Unfortunately 3 does not decrease the search space enough per iteration to be a worthwhile choice of "five". How can I count the number of element comparisons in the Quicksort algorithm? Which means it is at most 10cn. I want to be able to quit Finder but can't edit Finder's Info.plist after disabling SIP. Median-of-medians is a recursive algorithm which solves the more general selection problem: given an array $A$ of length $n$ (which we assume, for simplicity, has distinct elements) and an integer $k$, find the $k$'th smallest element (where $1 \leq k \leq n$). Similar logic for the number of elements m is bigger than. And that's your estimate of the overall median. Its logic is given in Wikipedia as: The chosen pivot is both less than and greater than half of the elements in the list of medians, which is around n/10 elements (1/2 * (n/5)) for each half. Median of Medians algorithm misunderstanding? Something I dont understand about median of medians algorithm, Generalizing the median of medians algorithm. It's not in CLRS unfortunately, and I don't have familiarity with other algorithms textbooks. One key step about this algorithm is to find an approximate median, and according to Wikipedia, we have the guarantee that this approximate median is greater than 30% of elements of the initial set. In particular, each recursive call will. Is it cheating if the proctor gives a student the answer key by mistake and the student doesn't report it? Medians and medoids. We have at least [g/2] groups (the group that its median is less than or equal to our pivot) that contain at least 3 element that is less than or equal to our pivot. The following code is my implementation of the quick select algorithm using Java. And yes, finding a median is a special case of selection, with the index being n/2. Sort each little set and identify the median element in this set. So my question is : where am I wrong?? Is that a correct interpretation? I'm struggling with the median of medians algorithm, and I think it's perhaps more of a semantics thing rather than a technical thing. Therefore, T(1) < 4*1. @m69 Yeah, I agree. In the previous post we said that our quickSelectSort was O (N^2) worst case. To learn more, see our tips on writing great answers. Making statements based on opinion; back them up with references or personal experience. You're going to take successive groups of five elements from your big list, find their median, and add it to a smaller list. (If you have some left over, you can ignore them.). Connecting three parallel LED strips to the same power supply, Effect of coal and natural gas burning on particulate matter pollution. Depend on our pivot, how many results we might have? In the above chart, our pivot (median of median) is in the green group. At this level, you obtain an exact median of the array you passed in. How to set a newcommand to be incompressible by justification? In the yellow group, there are 3 elements less than less or equal to our pivot, and in the purple group, there are 3 elements greater than or equal to our pivot. How to understand the complexity of medians of medians algorithm? This function too returns an exact result, not an approximation. Understanding "median of medians" algorithm, Explanation of the Median of Medians algorithm. If you see the "cross", you're on the right track. When we continuously expand this formula, we can find the rule. Stackoverflow is. Well, lets try. Use the median of medians algorithm to recursively determine the median of the set of all medians from the previous step. For example, the have an array with 15 items, we firstly group it into 3 groups, and find the median of each group, which are 8, 10 and 9. :-). In this case, g equals 3. Well, it turns out that 5 is optimal. Note that the algorithm used to find the approximate median is sometimes what people refer to when they say "median-of-medians", hence the confusion experienced by the OP I think. Modifying this Quicksort to always use the last element as the pivot, Explanation of the Median of Medians algorithm. Networks 45:125-142 19. Yes, it approximates medians at various levels, but the final output is exact. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Thanks for contributing an answer to Computer Science Stack Exchange! One of the reasons median-of-medians was such a big deal when it was discovered was that it was fully deterministic and worst-case efficient). Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missing, Ukkonen's suffix tree algorithm in plain English, Understanding "median of medians" algorithm, Image Processing: Algorithm Improvement for 'Coca-Cola Can' Recognition, Find running median from a stream of integers. Thats a Geometric series! Distance between points. Sort each sublist and determine its median directly. Otherwise, we need to find the (K -|LESS|-1)-th smallest item in GREATER. Firstly, we define T(n) as the following formula, T(n,k) means the expected number of comparisons to find the k-th smallest item in an array of length n, maximized over all arrays. Median of medians algorithm - which element to select as median for each group, Generalizing the median of medians algorithm. Where is it documented? To learn more, see our tips on writing great answers. Harry Potter and Detection of File Tampering, How To Develop First Web Page With Angular. Lets look at our example, we have a 4 length array. Oh I didn't realize it was in CLRS. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. If K = |LESS| + 1, our pivot is the answer! A fuzzy string matching algorithm for finding all occurrences from a set of strings in a large string, Median of distribution with memory constraint, Making use of one function to recursively find n/3 of another, finding a greedy algorithm that maximizes total energy of fruits subject to expiry dates. Are defenders behind an arrow slit attackable? . In all examples I've seen so far there already are the groups of the numbers divided , before the execution of the algorithm begins. Thanks for your reading, learning never ends! This makes it at least 3 numbers (2 numbers + the median itself) in each of those n / 10 small 5 element groups, that are smaller than m. Hence, m is at least bigger than 3n/10 numbers. Thus, it needs to operate within certain time bounds. What's the \synctex primitive? Do bracers of armor stack with magic armor enhancements and special abilities? What is a plain English explanation of "Big O" notation? Median of medians is an algorithm to select an approximate median as a pivot for a partitioning algorithm. Do non-Segwit nodes reject Segwit transactions with invalid signature? Questions: What about divided our array into groups that contain 3 elements? Bresenham line. We can easily find out that T(n) is a non-decreasing function of n, because as our array size increase, we need to execute more comparisons. Ready to optimize your JavaScript with Rust? Median Finding Algorithm. So what if we have n numbers..? In order to calculate T(n), the first component is after we randomly select a pivot, we need to compare our pivot with other items in our array, which result in n-1 comparisons. We want the index i that there are [n/2] numbers larger than ai. a linear-time algorithm to find the k'th element in an array (or in particular, find the median). General idea: Divide a problem into subprograms of the same kind; solve subprograms using the same approach and combine partial solution (if necessary). The 50-50 partition is given by the normal median, right? If p is between 0 and 1, we can have: The key property of this algorithm is n/5 + 7n/10 < n. And thats why our recursion works! In all the implementations I've seen, the median you find using median of medians is exact. Here is the pseudocode for median of medians algorithm (slightly modified to suit your example). (see the wikipedia page if my explanations are not clear). If you mix up the two, you will not get the expected result, as demonstrated in your example. Chans algorithm. Making statements based on opinion; back them up with references or personal experience. However, Median of Medians is a general-purpose selection algorithm, not merely a median-finding algorithm. ( Bound time- 7) If n>5, then partition the numbers into groups of 5. Even Wikipedia describes as an algorithm that approximates a median. The median-of-medians algorithm is separate from quickselect, so it shouldnt be making any recursive calls to quickselect. Here, we use the mathematical induction to prove that the expected number of comparisons for QuickSelect is at most 4n. apply a partitioning step on that median and use that to determine how to proceed from there. We do not currently allow content pasted from ChatGPT on Stack Overflow; read our policy here. For me, the easiest way to understand it is to just trust that recursion works and to trace through it only one layer deep, working under the assumption that all the recursive calls work, rather than trying to walk all the way down to the bottom of the recursion tree. The Median is an important measure (compared to the mean) for distorted data because the median is not so easily distorted. (You can see this by noting that you got back 27, which isn't the true median of that collection of values). Do you know of a textbook that describes the median of medians? So how big should "five" be? Why should Insertion Sort be used after threshold crossover in Merge Sort. Because we assume that at least 3/10 items are below our pivot, so the smallest value of |LESS| are 3n/10, and the largest value of |GREATER| is 7n/10. Asking for help, clarification, or responding to other answers. There are several ways to code this, based on e.g. As before, we define T(n,k) as the worse case time to find k-th smallest element in an array. When should i use streams vs just accessing the cloud firestore once in flutter? This means that for each of those smaller 5 element groups where m was bigger than its median, m is also bigger than two other numbers. Then, it takes those medians and puts them into a list and finds the median of that list. It corresponds to the cumulative percentage of 50%.The size of two arrays must be same, we will find the median of two separate arrays at first, then compare the separate medians to get an actual median of two lists.Input and OutputInput: Two sorted array are given. In the example above, we saw that if the median is k and you have m > k, then m is also bigger than 2 other numbers (that were themselves smaller than k). Not the answer you're looking for? Then how come it gives a 50-50 partition on average? However, when I look at actual implementations, e.g., in https://brilliant.org/wiki/median-finding-algorithm/, the algorithm they posted returns an exact median, but at each level of the recursion, you may have some approximate median generated from a sublist of medians. It might be easier to understand if explained as a base case and a recursive case. 2) The method you use does not return the median, it just return a number which is not so far from the median. What is this fallacy: Perfection is impossible, therefore imperfection should be overlooked. johndcook.com/blog/2009/06/23/tukey-median-ninther. I am working with the median-median algorithm or BFPRT algorithm and I seek to understand why would the partition of the array by $7$ blocks would work but with the $3$ fail? Find the median of medians takes us T(n/3), and in order to recurse on the larger side, we have: Received a 'behavior reminder' from manager. Median of Medians is an algorithm to find a good pivot point in sorting and selection algorithms.We first discuss how to find a median in an array of size N,. So I cannot understand how these groups are made. Firstly, what about using a sort algorithm and then find the middle index? Is it possible to hide or delete the new Toolbar in 13.1? Thus the search set decreases by at least 30%. Suppose we have g groups. In the first step, we have n/5 groups, for each group, it takes us O(1) to find the median of 5 items. A tag already exists with the provided branch name. Therefore, we give ourselves leeway by assuming the pivot can be somewhere that is roughly in the middle of our array. Should I give a brutally honest feedback on course evaluations? We do not currently allow content pasted from ChatGPT on Stack Overflow; read our policy here. Use this element as the pivot and proceed as in the quick-select algorithm. Do non-Segwit nodes reject Segwit transactions with invalid signature? Thanks for contributing an answer to Stack Overflow! I'm completely with your analysis up through the point where you get the medians of each of the blocks of five elements, when you're left with this collection of elements: You are correct that, at this point, we need to get the median of this collection of elements. We can use this algorithm to find the k-th smallest element in our array. At its most basic, the overall algorithm works like this: So this is where your calculation went wrong. Use the median of the medians from step 3 as the pivot. Cohen sutherland lineclip. Therefore, our final formula is: because n/3 + 2n/3 equals 1, our recursion cannot work in this example. Why does the USA not have a constitutional court? The median-of-medians algorithm is separate from quickselect, so it shouldn't be making any recursive calls to quickselect. If you make your groups of size. Can a prospective pilot be negated their certification because of too big/small hands? If our target is 3, 3 =|LESS| + 1, our pivot 4 is the answer. The median-of-medians algorithm is a deterministic linear-time selection algorithm. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. If he had met some scary fish, he would immediately return to the surface. I've always thought the median of medians algorithm as finding an approximate median $p$ such that $p$ is within $20\%$ of the true median $M$ in the sorted array. in code blocks, it would help. However, its pretty hard to achieve. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Optimal median of medians selection - 3 element blocks vs 5 element blocks? The one on brilliant.org was probably the best one I read, but I still would prefer a textbook read for this algo. Something I dont understand about median of medians algorithm. Earlier I was doing a search for median of medians in the book and could not find it. This recursion stops when medianOfMedians() is called for 25 elements or fewer, because then there are only 5 medians, and instead of using select() to find their median, it can use medianOfFive(). This is obvious. 1980s short story - disease of self absorption. Could you try to clarify the algorithms studied so far? Thanks for contributing an answer to Stack Overflow! As you see, the select() function recurses (unless the pivot happens to be the n-th element), but on ever smaller ranges of the array, so at some point (e.g. From this set of n /5 "baby" medians, apply the selection algorithm recursively to find the median of the baby medians. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Base case: T(1) = 0, when we have an array size of 1, we dont need to do anything! Lather, rinse, and repeat until you get down to less than five elements remaining. Median of Medians is an independent algorithm per se, however QuickSelect is one of the most common applications. And this finds the ith item in O (n) time. So I had the same confusion as this poster https://stackoverflow.com/questions/52461306/something-i-dont-understand-about-median-of-medians-algorithm and some others. and then the LESS and GREATER subarray have the same length. The purpose of those groups is to strip away elements that are surely lower or grater than the median of medians. T ( n) T ( n / 5) + T ( 7 n / 10) + O ( n). If the user adds a constant to every value, the . Finally, the 2nd smallest item in GREATER is our final answer. Is there any good technique that should I follow to find the number of elements its group should have ? :param arr::return: """ if arr is None or len (arr) == 0: return None: return select_pivot (arr, len (arr) // 2) def select_pivot (arr, k): """ Select a pivot corresponding to the kth largest element in the array:param arr: Array from which . Now if you get the median of those numbers (call it m), it is bigger than half of them and smaller than the other half (by definition of median!). Additionally, if you could put examples etc. Median of medians can be used as a pivot strategy in quicksort, yielding an optimal algorithm. Graham scan. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. @pepo I know it's not a good technique to post hyperlinks but I dont want to copy the site's content. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. That may be a good idea with an O(nlogn) time complexity, however, today we will look at two better algorithms, not only can achieve an O(n) time complexity, but also can be applied to a wider range of the problem. TabBar and TabView without Scaffold and with fixed Widget. Thats our pivot! . The idea is to use the "median of medians" algorithm twice and partition only after that. Finally, lets implement Deterministic Select in Java! @BlackVegetable I am in a little bit of a hurry now so I will edit the question in a couple of hours to be more specific! After finding the medians of those subarrays which for one . The median-of-medians algorithm computes an approximate median, namely a point that is guaranteed to be between the 30th and 70th percentiles (in the middle 4 deciles). CGAC2022 Day 10: Help Santa sort presents! If this new, smaller list is small enough, you can apply the base case, as described above. We were looking for the 4th element of 16, so now we look for the 4th element out of 7: Range of medians of five partitioned with pivot 1031 (depends on method): The smaller part has 2 elements, and the larger has 4, so now we look for the 4 - 2 - 1 = 1st element out of 4: Range of medians of five partitioned with pivot 1043 (depends on method): The smaller part has only one element, and we were looking for the first element, so we can return the small element 1038. Hence, the pivot is less than 3(n/10) elements outside the block, and greater than another 3(n/10) elements outside the block. The Median of medians approach is very popular in quicksort type partitioning algorithms to yield a fairly good pivot, such that it partitions the array uniformly. The best answers are voted up and rise to the top, Not the answer you're looking for? I think I cannot apply mater theorem to the expression above and wikipedia says I should use induction but I don't know How.. algorithm; sorting; The cause of your confusion about the median-of-medians algorithm is that, while median-of-medians returns an approximate result within 20% of the actual median, at some stages in the algorithm we also need to calculate exact medians. We have 8 possible results, the length of the new array that we recurse in has 4 possible value, which is n-1, n-2, n-3, and n/2. So when you're left with the medians of each group, as you were before, you should just trust that when you need to get the median by a recursive call, you end up with the true median. Hm, then the Wikipedia article is at best confusing and possibly incorrect. What if we select the median as our pivot? Klose A, Gortz S (2007) A branch-and-price algorithm for the capacitated facility location . This algorithm is, in my opinion, something that's way too complicated to actually trace through by hand. So that is the idea. The Median of medians approach is very popular in quicksort type partitioning algorithms to yield a fairly good pivot, such that it partitions the array uniformly. Why is the approximate median is in my case not greater than 30% of elements???? Solution 1. However I have some problem in calculating time complexity of median of medians algorithm . If our g=target is 5, we already find our that our target is not in LESS, and its not our pivot, so we already have (1 + |LESS|) items smaller than our target. Assume that items in our array are all distinct, which is for simplicity. We'll go into more detail below. Should I give a brutally honest feedback on course evaluations? If we have an array with length 8, whats the possible result of |LESS| and GREATER? I had thought (up until reading your post) that the approximate median is within 20% of the median of the INITIAL ARRAY (i.e., at the very beginning of the program), but it's actually within 20% of the median of the array you passed in, which is not the initial array when you recurse more than 1 level deep. Find centralized, trusted content and collaborate around the technologies you use most. T(n) equals n-1 (compare each item and our pivot) plus the expected T(i), which is our recursion part. Is there a text book that this algorithm is in? Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. I am finding it difficult to understand the logic. I believe some people call median of median the algorithm which selects an approximate median in linear time, and some people mean what you get when you combine that with quickselect, i.e. It appeared most sensible to us to use the same algorithm as in the reference. And it generally needs to be odd, unless you want to spend cycles splitting the difference between elements. Connect and share knowledge within a single location that is structured and easy to search. CGAC2022 Day 10: Help Santa sort presents! Partition the items in 2 bags and call the algorithm again on one of the 2 bags. Then we find the median of these three medians, which is 9. Here I am going to explain the third row: The right-hand side is the average of i from n/2 to n-1. MOSFET is getting very hot at high frequency PWM. $\begingroup$ I believe some people call median of median the algorithm which selects an approximate median in linear time, and some people mean what you get when you combine that with quickselect, i.e. When you were working through your analysis, you attempted to get the median of this set of values by, once again, splitting the input into blocks of size five and taking the median of each. 1. Input array (125 values, 25 groups of five): Medians of five partitioned with pivot 27 (depends on method): The smaller group has 8 elements, the larger group 16 elements. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. But this number is greater or equals than only 27 elements. Debian/Ubuntu - Is there a man page listing all the version codenames/numbers? Median of medians confusion -- the "approximate" median part. What's the \synctex primitive? To find this approximate median, we compute the median of each group of 5 elements, we gather these medians in a new set, and we recompute the medians until the obtained set have least than 5 elements. The size of the groups is always 5, hence you end with. We have four possible results of |LESS| and |GREATER| group. Asking for help, clarification, or responding to other answers. In the United States, must state courts follow rulings by federal courts of appeals? This algorithm is famously tricky to understand. A discussion of the Quick-Select algorithm. Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. The difference is that quickselect returns the actual median, not an approximate median. The median is a number which partitions the array into the upper and lower half. So instead of: T(n) <= T(n/3) + T(2n/3) + O(n) T(n) = O(nlogn) one gets: T(n) <= T(n/9) + T(7n/9) + O(n) T(n) = Theta(n) You really need to trust that, since each recursive call you're making works on a smaller array than what you started with, each recursive call will indeed do what it says to do. Quicksort with median of medians is considered practical Noriyuki Kurosawa March 9, 2022 The linear pivot selection algorithm, known as median-of-medians, makes the worst case complexity of quicksort be O(nlnn). Learn on the go with our new app. median of medians QuickSelect pivot. Then we compare each item in this array with our pivot and put these items in two different subarray. At what point in the prequels is it revealed that Palpatine is Darth Sidious? Just another question, how does this method guarantee that this number will be the median? TypeError: unsupported operand type(s) for *: 'IntVar' and 'float'. Love podcasts or audiobooks? Q J Belg Fr Ital Oper Res Soc 1(4):319-340 18. Each of these elements is a median of 5, making it less than 2 other elements and greater than 2 other elements outside the block. In this mini-lecture we go into how the algorithm works overall, and how we enhance the algorithm using the media. We can firstly choose a random element ai in the array, and call it our pivot. This all sounds fairly straightforward, but where it becomes complicated is that the function select() calls medianOfMedians() to get a first estimate of the median, which it then uses to calculate the exact median, so you get a two-way recursion where two functions call each other. The purpose of those groups is to strip away elements that are surely lower or grater than the median of medians. In the second step, the size of the median finding is reduced, which will take us T(n/5). Now if you have a number n, if n > 3, then it is bigger than at least half of the numbers above. In cluster analysis, the k-medians clustering algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in k-means clustering, is replaced by maximising the distance between cluster-medians. 2022/9/10 2 Divide and Conquer The most-well known algorithm design strategy. The above algorithm use randomness (randomly select pivot), now we look at how to perform O(n) comparisons without use randomness. This is a method of robust regression. Making statements based on opinion; back them up with references or personal experience. How did muzzle-loaded rifled artillery solve the problems of the hand-held rifle? Help us identify new roles for community members, Proposing a Community-Specific Closure Reason for non-English content. I think I should go fix that. Find centralized, trusted content and collaborate around the technologies you use most. (Quickselect is a randomized selection algorithm that chooses pivots at random. Therefore: c is a constant that greater than 0. Area of polygon. Its described in CLRS and on Wikipedia, and probably in many other lecture notes and slides. two elements) finding the n-th element will become trivial, and recursing further is no longer needed. LESS|,GREATER) = (0,3) or (1,2) or (2,1) or (3,0). How many transistors at minimum do you need to build a general-purpose computer? It is easily solvable in O(n log n) time via sorting and the Median of Me. The algorithm works by dividing a list into sublists and then determines the approximate median in each of the sublists. How would you create a standalone widget from this widget tree? You divide the whole set of numbers to groups of five, first five numbers will form the first group, next five will be the next group etc., last group will possibly have less than five elements. Like I said before, we are going to recurse on the larger part, which means, we recurse on 3, and then 2, then 2, and finally find our result in 3. Can somebody explain it a bit lucidly for me. Counterexamples to differentiation under integral sign, revisited. If this seems confusing, don't worry - you're in really good company. It is a divide and conquer algorithm in that, it returns a pivot that in the worst case will divide a list of unsorted elements into sub-problems of size 3n 10 3 n 10 and 7n 10 7 n 10 assuming we choose a sublist size of 5. a sorting network or insertion sort. It should work with any odd sized groups (greater than 1 ofc). Is it correct to say "The glue on the back of the sticker is dying down so I can not stick the sticker to the wall"? 2. And 27/125 = 21.6% < 30%!! Or am I operating under a false premise in thinking that Median of Medians finds an approximate median to the ORIGINAL array? After creating an array with the median-of-fives, you then used the median-of-medians function again on this array, which gives you an approximation of the median (27), but here you need the actual median (1038). MOSFET is getting very hot at high frequency PWM. I believe it still remains open now. The pseudocode in wikipedia fails to portray the inner workings of the selectIdx function call.. I've added comments to the code for explanation. If n < 3, then it is smaller than at least half of the numbers above. Find the median of medians takes us T(n/3), and in order to recurse on the larger side, we have: There are at least n/3 items below our pivot, and the above part is 2n/3. Lets look at a specific example, suppose our array is [1, 3, 5, 4, 10, 6], and 4 is randomly select as our pivot. Hello I am trying to understand how the median of medians algorithm works. That is, for each set of 5 numbers, you get their median. As you will see, 1038 is the exact median of the original 25 median-of-fives, and there are 62 smaller values in the original array of 125: which not only puts it in the 30~70% range, but means it is actually the exact median (note that this is a coincidence of this particular example). CGAC2022 Day 10: Help Santa sort presents! In this case, we get the median of the set. Is there a higher analog of "category with all same side inverses is a groupoid"? a linear-time algorithm to find the k'th element in an array (or in particular, find the median). Thats definitely perfect! Median of Medians Algorithm. Can someone clarify the difference between Quicksort and Randomized Quicksort? Ray. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. Connect and share knowledge within a single location that is structured and easy to search. Asking for help, clarification, or responding to other answers. Effect of coal and natural gas burning on particulate matter pollution. Since we are dividing the subarray in an recursive manner, I think that the Time complexity of the algorithm should be O (nlogn). If you make your groups of size 2k+1, then in each group there are at least k elements smaller or k elements bigger than the median of medians, which leaves you with . We were looking for the middle 13th element out of 25, so now we look for the 13 - 8 - 1 = 4th element out of 16: Range of medians of five partitioned with pivot 1058 (depends on method): The smaller group has 7 elements. To be more specific at the examples studied so far, is stated that there are 9 groups of 5 numbers each, for example aka 45 numbers, or 4 groups of 10 numbers aka 40 numbers at all. Why does my stock Samsung Galaxy phone/tablet lack some features compared to other Samsung Galaxy models? To get the median, you need to count how many number are greater than your pseudo-median, if a majority is greater, repeat the algorithm with the numbers greater than the pseudo-median, else repeat with the other numbers. Connect and share knowledge within a single location that is structured and easy to search. Select the middle elements (the medians). Someone showed the complexity analysis over at the Wikipedia page for this topic. Are defenders behind an arrow slit attackable? 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