceptable overhead and a carefully crafted merge sort could have considerable running time improvement. This opportunity might change if fast memory becomes very cheap again, or if exotic architectures like the Tera MTA become commonplace.

Designing a merge sort to perform optimally often requires adjustment to available hardware, eg. number of tape drives, or size and speed of the relevant cache memory levels.

Comparison with other sort algorithms

Although heap sort has the same time bounds as merge sort, it requires only Ω(1) auxilary space instead of merge sort's Ω(n), and is consequently often faster in practical implementations. Quicksort, however, is considered by many to be the fastest general-purpose sort algorithm in practice. Its average-case complexity is O(n log n), with a much smaller coefficient, in good implementations, than merge sort's, even though it is quadratic in the worst case. On the plus side, merge sort is a stable sort, parallelizes better, operates easily on linked lists, and is more efficient at handling slow-to-access sequential media.

As of Perl 5.8, merge sort is its default sorting algorithm (it was quicksort in previous versions of Perl).

External Resources

• Dictionary of Algorithms and Data Structures: Merge sort (http://www.nist.gov/dads/HTML/mergesort.html)
• Power Programming - Merge Sort (http://www.bitesizeinc.net/power.program.merge.sort.html)
• Merge Sort Algorithm Simulation (http://www.geocities.com/SiliconValley/Program/2864/File/Merge1/mergesort.html)

Sample implementations

See the Merge algorithm article for the respective implementations of the merge functions referenced in the examples below.

FL

  def sort == tree:⟨merge,⟨⟩⟩ ° α:[id]

Haskell

sort :: Ord a => [a] -> [a]

sort []         =  []
sort [x]        =  [x]
sort xs         =  merge (sort ys) (sort zs)
  where (ys,zs) =  splitAt (length xs `div` 2) xs

Python

def sort(array):
  if len(array) <= 1: return array
  mid = len(array) // 2
  return merge (sort(array[0:mid]), sort(array[mid:]))

Miranda

  sort []    = []
  sort [x]   = [x]
  sort array = merge (sort left) (sort right)
               where
                 left  = [array!y | y <- [0..mid]]
                 right = [array!y | y <- [(mid+1)..max]]
                 max   = #array - 1
                 mid   = max div 2

C / C++ / Java

void Sort(float array[], int begin, int end) 
{
 int mid;  
 if (begin == end)     
   return; 
 if (begin == end - 1) 
   return; 
 mid = (begin + end) / 2;
 Sort(array, begin, mid);
 Sort(array, mid, end);
 Merge(array, begin, mid , end);
}