First, we want to set up a recurrence relation. If we consider the best-case for the quicksort, that is, that each partition splits the collection in 1/2 (not a valid assumption), then (look at a recursive quicksort algorithm):
The partition requires about n comparisons. It is then left to sort 2 collections of size n/2. So:
T(n) = 2*T(n/2) + n
Use repeated substitution, or the "master theorem" or whatever it's called, a generalisation of closed forms, or draw a tree, or use the accounting method, or guess at the answer, prove it by induction.
Now, we need to consider the worst case. The partition still costs the same (about n comparisons), but the partition didn't yield 2 nice-sized sub-collections to sort. Instead, we have a single sub-array to sort, of size n-1 . So now the recurrence relation looks like:
T(n) = T(n-1) + n
Solve this in your favorite way.
Now, on average, how good is your algorithm? That depends on how well you choose your pivot. You can do some reasoning, or, you can run some nice experiments.
Have fun.
you can find an example in this link ww.computing.dcu.ie/~away/CA313/space.pdfgood luck
Polynomial vs non polynomial time complexity
The answer depends on what information you have from which you wish to calculate time.
That means that the running time of a program is proportional to some power of the input size.
Quality of solution
The time complexity of the quicksort algorithm is O(n log n) in the average case and O(n2) in the worst case.
The worst-case time complexity of quicksort is O(n2), where n is the number of elements in the array being sorted.
The time complexity of quicksort when the first element is chosen as the pivot is O(n2) in the worst-case scenario.
The Big O notation of Quicksort algorithm is O(n log n) in terms of time complexity.
The time complexity of Quicksort algorithm is O(n log n) in terms of Big O notation.
The recurrence relation for the quicksort algorithm is T(n) T(k) T(n-k-1) O(n), where k is the position of the pivot element. This relation affects the time complexity of quicksort by determining the number of comparisons and swaps needed to sort the elements. The average time complexity of quicksort is O(n log n), but in the worst-case scenario, it can be O(n2) if the pivot selection is not optimal.
The memory complexity of the quicksort algorithm is O(log n) in the best and average cases, and O(n) in the worst case.
The space complexity of the quicksort algorithm is O(log n) in the best and average cases, and O(n) in the worst case.
Quicksort is generally more efficient than heapsort for large datasets due to its average time complexity of O(n log n) compared to heapsort's O(n log n) worst-case time complexity.
Quicksort is generally more efficient than heapsort for large datasets due to its average-case time complexity of O(n log n) compared to heapsort's O(n log n) worst-case time complexity.
Quicksort's time complexity is O(n log n) because it divides the input array into smaller subarrays and recursively sorts them. The partitioning step takes O(n) time, and on average, the algorithm splits the array into two equal parts. This results in a logarithmic number of levels in the recursion tree, leading to a time complexity of O(n log n).
For small datasets, insertion sort is generally more efficient than quicksort. This is because insertion sort has a lower overhead and performs well on small lists due to its simplicity and low time complexity.