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7.7. The Merge Sort¶
We now turn our attention to using a divide and conquer strategy as a
way to improve the performance of sorting algorithms. The first
algorithm we will study is the merge sort. Merge sort is a recursive
algorithm that continually splits a vector in half. If the vector is empty
or has one item, it is sorted by definition (the base case). If the vector
has more than one item, we split the vector and recursively invoke a merge
sort on both halves. Once the two halves are sorted, the fundamental
operation, called a merge, is performed. Merging is the process of
taking two smaller sorted vectors and combining them together into a
single, sorted, new vector. Figure 10 shows our familiar example
vector as it is being split by
mergeSort. Figure 11 shows
the simple vectors, now sorted, as they are merged back together.
mergeSort function shown in ActiveCode 1 begins by asking the
base case question. If the length of the vector is less than or equal to
one, then we already have a sorted vector and no more processing is
necessary. If, on the other hand, the length is greater than one, then we utilize
the vector intializer using .begin to extract the left and right halves.
This is similar to using the Python
slice operation to extract the left and right
halves. It is important to note that the vector may not have an even
number of items. That does not matter, as the lengths will differ by at
mergeSort function is invoked on the left half and the
right half (lines 8–9), it is assumed they are sorted. The rest of the
function (lines 11–31) is responsible for merging the two smaller sorted
vectors into a larger sorted vector. Notice that the merge operation places
the items back into the original vector (
avector) one at a time by
repeatedly taking the smallest item from the sorted vectors.
mergeSort function has been augmented with a
The visualization above allows you to step through the algorithm. Red bars represent the element being looked at and blue represents the last element to look at during a pass.
The visualization above highlights the individual components of the algorithm itself. The arrows on the bottom indicate the left, middle, and right portions that the algorithm is currently examining. Left and right components are indicated by the color brown, while the middle is indicated by orange. Look for the “divide-and-conquer” aspect of the algorithm here.
In order to analyze the
mergeSort function, we need to consider the
two distinct processes that make up its implementation. First, the vector
is split into halves. We already computed (in a binary search) that we
can divide a vector in half \(\log n\) times where n is the
length of the vector. The second process is the merge. Each item in the
vector will eventually be processed and placed on the sorted vector. So the
merge operation which results in a vector of size n requires n
operations. The result of this analysis is that \(\log n\) splits,
each of which costs \(n\) for a total of \(n\log n\)
operations. A merge sort is an \(O(n\log n)\) algorithm.
Recall that the slicing operator is \(O(k)\) where k is the size
of the slice. In order to guarantee that
mergeSort will be
\(O(n\log n)\) we will need to remove the slice operator. Again,
this is possible if we simply pass the starting and ending indices along
with the vector when we make the recursive call. We leave this as an
It is important to notice that the
mergeSort function requires extra
space to hold the two halves as they are extracted with the slicing
operations. This additional space can be a critical factor if the vector
is large and can make this sort problematic when working on large data
sets. In the case with using lists in python, the space complexity is \(O(log n)\).