All of these sorting algorithms have an average case of O(n log n), so I would just like to know how I would be able to differentiate between these three sorting algorithms if I could run tests but not know which sorting algorithm was being run.
another difference between Heap and Merge sort you may want to concern is, Heap is not stable sort, but Mergesort is.
here is a table(link below), you could find (almost) any information about comparison sort algorithms you want.
https://en.wikipedia.org/wiki/Sorting_algorithm#Comparison_of_algorithms
heapsort is a inplace sorting algorithm , we don't need extra storage to sort the elements but mergesort is not inplace sorting algorithm , we required extra storage , in merge procedure , to sort the elements.The worst case running time of quicksort is O(n^2) that differentiate it form heapsort and mergesort
There are many cases in which performance of these algorithms are different.
For example.
if all input element are same.
then, heapsort will run in O(n) time
quicksort will run in O(n^2) time. (if last element is a pivote element)
and,
mergesort is going to take O(logn) time.
Related
What kind of data input are the following sorting algorithms efficient on/not efficient on? Quicksort, Mergesort, Heapsort, Insertion sort etc.
I know there are at least 2 factors that affect the performance of a sorting algorithm: 1) The size of the input, and 2) whether or not the data is already mostly sorted. But I don't know exactly how these factors affect the efficiency of the algorithms.
I'd like to study this in detail, so if there are any sources/links that you can point me to, that'd be great.
Assuming quicksort is based on Hoare partition scheme (middle value as pivot), then it won't degrade to worst case time complexity of O(n^2) for almost sorted data.
https://en.wikipedia.org/wiki/Quicksort#Hoare_partition_scheme
Mergesort always does n ⌈log2(n)⌉ moves. If data is already sorted, then the number of compares is about (⌈n ⌈log2(n)⌉)/2.
Heapsort time complexity remains about the same (duplicates may reduce running time).
Insertion sort is the only sort in this list that is faster if the data is nearly sorted, but it's time complexity is O(n^2). I'm thinking that for nearly sorted data, the time complexity would be ~ O(m n), where m is the number of elements out of place.
Variations of natural merge sort, which might use insertion sort on small runs while scanning and identifying already sorted runs, would have time complexity O(n) on already sorted data.
When building a sorting algorithm to sort an array, how many n elements in the array is quick sort faster that Insertion sort? I know that Quick sort is good for more elements and that Insertion sort is great for smaller size. But was wondering around what size is Quick Sort a far better option than Insertion Sort?
These algorithms depend on more than just the size of the arrays to determine their run time. For quicksort, the pivot your algorithm selects can have a significant effect on runtime. If the pivot is consistently the greatest or least element, then the quicksort takes O(n^2). Insertion sort is also influenced by factors besides array size. If you are inserting elements in order, the algorithm might allow for a runtime of O(n) regardless of array size. However, if you are inserting in reverse-order, this algorithm will take O(n^2). Due to these factors, there is no size n for which one algorithm is guaranteed to perform better than the other. If you are concerned with the runtimes of sorting algorithms for large arrays, you should check out heapsort or mergesort, they are both O(n log n) and are much faster!
I revisited insertion sort algorithm and noticed something funny.
One obviously shouldn't use an array with this sort, as upon insertion, one will have to shift all subsequent elements O(n^2 log(n)). However a linked list is also not good here, since we preferably find the right placement using binary search, which isn't possible for a simple linked list (so we end up with O(n^2)).
Which makes me wonder: what is a data structure on which this sorting algorithm provides its premise of O(nlog(n)) complexity?
From where did you get the premise of O(n log n)? Wikipedia disagrees, as does my own experience. The premises of the insertion sort include components that are O(n) for each of the n elements.
Also, I believe that your claim of O(n^2 log n) is incorrect. The binary search is log n, and the ensuing "move sideways" is n, but these two steps are in succession, not nested. The result is n + log n, not a multiplication. The result is the expected O(n^2).
If you use a gapped array and a binary search to figure out where to insert things, then with high probability your sort will be O(n log(n)). See https://en.wikipedia.org/wiki/Library_sort for details.
However this is not as efficient as a wide variety of other sorts that are widely implemented. So this knowledge is only of theoretical interest.
Insertion sort is defined over array or list, if you use some other data structure, then it will be another algorithm.
Of course if you use a BST, insertion and search would be O(log(n)) and your overall complexity would be O(n.log(n)) on the average (remind that it will be O(n^2) in the worst), but this will be no more an insertion sort but a tree sort. If you use an AVL tree, then you get the O(n.log(n)) worst case complexity.
In insertion sort the best case scenario is when the sequence is already sorted and that takes Linear time and in the worst case takes O(n^2) time. I do not know how you got the logarithmic part in the complexity.
Quick sort has worst case time complexity as O(n^2) while others like heap sort and merge sort has worst case time complexity as O(n log n) ..still quick sort is considered as more fast...Why?
On a side note, if sorting an array of integers, then counting / radix sort is fastest.
In general, merge sort does more moves but fewer compares than quick sort. The typical implementation of merge sort uses a temp array of the same size as the original array, or 1/2 the size (sort 2nd half into second half, sort first half into temp array, merge temp array + 2nd half into original array), so it needs more space than quick sort which optimally only needs log2(n) levels of nesting, and to avoid worst case nesting, a nesting check may be used and quick sort changed to heap sort, (this is called introsort).
If the compare overhead is greater than the move overhead, then merge sort is faster. A common example where compares take longer than moves would be sorting an array of pointers to strings. Only the (4 or 8 byte) pointers are moved, while the strings may be significantly larger (and similar for a large number of strings).
If there is significant pre-ordering of the data to be sorted, then timsort (fixed sized runs) or a "natural" merge sort (variable sized runs) will be faster.
While it is true that quicksort has worst case time complexity of O(n^2), as long as the quicksort implementation properly randomizes the input, its average case (expected) running time is O(n log n).
Additionally, the constant factors hidden by the asymptotic notation that do matter in practice are pretty small as compared to other popular choices such as merge sort. Thus, in expectation, quicksort will outperform other O(n log n) comparison sorts despite the less savory worst case bounds
Not exactly like that. Quicksort is the best in most cases, however it's pesimistic time complexity can be O(n^2), it doesn't mean it always is. The issue lies in choosing the right point of pivot, if you choose it correctly you have time complexity O(n log n).
In addition, quicksort is one of the cheapest/easiest in implementation.
Quicksort is better than mergesort in many cases. But when might mergesort be better than quicksort?
For example, mergesort works better when all data cannot be loaded to memory at once. Are there any other cases?
Answers to the suggested duplicate question list advantages of using quicksort over mergesort. I'm asking about the possible cases and applications where mergesort would be better than quicksort.
Both quicksort and mergesort can work just fine if you can't fit all data into memory at once. You can implement quicksort by choosing a pivot, then streaming elements in from disk into memory and writing elements into one of two different files based on how that element compares to the pivot. If you use a double-ended priority queue, you can actually do this even more efficiently by fitting the maximum number of possible elements into memory at once.
Mergesort is worst-case O(n log n). That said, you can easily modify quicksort to produce the introsort algorithm, a hybrid between quicksort, insertion sort, and heapsort, that's worst-case O(n log n) but retains the speed of quicksort in most cases.
It might be helpful to see why quicksort is usually faster than mergesort, since if you understand the reasons you can pretty quickly find some cases where mergesort is a clear winner. Quicksort usually is better than mergesort for two reasons:
Quicksort has better locality of reference than mergesort, which means that the accesses performed in quicksort are usually faster than the corresponding accesses in mergesort.
Quicksort uses worst-case O(log n) memory (if implemented correctly), while mergesort requires O(n) memory due to the overhead of merging.
There's one scenario, though, where these advantages disappear. Suppose you want to sort a linked list of elements. The linked list elements are scattered throughout memory, so advantage (1) disappears (there's no locality of reference). Second, linked lists can be merged with only O(1) space overhead instead of O(n) space overhead, so advantage (2) disappears. Consequently, you usually will find that mergesort is a superior algorithm for sorting linked lists, since it makes fewer total comparisons and isn't susceptible to a poor pivot choice.
A single most important advantage of merge sort over quick sort is its stability: the elements compared equal retain their original order.
MergeSort is stable by design, equal elements keep their original order.
MergeSort is well suited to be implemented parallel (multithreading).
MergeSort uses (about 30%) less comparisons than QuickSort. This is an often overlooked advantage, because a comparison can be quite expensive (e.g. when comparing several fields of database rows).
Quicksort is average case O(n log n), but has a worst case of O(n^2). Mergesort is always O(n log n). Besides the asymptotic worst case and the memory-loading of mergesort, I can't think of another reason.
Scenarios when quicksort is worse than mergesort:
Array is already sorted.
All elements in the array are the same.
Array is sorted in reverse order.
Take mergesort over quicksort if you don't know anything about the data.
Merge sort has a guaranteed upper limit of O(N log2N). Quick sort has such limit, too, but it is much higher - it is O(N2). When you need a guaranteed upper bound on the timing of your code, use merge sort over quick sort.
For example, if you write code for a real-time system that relies on sorting, merge sort would be a better choice.
Merge Sort Worst case complexity is O(nlogn) whereas Quick Sort worst case is O(n^2).
Merge Sort is a stable sort which means that the same element in an array maintain their original positions with respect to each other.