# How many number of comparisons are required in insertion sort

Insertion sort is a simple sorting algorithm that works similar to the way you sort playing cards in your hands. The array is virtually split into a sorted and an unsorted part. Values from the unsorted part are picked and placed at the correct position in the sorted part.

Algorithm To sort an array of size n in ascending order: 1: Iterate from arr to arr[n] over the array. Move the greater elements one position up to make space for the swapped element. Example: Another Example: 1211, 13, 5, 6. Since 11 is smaller than 12, move 12 and insert 11 before 12 11, 1213, 5, 6.

### Number of comparisons in an insertion sort

I-1] are smaller than 13 11, 12, 135, 6. Boundary Cases : Insertion sort takes maximum time to sort if elements are sorted in reverse order. And it takes minimum time Order of n when elements are already sorted. Uses: Insertion sort is used when number of elements is small. It can also be useful when input array is almost sorted, only few elements are misplaced in complete big array. What is Binary Insertion Sort?

We can use binary search to reduce the number of comparisons in normal insertion sort. Binary Insertion Sort uses binary search to find the proper location to insert the selected item at each iteration.

In normal insertion, sorting takes O i at ith iteration in worst case. We can reduce it to O logi by using binary search. The algorithm, as a whole, still has a running worst case running time of O n2 because of the series of swaps required for each insertion. Refer this for implementation. How to implement Insertion Sort for Linked List?By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information.

I want to find the total comparisons for sorting n elements in an array using different sorting algorithms.

## Comparison among Bubble Sort, Selection Sort and Insertion Sort

I don't want to do it manually in case the number of elements in the array is considerably large. Is there a "formula" to calculate the comparisons for each of the sorting algorithms listed below if for example there is 8 elements in an array containing the following elements [3,24,66,34,8,-5,42,80]?

How can I find the comparisons for each? This is not an easy task, as it can depend on details of the algorithm implementation, and also is not a pure function of n. Actually, what you get is a distribution of values of the number of comparisons, depending on the permutation of the input. Usually, one distinguishes the best case least number of comparisonthe worst case largest number and the average case mathematical expectation when you assume the respective probabilities of the input permutations.

These numbers can be obtained by reasoning on the program, but this is usually a difficult task even daunting for the average caseoften solved with approximations. Anyway, you can obtain it empirically by instrumenting your program: declare a counter variable, and increment it at the same time as a comparison is made. Even though n will remain modest, you will observe the best and worst cases, and with more care, the central trend and the spread.

This should be instructive. It is impossible, as the exact number depends on the input. This why you have optimistic complexity, pessimistic, and average sometimes also called expected.

On the other hand optimistic case for bubble sort is O n. The only thing you can do is to compute it per problem instance, for example by tapping into comparison function. Although, I am not sure if per-instance values are very meaningful. Usually people do not make this kind of calculation.

They are interested in evaluating the complexity of the algorithm, i. For instance, merge sort grows in average with O n log n. This means that the number of comparisons of merge sort is not worse than n log n where n is the size of the input. There are some methods to arrive to this expression, namely master theorem or tree method.

Actually, one can prove that no algorithm based only on comparisons cannot make less comparisons than n log n. This is the so-called comparison model! However, sorting can be done in linear time, depending on the type of your set, for instance using counting sort - a kind of histogram.A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation often a "less than or equal to" operator or a three-way comparison that determines which of two elements should occur first in the final sorted list.

The only requirement is that the operator forms a total preorder over the data, with:. In a stable sortthe input order determines the sorted order in this case. A metaphor for thinking about comparison sorts is that someone has a set of unlabelled weights and a balance scale. Their goal is to line up the weights in order by their weight without any information except that obtained by placing two weights on the scale and seeing which one is heavier or if they weigh the same.

There are fundamental limits on the performance of comparison sorts. This is a consequence of the limited information available through comparisons alone — or, to put it differently, of the vague algebraic structure of totally ordered sets. In this sense, mergesort, heapsort, and introsort are asymptotically optimal in terms of the number of comparisons they must perform, although this metric neglects other operations. Non-comparison sorts such as the examples discussed below can achieve O n performance by using operations other than comparisons, allowing them to sidestep this lower bound assuming elements are constant-sized.

Comparison sorts may run faster on some lists; many adaptive sorts such as insertion sort run in O n time on an already-sorted or nearly-sorted list. Real-world measures of sorting speed may need to take into account the ability of some algorithms to optimally use relatively fast cached computer memoryor the application may benefit from sorting methods where sorted data begins to appear to the user quickly and then user's speed of reading will be the limiting factor as opposed to sorting methods where no output is available until the whole list is sorted.

Despite these limitations, comparison sorts offer the notable practical advantage that control over the comparison function allows sorting of many different datatypes and fine control over how the list is sorted. For example, reversing the result of the comparison function allows the list to be sorted in reverse; and one can sort a list of tuples in lexicographic order by just creating a comparison function that compares each part in sequence:.

Balanced ternary notation allows comparisons to be made in one step, whose result will be one of "less than", "greater than" or "equal to". Comparison sorts generally adapt more easily to complex orders such as the order of floating-point numbers. Additionally, once a comparison function is written, any comparison sort can be used without modification; non-comparison sorts typically require specialized versions for each datatype.

This flexibility, together with the efficiency of the above comparison sorting algorithms on modern computers, has led to widespread preference for comparison sorts in most practical work.

When the keys form a small compared to n range, counting sort is an example algorithm that runs in linear time. Other integer sorting algorithms, such as radix sortare not asymptotically faster than comparison sorting, but can be faster in practice. This bound is asymptotically tight. Given a list of distinct numbers we can assume this because this is a worst-case analysisthere are n factorial permutations exactly one of which is the list in sorted order.

The sort algorithm must gain enough information from the comparisons to identify the correct permutation.Those code lines from your quick sort algorithm does 1 swap. Changing places of numbers 2 and 1. But you are counting as 3 swaps. So you have to fix it.

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Mike Stein wrote: for my particular project I have to count every assignment e. Mike Stein wrote: Since I need it to count when both true and false it is redundant to have the counter in the if statement!

You don't need to increment the number of comparisons when something is true or false. You need to increment the number of comparisons when a comparison takes place. Forum: Beginning Java. Count comparisons and assignments in three sorting algorithms. Mike Stein. I am meant to only count the swaps and comparisons that involve anything other than indexes as they are too fast to really matter according to the professor.

Could anyone please look over my work and let me know if my counters are in the right position. Makes me think that my counters are somehow out of place or that I don't have enough of them. Liutauras Vilda.

I like Hello Mike, Had a change to have a look only at the quick sort at the moment. Actually over here, is just a one "visible" swap. Technically, I do agree that we could state swaps count as you did. In fact, I still think your professor expects as a 1 swap over here. Please clarify. How many swaps you considering you need to make? It is important to make it clear, so someone could answer your main question correctly. Mike Stein wrote: Just one swap would be needed. Liutauras, I understand completely that this should be considered one swap. However, for my particular project I have to count every assignment e. The only assignments and comparisons that I don't count involve indexes. Trouble is, I'm not quite sure that I spotted or counted all of these operations in my sorting classes. As it is hell confusing moves, swaps, compares, comparisons Beside that, there are misplaced positions.

More better, on a separate line for better readability.The selection sort algorithm sorts an array by repeatedly finding the minimum element considering ascending order from unsorted part and putting it at the beginning. If this activity does not load, try refreshing your browser. Also, this page requires javascript.

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Please visit using a browser with javascript enabled. If loading fails, click here to try again Question 1 Consider a situation where swap operation is very costly. Which of the following sorting algorithms should be preferred so that the number of swap operations are minimized in general? Please wait while the activity loads.

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Consider a situation where swap operation is very costly. Question 1 Explanation:. Selection sort makes O n swaps which is minimum among all sorting algorithms mentioned above. Question 2. Which sorting algorithm will take least time when all elements of input array are identical? Consider typical implementations of sorting algorithms.

Question 2 Explanation:. The insertion sort will take n time when input array is already sorted. Question 3. Which of the following sorting algorithms has the lowest worst-case complexity? Question 3 Explanation:. Question 4. Which is the correct order of the following algorithms with respect to their time Complexity in the best case? Question 4 Explanation:. Question 5.

## Comparison sort

Which one of the following in-place sorting algorithms needs the minimum number of swaps? Question 5 Explanation:. Selection Sort is an in-place algorithm having minimum number of swaps.

It works on greedy approach and takes O n swaps to sort the array of n elements. Question 6. How many comparisons are needed to sort an array of length 5 if a straight selection sort is used and array is already in the opposite order?Forum: Beginning Java. Insertion sort counting how many comparisons. Jay Fischer. Hey guys, I'm new to coderanch and also java.

I hope this question belongs in this section. The correct answer is 6 Copies and 3 Comparisons. But I get 6 Copies and 2 Comprisons with my code. I can't seem to spot where I messed up, could anybody give me some advice? Thank you in advance. Winston Gutkowski. I like Call me Jay wrote: The correct answer is 6 Copies and 3 Comparisons. Says who? As I recall, insertion sort is extremely good when data are already partially sorted - as yours are. Try them in reverse order and see if you get a different result.

You're also not likely to get an accurate reading from only 3 items. Try 10 and see if your results are as predicted - and try them with some corner cases like reverse order and "almost sorted", along with random arrangements. That'll give you a much better result set to work with. I'm pretty sure the code for insertion sort is right and properly working.

Stephan van Hulst. Saloon Keeper. The comp variable is only incremented if the comparison evaluates to true. If the comparison evaluates to false, you don't increment the counter, but that doesn't mean the comparison didn't happen. Campbell Ritchie. That could power this tiny ad:. Boost this thread!It always maintains a sorted sublist in the lower positions of the list. Figure 4 shows the insertion sorting process. The shaded items represent the ordered sublists as the algorithm makes each pass.

As we look back into the already sorted sublist, we shift those items that are greater to the right. When we reach a smaller item or the end of the sublist, the current item can be inserted.

Figure 5 shows the fifth pass in detail. At this point in the algorithm, a sorted sublist of five items consisting of 17, 26, 54, 77, and 93 exists. We want to insert 31 back into the already sorted items. The first comparison against 93 causes 93 to be shifted to the right. When the item 26 is encountered, the shifting process stops and 31 is placed in the open position. Now we have a sorted sublist of six items. Line 8 performs the shift operation that moves a value up one position in the list, making room behind it for the insertion.

Remember that this is not a complete exchange as was performed in the previous algorithms. However, in the best case, only one comparison needs to be done on each pass. This would be the case for an already sorted list.

One note about shifting versus exchanging is also important. In general, a shift operation requires approximately a third of the processing work of an exchange since only one assignment is performed. In benchmark studies, insertion sort will show very good performance. Social Runestone in social media:. Each pass produces a longer sorted list.

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