Well, I want to use Quick sort on given 3 values, doesn't matter what values, how can I get to the worst case which is 9 operations?
Can anyone draw a tree and show how it show nlogn and n^2 operations? I've tried to find on the internet, but I still didnt manage to draw one properly to show that.
The worst case complexity of quick sort depends on the chosen pivot. If the pivot chosen is the leftmost or the rightmost element. Then the worst case complexity will occur in the following cases:
1) Array is already sorted in same order.
2) Array is already sorted in reverse order.
3) All elements are same (special case of case 1 and 2).
Since these cases occur very frequently, the pivot is chosen randomly. By choosing pivot randomly the chances of worst case are reduced.
The analysis of quicksort algorithm is explained in this blogpost by khan academy.
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Hello all this is my very first question here. I am new to datastructure and algorithms my teacher asked me to compare time complexity of different algorithms including: merge sort, heap sort, insertion sort, and quick sort. I search over internet and find out that quick sort is the fastest of all but my version of quick sort is the slowest of all (it sort 100 random integers in almost 1 second while my other sorting algorithms took almost 0 second). I tweak my quick sort logic many times (taking first value as pivot than tried to take middle value as pivot but in vain) I finally search the code over internet and there was not much difference in my code and code on internet. Now I really am confused that if this is behaviour of quick sort is natural (I mean whatever your logic is you are gonna get same results.) or there are some specific situations where you should use quick sort. In the end I know my question is not clear (I don't know how to ask besides my english is also not very good.) I hope someone can help me I really wanted to attach picture of awkward result I am having but I can't (reputation < 10).
Theoretically, quicksort is supposed to be the fastest algorithm for sorting, with a runtime of O(nlogn). It's worst case would be O(n^2), but only occurs if there are repeated values are equal to the pivot.
In your situation, I can only assume that your pivot value is not ideal in your data array, but is still able to sort the values using that pivot. Otherwise, your quicksort implementation is unfortunately incorrect.
Quicksort has O(n^2) worst-case runtime and O(nlogn) average case runtime. A good reason why Quicksort is so fast in practice compared to most other O(nlogn) algorithms such as Heapsort, is because it is relatively cache-efficient. Its running time is actually O(n/Blog(n/B)), where B is the block size. Heapsort, on the other hand, doesn't have any such speedup: it's not at all accessing memory cache-efficiently.
The value you choose as pivot may not be appropriate hence your sorting may be taking some time.You can avoid quicksort’s worst-case run time of O(n^2) almost entirely by using an appropriate choice of the pivot – such as picking it at random.
Also , the best and worst case often are extremes rarely occurring in practice.But any average case analysis assume some distribution of inputs. For sorting, the typical choice is the random permutation model (as assumed on Wikipedia).
I know that quicksort is not stable method, namely for equal elements, maybe member of array will not be placed at correct position, I need example of array (in which elements are repeated several times) quicksort does not work (need for example three way of partitioning method). I could not be able to find such example of array in internet and could you help me?
sure I can use other kind of sorting methods for this problem (like heap sort ,merge sort, etc), but my attitude is to know in real world example, what kind of data contains risk for quicksort, because as know it is one most useful method and is used often
Quicksort shouldn't crash no matter what array it is given.
When a sorting algorithm is called 'stable' or 'not stable', it does not refer to safety of the algorithm or whether or not it crashes. It is related to maintaining relative order of elements that have the same key.
As a brief example, if you have:
[9, 5, 7, 5, 1]
Then a 'stable' sorting algorithm should guarantee that in the sorted array the first 5 is still placed before the second 5. Even though for this trivial example there is no difference, there are examples in which it makes a difference, such as when sorting a table based on one column (you want the other columns to stay in the same order as before).
See more here: http://en.wikipedia.org/wiki/Stable_sort#Stability
I read that it's possible to make quicksort run at O(nlogn)
the algorithm says on each step choose the median as a pivot
but, suppose we have this array:
10 8 39 2 9 20
which value will be the median?
In math if I remember correct the median is (39+2)/2 = 41/2 = 20.5
I don't have a 20.5 in my array though
thanks in advance
You can choose either of them; if you consider the input as a limit, it does not matter as it scales up.
We're talking about the exact wording of the description of an algorithm here, and I don't have the text you're referring to. But I think in context by "median" they probably meant, not the mathematical median of the values in the list, but rather the middle point in the list, i.e. the median INDEX, which in this cade would be 3 or 4. As coffNjava says, you can take either one.
The median is actually found by sorting the array first, so in your example, the median is found by arranging the numbers as 2 8 9 10 20 39 and the median would be the mean of the two middle elements, (9+10)/2 = 9.5, which doesn't help you at all. Using the median is sort of an ideal situation, but would work if the array were at least already partially sorted, I think.
With an even numbered array, you can't find an exact pivot point, so I believe you can use either of the middle numbers. It'll throw off the efficiency a bit, but not substantially unless you always ended up sorting even arrays.
Finding the median of an unsorted set of numbers can be done in O(N) time, but it's not really necessary to find the true median for the purposes of quicksort's pivot. You just need to find a pivot that's reasonable.
As the Wikipedia entry for quicksort says:
In very early versions of quicksort, the leftmost element of the partition would often be chosen as the pivot element. Unfortunately, this causes worst-case behavior on already sorted arrays, which is a rather common use-case. The problem was easily solved by choosing either a random index for the pivot, choosing the middle index of the partition or (especially for longer partitions) choosing the median of the first, middle and last element of the partition for the pivot (as recommended by R. Sedgewick).
Finding the median of three values is much easier than finding it for the whole collection of values, and for collections that have an even number of elements, it doesn't really matter which of the two 'middle' elements you choose as the potential pivot.
While working with Quick Sort algorithm I wondered whether any formula or some kind of stuff might be available for finding the no of passes that a particular set of values may take to completely sorted in ascending order.
Is there any formula to calculate the no of passes that a Quick Sort algorithm will take?
Any given set of values will have a different number of operations, based on pivot value selection method, and the actual values being sorted.
So...no, unless the approximations of 'between O(N log(N)) and O(N^2)' is good enough.
That one has to qualify the average versus worst case should be enough to show that the only way to determine the number of operations is to actually run the quicksort.
I've read that the pivote can be the median of 3 numbers, bottom, middle and top. But, could that generate overflow? What happens if the median returns a value larger than the array size?
I assume that the this choice is by assuming that they array values can't be longer than the array size.
I think I'm confused at what a pivote really is.
The pivot is just the value that you compare other values against - lower values go the left, higher to the right. The pivot can be chosen by taking any of the existing values in the array. If the array is completely unsorted, it won't matter which value you choose. If it is somewhat sorted, you should choose a value from the middle of the array.
UPDATE: Some reading informs me that a better pivot choice may be to choose the median value of 3 values in the array (such as middle, bottom and top or 3 random positions). Some people advocate taking the median of 5 values. The worst-case performance of quicksort occurs when pivot is close to the smallest or largest value in the array, and this tactic is intended to defend against that occurring. This is just an optimisation for certain kinds of data - it is not a necessity.