Timing Your Code

When comparing two sorting algorithms in Python, it’s always informative to look at how long each one takes to run. The specific time each algorithm takes will be partly determined by your hardware, but you can still use the proportional time between executions to help you decide which implementation is more time efficient.

In this section, you’ll focus on a practical way to measure the actual time it takes to run to your sorting algorithms using thetimeit module. For more information on the different ways you can time the execution of code in Python, check outPython Timer Functions: Three Ways to Monitor Your Code.

Here’s a function you can use to time your algorithms:

 1fromrandomimportrandint 2fromtimeitimportrepeat 3 4defrun_sorting_algorithm(algorithm,array): 5# Set up the context and prepare the call to the specified 6# algorithm using the supplied array. Only import the 7# algorithm function if it's not the built-in `sorted()`. 8setup_code=f"from __main__ import{algorithm}" \ 9ifalgorithm!="sorted"else""1011stmt=f"{algorithm}({array})"1213# Execute the code ten different times and return the time14# in seconds that each execution took15times=repeat(setup=setup_code,stmt=stmt,repeat=3,number=10)1617# Finally, display the name of the algorithm and the18# minimum time it took to run19print(f"Algorithm:{algorithm}. Minimum execution time:{min(times)}")

In this example,run_sorting_algorithm() receives the name of the algorithm and the input array that needs to be sorted. Here’s a line-by-line explanation of how it works:

• Line 8 imports the name of the algorithm using the magic ofPython’s f-strings. This is so thattimeit.repeat() knows where to call the algorithm from. Note that this is only necessary for the custom implementations used in this tutorial. If the algorithm specified is the built-insorted(), then nothing will be imported.

• Line 11 prepares the call to the algorithm with the supplied array. This is the statement that will be executed and timed.

• Line 15 callstimeit.repeat() with the setup code and the statement. This will call the specified sorting algorithm ten times, returning the number of seconds each one of these executions took.

• Line 19 identifies the shortest time returned and prints it along with the name of the algorithm.

Here’s an example of how to userun_sorting_algorithm() to determine the time it takes to sort an array of ten thousand integer values usingsorted():

21ARRAY_LENGTH=100002223if__name__=="__main__":24# Generate an array of `ARRAY_LENGTH` items consisting25# of random integer values between 0 and 99926array=[randint(0,1000)foriinrange(ARRAY_LENGTH)]2728# Call the function using the name of the sorting algorithm29# and the array you just created30run_sorting_algorithm(algorithm="sorted",array=array)

If you save the above code in asorting.py file, then you can run it from theterminal and see its output:

$pythonsorting.pyAlgorithm: sorted. Minimum execution time: 0.010945824000000007

Remember that the time in seconds of every experiment depends in part on the hardware you use, so you’ll likely see slightly different results when running the code.

Measuring Efficiency With Big O Notation

The specific time an algorithm takes to run isn’t enough information to get the full picture of itstime complexity. To solve this problem, you can use Big O (pronounced “big oh”) notation. Big O is often used to compare different implementations and decide which one is the most efficient, skipping unnecessary details and focusing on what’s most important in the runtime of an algorithm.

The time in seconds required to run different algorithms can be influenced by several unrelated factors, including processor speed or available memory. Big O, on the other hand, provides a platform to express runtime complexity in hardware-agnostic terms. With Big O, you express complexity in terms of how quickly your algorithm’s runtime grows relative to the size of the input, especially as the input grows arbitrarily large.

Assuming thatn is the size of the input to an algorithm, the Big O notation represents the relationship betweenn and the number of steps the algorithm takes to find a solution. Big O uses a capital letter “O” followed by this relationship inside parentheses. For example,O(n) represents algorithms that execute a number of steps proportional to the size of their input.

Although this tutorial isn’t going to dive very deep into the details of Big O notation, here are five examples of the runtime complexity of different algorithms:

This tutorial covers the Big O runtime complexity of each of the sorting algorithms discussed. It also includes a brief explanation of how to determine the runtime on each particular case. This will give you a better understanding of how to start using Big O to classify other algorithms.

Implementing Bubble Sort in Python

Here’s an implementation of a bubble sort algorithm in Python:

 1defbubble_sort(array): 2n=len(array) 3 4foriinrange(n): 5# Create a flag that will allow the function to 6# terminate early if there's nothing left to sort 7already_sorted=True 8 9# Start looking at each item of the list one by one,10# comparing it with its adjacent value. With each11# iteration, the portion of the array that you look at12# shrinks because the remaining items have already been13# sorted.14forjinrange(n-i-1):15ifarray[j]>array[j+1]:16# If the item you're looking at is greater than its17# adjacent value, then swap them18array[j],array[j+1]=array[j+1],array[j]1920# Since you had to swap two elements,21# set the `already_sorted` flag to `False` so the22# algorithm doesn't finish prematurely23already_sorted=False2425# If there were no swaps during the last iteration,26# the array is already sorted, and you can terminate27ifalready_sorted:28break2930returnarray

Since this implementation sorts the array in ascending order, each step “bubbles” the largest element to the end of the array. This means that each iteration takes fewer steps than the previous iteration because a continuously larger portion of the array is sorted.

The loops inlines 4 and 10 determine the way the algorithm runs through the list. Notice howj initially goes from the first element in the list to the element immediately before the last. During the second iteration,j runs until two items from the last, then three items from the last, and so on. At the end of each iteration, the end portion of the list will be sorted.

As the loops progress,line 15 compares each element with its adjacent value, andline 18 swaps them if they are in the incorrect order. This ensures a sorted list at the end of the function.

To properly analyze how the algorithm works, consider a list with values[8, 2, 6, 4, 5]. Assume you’re usingbubble_sort() from above. Here’s a figure illustrating what the array looks like at each iteration of the algorithm:

Now take a step-by-step look at what’s happening with the array as the algorithm progresses:

1. The code starts by comparing the first element,8, with its adjacent element,2. Since8 > 2, the values are swapped, resulting in the following order:[2, 8, 6, 4, 5].

2. The algorithm then compares the second element,8, with its adjacent element,6. Since8 > 6, the values are swapped, resulting in the following order:[2, 6, 8, 4, 5].

3. Next, the algorithm compares the third element,8, with its adjacent element,4. Since8 > 4, it swaps the values as well, resulting in the following order:[2, 6, 4, 8, 5].

4. Finally, the algorithm compares the fourth element,8, with its adjacent element,5, and swaps them as well, resulting in[2, 6, 4, 5, 8]. At this point, the algorithm completed the first pass through the list (i = 0). Notice how the value8 bubbled up from its initial location to its correct position at the end of the list.

5. The second pass (i = 1) takes into account that the last element of the list is already positioned and focuses on the remaining four elements,[2, 6, 4, 5]. At the end of this pass, the value6 finds its correct position. The third pass through the list positions the value5, and so on until the list is sorted.

Measuring Bubble Sort’s Big O Runtime Complexity

Your implementation of bubble sort consists of two nestedfor loops in which the algorithm performsn - 1 comparisons, thenn - 2 comparisons, and so on until the final comparison is done. This comes at a total of(n - 1) + (n - 2) + (n - 3) + … + 2 + 1 = n(n-1)/2 comparisons, which can also be written as½n² - ½n.

You learned earlier that Big O focuses on how the runtime grows in comparison to the size of the input. That means that, in order to turn the above equation into the Big O complexity of the algorithm, you need to remove the constants because they don’t change with the input size.

Doing so simplifies the notation ton² - n. Sincen² grows much faster thann, this last term can be dropped as well, leaving bubble sort with an average- and worst-case complexity ofO(n²).

In cases where the algorithm receives an array that’s already sorted—and assuming the implementation includes thealready_sorted flag optimization explained before—the runtime complexity will come down to a much betterO(n) because the algorithm will not need to visit any element more than once.

O(n), then, is the best-case runtime complexity of bubble sort. But keep in mind that best cases are an exception, and you should focus on the average case when comparing different algorithms.

Timing Your Bubble Sort Implementation

Using yourrun_sorting_algorithm() from earlier in this tutorial, here’s the time it takes for bubble sort to process an array with ten thousand items.Line 8 replaces the name of the algorithm and everything else stays the same:

 1if__name__=="__main__": 2# Generate an array of `ARRAY_LENGTH` items consisting 3# of random integer values between 0 and 999 4array=[randint(0,1000)foriinrange(ARRAY_LENGTH)] 5 6# Call the function using the name of the sorting algorithm 7# and the array you just created 8run_sorting_algorithm(algorithm="bubble_sort",array=array)

You can now run the script to get the execution time ofbubble_sort:

$pythonsorting.pyAlgorithm: bubble_sort. Minimum execution time: 73.21720498399998

It took73 seconds to sort the array with ten thousand elements. This represents the fastest execution out of the ten repetitions thatrun_sorting_algorithm() runs. Executing this script multiple times will produce similar results.

Analyzing the Strengths and Weaknesses of Bubble Sort

The main advantage of the bubble sort algorithm is itssimplicity. It is straightforward to both implement and understand. This is probably the main reason why most computer science courses introduce the topic of sorting using bubble sort.

As you saw before, the disadvantage of bubble sort is that it isslow, with a runtime complexity ofO(n²). Unfortunately, this rules it out as a practical candidate for sorting large arrays.

Implementing Insertion Sort in Python

The insertion sort algorithm works exactly like the example with the deck of cards. Here’s the implementation in Python:

 1definsertion_sort(array): 2# Loop from the second element of the array until 3# the last element 4foriinrange(1,len(array)): 5# This is the element we want to position in its 6# correct place 7key_item=array[i] 8 9# Initialize the variable that will be used to10# find the correct position of the element referenced11# by `key_item`12j=i-11314# Run through the list of items (the left15# portion of the array) and find the correct position16# of the element referenced by `key_item`. Do this only17# if `key_item` is smaller than its adjacent values.18whilej>=0andarray[j]>key_item:19# Shift the value one position to the left20# and reposition j to point to the next element21# (from right to left)22array[j+1]=array[j]23j-=12425# When you finish shifting the elements, you can position26# `key_item` in its correct location27array[j+1]=key_item2829returnarray

Unlike bubble sort, this implementation of insertion sort constructs the sorted list by pushing smaller items to the left. Let’s break downinsertion_sort() line by line:

• Line 4 sets up the loop that determines thekey_item that the function will position during each iteration. Notice that the loop starts with the second item on the list and goes all the way to the last item.

• Line 7 initializeskey_item with the item that the function is trying to place.

• Line 12 initializes a variable that will consecutively point to each element to the left ofkey item. These are the elements that will be consecutively compared withkey_item.

• Line 18 compareskey_item with each value to its left using awhile loop, shifting the elements to make room to placekey_item.

• Line 27 positionskey_item in its correct place after the algorithm shifts all the larger values to the right.

Here’s a figure illustrating the different iterations of the algorithm when sorting the array[8, 2, 6, 4, 5]:

Now here’s a summary of the steps of the algorithm when sorting the array:

1. The algorithm starts withkey_item = 2 and goes through the subarray to its left to find the correct position for it. In this case, the subarray is[8].

2. Since2 < 8, the algorithm shifts element8 one position to its right. The resultant array at this point is[8, 8, 6, 4, 5].

3. Since there are no more elements in the subarray, thekey_item is now placed in its new position, and the final array is[2, 8, 6, 4, 5].

4. The second pass starts withkey_item = 6 and goes through the subarray located to its left, in this case[2, 8].

5. Since6 < 8, the algorithm shifts 8 to its right. The resultant array at this point is[2, 8, 8, 4, 5].

6. Since6 > 2, the algorithm doesn’t need to keep going through the subarray, so it positionskey_item and finishes the second pass. At this time, the resultant array is[2, 6, 8, 4, 5].

7. The third pass through the list puts the element4 in its correct position, and the fourth pass places element5 in the correct spot, leaving the array sorted.

Measuring Insertion Sort’s Big O Runtime Complexity

Similar to your bubble sort implementation, the insertion sort algorithm has a couple of nested loops that go over the list. The inner loop is pretty efficient because it only goes through the list until it finds the correct position of an element. That said, the algorithm still has anO(n²) runtime complexity on the average case.

The worst case happens when the supplied array is sorted in reverse order. In this case, the inner loop has to execute every comparison to put every element in its correct position. This still gives you anO(n²) runtime complexity.

The best case happens when the supplied array is already sorted. Here, the inner loop is never executed, resulting in anO(n) runtime complexity, just like the best case of bubble sort.

Although bubble sort and insertion sort have the same Big O runtime complexity, in practice, insertion sort is considerably more efficient than bubble sort. If you look at the implementation of both algorithms, then you can see how insertion sort has to make fewer comparisons to sort the list.

Timing Your Insertion Sort Implementation

To prove the assertion that insertion sort is more efficient than bubble sort, you can time the insertion sort algorithm and compare it with the results of bubble sort. To do this, you just need to replace the call torun_sorting_algorithm() with the name of your insertion sort implementation:

 1if__name__=="__main__": 2# Generate an array of `ARRAY_LENGTH` items consisting 3# of random integer values between 0 and 999 4array=[randint(0,1000)foriinrange(ARRAY_LENGTH)] 5 6# Call the function using the name of the sorting algorithm 7# and the array we just created 8run_sorting_algorithm(algorithm="insertion_sort",array=array)

You can execute the script as before:

$pythonsorting.pyAlgorithm: insertion_sort. Minimum execution time: 56.71029764299999

Notice how the insertion sort implementation took around17 fewer seconds than the bubble sort implementation to sort the same array. Even though they’re bothO(n²) algorithms, insertion sort is more efficient.

Analyzing the Strengths and Weaknesses of Insertion Sort

Just like bubble sort, the insertion sort algorithm is very uncomplicated to implement. Even though insertion sort is anO(n²) algorithm, it’s also much more efficient in practice than other quadratic implementations such as bubble sort.

There are more powerful algorithms, including merge sort and Quicksort, but these implementations are recursive and usually fail to beat insertion sort when working on small lists. Some Quicksort implementations even use insertion sort internally if the list is small enough to provide a faster overall implementation.Timsort also uses insertion sort internally to sort small portions of the input array.

That said, insertion sort is not practical for large arrays, opening the door to algorithms that can scale in more efficient ways.

Implementing Merge Sort in Python

The implementation of the merge sort algorithm needs two different pieces:

1. A function that recursively splits the input in half
2. A function that merges both halves, producing a sorted array

Here’s the code to merge two different arrays:

 1defmerge(left,right): 2# If the first array is empty, then nothing needs 3# to be merged, and you can return the second array as the result 4iflen(left)==0: 5returnright 6 7# If the second array is empty, then nothing needs 8# to be merged, and you can return the first array as the result 9iflen(right)==0:10returnleft1112result=[]13index_left=index_right=01415# Now go through both arrays until all the elements16# make it into the resultant array17whilelen(result)<len(left)+len(right):18# The elements need to be sorted to add them to the19# resultant array, so you need to decide whether to get20# the next element from the first or the second array21ifleft[index_left]<=right[index_right]:22result.append(left[index_left])23index_left+=124else:25result.append(right[index_right])26index_right+=12728# If you reach the end of either array, then you can29# add the remaining elements from the other array to30# the result and break the loop31ifindex_right==len(right):32result+=left[index_left:]33break3435ifindex_left==len(left):36result+=right[index_right:]37break3839returnresult

merge() receives two different sorted arrays that need to be merged together. The process to accomplish this is straightforward:

• Lines 4 and 9 check whether either of the arrays is empty. If one of them is, then there’s nothing to merge, so the function returns the other array.

• Line 17 starts awhile loop that ends whenever the result contains all the elements from both of the supplied arrays. The goal is to look into both arrays and combine their items to produce a sorted list.

• Line 21 compares the elements at the head of both arrays, selects the smaller value, andappends it to the end of the resultant array.

• Lines 31 and 35 append any remaining items to the result if all the elements from either of the arrays were already used.

With the above function in place, the only missing piece is a function that recursively splits the input array in half and usesmerge() to produce the final result:

41defmerge_sort(array):42# If the input array contains fewer than two elements,43# then return it as the result of the function44iflen(array)<2:45returnarray4647midpoint=len(array)//24849# Sort the array by recursively splitting the input50# into two equal halves, sorting each half and merging them51# together into the final result52returnmerge(53left=merge_sort(array[:midpoint]),54right=merge_sort(array[midpoint:]))

Here’s a quick summary of the code:

• Line 44 acts as the stopping condition for the recursion. If the input array contains fewer than two elements, then the function returns the array. Notice that this condition could be triggered by receiving either a single item or an empty array. In both cases, there’s nothing left to sort, so the function should return.

• Line 47 computes the middle point of the array.

• Line 52 callsmerge(), passing both sorted halves as the arrays.

Notice how this function calls itselfrecursively, halving the array each time. Each iteration deals with an ever-shrinking array until fewer than two elements remain, meaning there’s nothing left to sort. At this point,merge() takes over, merging the two halves and producing a sorted list.

Take a look at a representation of the steps that merge sort will take to sort the array[8, 2, 6, 4, 5]:

The figure uses yellow arrows to represent halving the array at each recursion level. The green arrows represent merging each subarray back together. The steps can be summarized as follows:

1. The first call tomerge_sort() with[8, 2, 6, 4, 5] definesmidpoint as2. Themidpoint is used to halve the input array intoarray[:2] andarray[2:], producing[8, 2] and[6, 4, 5], respectively.merge_sort() is then recursively called for each half to sort them separately.

2. The call tomerge_sort() with[8, 2] produces[8] and[2]. The process repeats for each of these halves.

3. The call tomerge_sort() with[8] returns[8] since that’s the only element. The same happens with the call tomerge_sort() with[2].

4. At this point, the function starts merging the subarrays back together usingmerge(), starting with[8] and[2] as input arrays, producing[2, 8] as the result.

5. On the other side,[6, 4, 5] is recursively broken down and merged using the same procedure, producing[4, 5, 6] as the result.

6. In the final step,[2, 8] and[4, 5, 6] are merged back together withmerge(), producing the final result:[2, 4, 5, 6, 8].

Measuring Merge Sort’s Big O Complexity

To analyze the complexity of merge sort, you can look at its two steps separately:

1. merge() has a linear runtime. It receives two arrays whose combined length is at mostn (the length of the original input array), and it combines both arrays by looking at each element at most once. This leads to a runtime complexity ofO(n).

2. The second step splits the input array recursively and callsmerge() for each half. Since the array is halved until a single element remains, the total number of halving operations performed by this function islog₂n. Sincemerge() is called for each half, we get a total runtime ofO(n log₂n).

Interestingly,O(n log₂n) is the best possible worst-case runtime that can be achieved by a sorting algorithm.

Timing Your Merge Sort Implementation

To compare the speed of merge sort with the previous two implementations, you can use the same mechanism as before and replace the name of the algorithm inline 8:

 1if__name__=="__main__": 2# Generate an array of `ARRAY_LENGTH` items consisting 3# of random integer values between 0 and 999 4array=[randint(0,1000)foriinrange(ARRAY_LENGTH)] 5 6# Call the function using the name of the sorting algorithm 7# and the array you just created 8run_sorting_algorithm(algorithm="merge_sort",array=array)

You can execute the script to get the execution time ofmerge_sort:

$pythonsorting.pyAlgorithm: merge_sort. Minimum execution time: 0.6195857160000173

Compared to bubble sort and insertion sort, the merge sort implementation is extremely fast, sorting the ten-thousand-element array in less than a second!

Analyzing the Strengths and Weaknesses of Merge Sort

Thanks to its runtime complexity ofO(n log₂n), merge sort is a very efficient algorithm that scales well as the size of the input array grows. It’s also straightforward toparallelize because it breaks the input array into chunks that can be distributed and processed in parallel if necessary.

That said, for small lists, the time cost of the recursion allows algorithms such as bubble sort and insertion sort to be faster. For example, running an experiment with a list of ten elements results in the following times:

Algorithm: bubble_sort. Minimum execution time: 0.000018774999999998654Algorithm: insertion_sort. Minimum execution time: 0.000029786000000000395Algorithm: merge_sort. Minimum execution time: 0.00016983000000000276

Both bubble sort and insertion sort beat merge sort when sorting a ten-element list.

Another drawback of merge sort is that it creates copies of the array when calling itself recursively. It also creates a new list insidemerge() to sort and return both input halves. This makes merge sort use much more memory than bubble sort and insertion sort, which are both able to sort the list in place.

Due to this limitation, you may not want to use merge sort to sort large lists in memory-constrained hardware.

Implementing Quicksort in Python

Here’s a fairly compact implementation of Quicksort:

 1fromrandomimportrandint 2 3defquicksort(array): 4# If the input array contains fewer than two elements, 5# then return it as the result of the function 6iflen(array)<2: 7returnarray 8 9low,same,high=[],[],[]1011# Select your `pivot` element randomly12pivot=array[randint(0,len(array)-1)]1314foriteminarray:15# Elements that are smaller than the `pivot` go to16# the `low` list. Elements that are larger than17# `pivot` go to the `high` list. Elements that are18# equal to `pivot` go to the `same` list.19ifitem<pivot:20low.append(item)21elifitem==pivot:22same.append(item)23elifitem>pivot:24high.append(item)2526# The final result combines the sorted `low` list27# with the `same` list and the sorted `high` list28returnquicksort(low)+same+quicksort(high)

Here’s a summary of the code:

• Line 6 stops the recursive function if the array contains fewer than two elements.

• Line 12 selects thepivot element randomly from the list and proceeds to partition the list.

• Lines 19 and 20 put every element that’s smaller thanpivot into the list calledlow.

• Lines 21 and 22 put every element that’s equal topivot into the list calledsame.

• Lines 23 and 24 put every element that’s larger thanpivot into the list calledhigh.

• Line 28 recursively sorts thelow andhigh lists and combines them along with the contents of thesame list.

Here’s an illustration of the steps that Quicksort takes to sort the array[8, 2, 6, 4, 5]:

The yellow lines represent the partitioning of the array into three lists:low,same, andhigh. The green lines represent sorting and putting these lists back together. Here’s a brief explanation of the steps:

1. Thepivot element is selected randomly. In this case,pivot is6.

2. The first pass partitions the input array so thatlow contains[2, 4, 5],same contains[6], andhigh contains[8].

3. quicksort() is then called recursively withlow as its input. This selects a randompivot and breaks the array into[2] aslow,[4] assame, and[5] ashigh.

4. The process continues, but at this point, bothlow andhigh have fewer than two items each. This ends the recursion, and the function puts the array back together. Adding the sortedlow andhigh to either side of thesame list produces[2, 4, 5].

5. On the other side, thehigh list containing[8] has fewer than two elements, so the algorithm returns the sortedlow array, which is now[2, 4, 5]. Merging it withsame ([6]) andhigh ([8]) produces the final sorted list.

Selecting thepivot Element

Why does the implementation above select thepivot element randomly? Wouldn’t it be the same to consistently select the first or last element of the input list?

Because of how the Quicksort algorithm works, the number of recursion levels depends on wherepivot ends up in each partition. In the best-case scenario, the algorithm consistently picks themedian element as thepivot. That would make each generated subproblem exactly half the size of the previous problem, leading to at mostlog₂n levels.

On the other hand, if the algorithm consistently picks either the smallest or largest element of the array as thepivot, then the generated partitions will be as unequal as possible, leading ton-1 recursion levels. That would be the worst-case scenario for Quicksort.

As you can see, Quicksort’s efficiency often depends on thepivot selection. If the input array is unsorted, then using the first or last element as thepivot will work the same as a random element. But if the input array is sorted or almost sorted, using the first or last element as thepivot could lead to a worst-case scenario. Selecting thepivot at random makes it more likely Quicksort will select a value closer to the median and finish faster.

Another option for selecting thepivot is tofind the median value of the array and force the algorithm to use it as thepivot. This can be done inO(n) time. Although the process is little bit more involved, using the median value as thepivot for Quicksort guarantees you will have the best-case Big O scenario.

Measuring Quicksort’s Big O Complexity

With Quicksort, the input list is partitioned in linear time,O(n), and this process repeats recursively an average oflog₂n times. This leads to a final complexity ofO(n log₂n).

That said, remember the discussion about how the selection of thepivot affects the runtime of the algorithm. TheO(n) best-case scenario happens when the selectedpivot is close to the median of the array, and anO(n²) scenario happens when thepivot is the smallest or largest value of the array.

Theoretically, if the algorithm focuses first on finding the median value and then uses it as thepivot element, then the worst-case complexity will come down toO(n log₂n). The median of an array can be found in linear time, and using it as thepivot guarantees the Quicksort portion of the code will perform inO(n log₂n).

By using the median value as thepivot, you end up with a final runtime ofO(n) + O(n log₂n). You can simplify this down toO(n log₂n) because the logarithmic portion grows much faster than the linear portion.

Timing Your Quicksort Implementation

By now, you’re familiar with the process for timing the runtime of the algorithm. Just change the name of the algorithm inline 8:

 1if__name__=="__main__": 2# Generate an array of `ARRAY_LENGTH` items consisting 3# of random integer values between 0 and 999 4array=[randint(0,1000)foriinrange(ARRAY_LENGTH)] 5 6# Call the function using the name of the sorting algorithm 7# and the array you just created 8run_sorting_algorithm(algorithm="quicksort",array=array)

You can execute the script as you have before:

$pythonsorting.pyAlgorithm: quicksort. Minimum execution time: 0.11675417600002902

Not only does Quicksort finish in less than one second, but it’s also much faster than merge sort (0.11 seconds versus0.61 seconds). Increasing the number of elements specified byARRAY_LENGTH from10,000 to1,000,000 and running the script again ends up with merge sort finishing in97 seconds, whereas Quicksort sorts the list in a mere10 seconds.

Analyzing the Strengths and Weaknesses of Quicksort

True to its name, Quicksort isvery fast. Although its worst-case scenario is theoreticallyO(n²), in practice, a good implementation of Quicksort beats most other sorting implementations. Also, just like merge sort, Quicksort is straightforward toparallelize.

One of Quicksort’s main disadvantages is the lack of a guarantee that it will achieve the average runtime complexity. Although worst-case scenarios are rare, certain applications can’t afford to risk poor performance, so they opt for algorithms that stay withinO(n log₂n) regardless of the input.

Just like merge sort, Quicksort also trades off memory space for speed. This may become a limitation for sorting larger lists.

A quick experiment sorting a list of ten elements leads to the following results:

Algorithm: bubble_sort. Minimum execution time: 0.0000909000000000014Algorithm: insertion_sort. Minimum execution time: 0.00006681900000000268Algorithm: quicksort. Minimum execution time: 0.0001319930000000004

The results show that Quicksort also pays the price of recursion when the list is sufficiently small, taking longer to complete than both insertion sort and bubble sort.

Implementing Timsort in Python

In this section, you’ll create a barebones Python implementation that illustrates all the pieces of the Timsort algorithm. If you’re interested, you can also check out theoriginal C implementation of Timsort.

The first step in implementing Timsort is modifying the implementation ofinsertion_sort() from before:

 1definsertion_sort(array,left=0,right=None): 2ifrightisNone: 3right=len(array)-1 4 5# Loop from the element indicated by 6# `left` until the element indicated by `right` 7foriinrange(left+1,right+1): 8# This is the element we want to position in its 9# correct place10key_item=array[i]1112# Initialize the variable that will be used to13# find the correct position of the element referenced14# by `key_item`15j=i-11617# Run through the list of items (the left18# portion of the array) and find the correct position19# of the element referenced by `key_item`. Do this only20# if the `key_item` is smaller than its adjacent values.21whilej>=leftandarray[j]>key_item:22# Shift the value one position to the left23# and reposition `j` to point to the next element24# (from right to left)25array[j+1]=array[j]26j-=12728# When you finish shifting the elements, position29# the `key_item` in its correct location30array[j+1]=key_item3132returnarray

This modified implementation adds a couple of parameters,left andright, that indicate which portion of the array should be sorted. This allows the Timsort algorithm to sort a portion of the array in place. Modifying the function instead of creating a new one means that it can be reused for both insertion sort and Timsort.

Now take a look at the implementation of Timsort:

 1deftimsort(array): 2min_run=32 3n=len(array) 4 5# Start by slicing and sorting small portions of the 6# input array. The size of these slices is defined by 7# your `min_run` size. 8foriinrange(0,n,min_run): 9insertion_sort(array,i,min((i+min_run-1),n-1))1011# Now you can start merging the sorted slices.12# Start from `min_run`, doubling the size on13# each iteration until you surpass the length of14# the array.15size=min_run16whilesize<n:17# Determine the arrays that will18# be merged together19forstartinrange(0,n,size*2):20# Compute the `midpoint` (where the first array ends21# and the second starts) and the `endpoint` (where22# the second array ends)23midpoint=start+size-124end=min((start+size*2-1),(n-1))2526# Merge the two subarrays.27# The `left` array should go from `start` to28# `midpoint + 1`, while the `right` array should29# go from `midpoint + 1` to `end + 1`.30merged_array=merge(31left=array[start:midpoint+1],32right=array[midpoint+1:end+1])3334# Finally, put the merged array back into35# your array36array[start:start+len(merged_array)]=merged_array3738# Each iteration should double the size of your arrays39size*=24041returnarray

Although the implementation is a bit more complex than the previous algorithms, we can summarize it quickly in the following way:

• Lines 8 and 9 create small slices, or runs, of the array and sort them using insertion sort. You learned previously that insertion sort is speedy on small lists, and Timsort takes advantage of this. Timsort uses the newly introducedleft andright parameters ininsertion_sort() to sort the list in place without having to create new arrays like merge sort and Quicksort do.

• Line 16 merges these smaller runs, with each run being of size32 initially. With each iteration, the size of the runs is doubled, and the algorithm continues merging these larger runs until a single sorted run remains.

Notice how, unlike merge sort, Timsort merges subarrays that were previously sorted. Doing so decreases the total number of comparisons required to produce a sorted list. This advantage over merge sort will become apparent when running experiments using different arrays.

Finally,line 2 definesmin_run = 32. There are two reasons for using32 as the value here:

1. Sorting small arrays using insertion sort is very fast, andmin_run has a small value to take advantage of this characteristic. Initializingmin_run with a value that’s too large will defeat the purpose of using insertion sort and will make the algorithm slower.

2. Merging two balanced lists is much more efficient than merging lists of disproportionate size. Picking amin_run value that’s a power of two ensures better performance when merging all the different runs that the algorithm creates.

Combining both conditions above offers several options formin_run. The implementation in this tutorial usesmin_run = 32 as one of the possibilities.

Measuring Timsort’s Big O Complexity

On average, the complexity of Timsort isO(n log₂n), just like merge sort and Quicksort. The logarithmic part comes from doubling the size of the run to perform each linear merge operation.

However, Timsort performs exceptionally well on already-sorted or close-to-sorted lists, leading to a best-case scenario ofO(n). In this case, Timsort clearly beats merge sort and matches the best-case scenario for Quicksort. But the worst case for Timsort is alsoO(n log₂n), which surpasses Quicksort’sO(n²).

Timing Your Timsort Implementation

You can userun_sorting_algorithm() to see how Timsort performs sorting the ten-thousand-element array:

 1if__name__=="__main__": 2# Generate an array of `ARRAY_LENGTH` items consisting 3# of random integer values between 0 and 999 4array=[randint(0,1000)foriinrange(ARRAY_LENGTH)] 5 6# Call the function using the name of the sorting algorithm 7# and the array you just created 8run_sorting_algorithm(algorithm="timsort",array=array)

Now execute the script to get the execution time oftimsort:

$pythonsorting.pyAlgorithm: timsort. Minimum execution time: 0.5121690789999998

At0.51 seconds, this Timsort implementation is a full0.1 seconds, or 17 percent, faster than merge sort, though it doesn’t match the0.11 of Quicksort. It’s also a ridiculous 11,000 percent faster than insertion sort!

Now try to sort an already-sorted list using these four algorithms and see what happens. You can modify your__main__ section as follows:

 1if__name__=="__main__": 2# Generate a sorted array of ARRAY_LENGTH items 3array=[iforiinrange(ARRAY_LENGTH)] 4 5# Call each of the functions 6run_sorting_algorithm(algorithm="insertion_sort",array=array) 7run_sorting_algorithm(algorithm="merge_sort",array=array) 8run_sorting_algorithm(algorithm="quicksort",array=array) 9run_sorting_algorithm(algorithm="timsort",array=array)

If you execute the script now, then all the algorithms will run and output their corresponding execution time:

Algorithm: insertion_sort. Minimum execution time: 53.5485634999991Algorithm: merge_sort. Minimum execution time: 0.372304601Algorithm: quicksort. Minimum execution time: 0.24626494199999982Algorithm: timsort. Minimum execution time: 0.23350277099999994

This time, Timsort comes in at a whopping thirty-seven percent faster than merge sort and five percent faster than Quicksort, flexing its ability to take advantage of the already-sorted runs.

Notice how Timsort benefits from two algorithms that are much slower when used by themselves. The genius of Timsort is in combining these algorithms and playing to their strengths to achieve impressive results.

Analyzing the Strengths and Weaknesses of Timsort

The main disadvantage of Timsort is its complexity. Despite implementing a very simplified version of the original algorithm, it still requires much more code because it relies on bothinsertion_sort() andmerge().

One of Timsort’s advantages is its ability to predictably perform inO(n log₂n) regardless of the structure of the input array. Contrast that with Quicksort, which can degrade down toO(n²). Timsort is also very fast for small arrays because the algorithm turns into a single insertion sort.

For real-world usage, in which it’s common to sort arrays that already have some preexisting order, Timsort is a great option. Its adaptability makes it an excellent choice for sorting arrays of any length.