performance - How to find the kth largest element in an unsorted array of length n in O(n)?


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I believe there's a way to find the kth largest element in an unsorted array of length n in O(n). Or perhaps it's "expected" O(n) or something. How can we do this?


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    This is called finding the k-th order statistic. There's a very simple randomized algorithm (called quickselect) taking O(n) average time, O(n^2) worst case time, and a pretty complicated non-randomized algorithm (called introselect) taking O(n) worst case time. There's some info on Wikipedia, but it's not very good.

    Everything you need is in these powerpoint slides. Just to extract the basic algorithm of the O(n) worst-case algorithm (introselect):

    Select(A,n,i):
        Divide input into ⌈n/5⌉ groups of size 5.
    
        /* Partition on median-of-medians */
        medians = array of each group’s median.
        pivot = Select(medians, ⌈n/5⌉, ⌈n/10⌉)
        Left Array L and Right Array G = partition(A, pivot)
    
        /* Find ith element in L, pivot, or G */
        k = |L| + 1
        If i = k, return pivot
        If i < k, return Select(L, k-1, i)
        If i > k, return Select(G, n-k, i-k)
    

    It's also very nicely detailed in the Introduction to Algorithms book by Cormen et al.


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    If you want a true O(n) algorithm, as opposed to O(kn) or something like that, then you should use quickselect (it's basically quicksort where you throw out the partition that you're not interested in). My prof has a great writeup, with the runtime analysis: (reference)

    The QuickSelect algorithm quickly finds the k-th smallest element of an unsorted array of n elements. It is a RandomizedAlgorithm, so we compute the worst-case expected running time.

    Here is the algorithm.

    QuickSelect(A, k)
      let r be chosen uniformly at random in the range 1 to length(A)
      let pivot = A[r]
      let A1, A2 be new arrays
      # split into a pile A1 of small elements and A2 of big elements
      for i = 1 to n
        if A[i] < pivot then
          append A[i] to A1
        else if A[i] > pivot then
          append A[i] to A2
        else
          # do nothing
      end for
      if k <= length(A1):
        # it's in the pile of small elements
        return QuickSelect(A1, k)
      else if k > length(A) - length(A2)
        # it's in the pile of big elements
        return QuickSelect(A2, k - (length(A) - length(A2))
      else
        # it's equal to the pivot
        return pivot
    

    What is the running time of this algorithm? If the adversary flips coins for us, we may find that the pivot is always the largest element and k is always 1, giving a running time of

    T(n) = Theta(n) + T(n-1) = Theta(n2)

    But if the choices are indeed random, the expected running time is given by

    T(n) <= Theta(n) + (1/n) ∑i=1 to nT(max(i, n-i-1))

    where we are making the not entirely reasonable assumption that the recursion always lands in the larger of A1 or A2.

    Let's guess that T(n) <= an for some a. Then we get

    T(n) 
     <= cn + (1/n) ∑i=1 to nT(max(i-1, n-i))
     = cn + (1/n) ∑i=1 to floor(n/2) T(n-i) + (1/n) ∑i=floor(n/2)+1 to n T(i)
     <= cn + 2 (1/n) ∑i=floor(n/2) to n T(i)
     <= cn + 2 (1/n) ∑i=floor(n/2) to n ai

    and now somehow we have to get the horrendous sum on the right of the plus sign to absorb the cn on the left. If we just bound it as 2(1/n) ∑i=n/2 to n an, we get roughly 2(1/n)(n/2)an = an. But this is too big - there's no room to squeeze in an extra cn. So let's expand the sum using the arithmetic series formula:

    i=floor(n/2) to n i  
     = ∑i=1 to n i - ∑i=1 to floor(n/2) i  
     = n(n+1)/2 - floor(n/2)(floor(n/2)+1)/2  
     <= n2/2 - (n/4)2/2  
     = (15/32)n2

    where we take advantage of n being "sufficiently large" to replace the ugly floor(n/2) factors with the much cleaner (and smaller) n/4. Now we can continue with

    cn + 2 (1/n) ∑i=floor(n/2) to n ai,
     <= cn + (2a/n) (15/32) n2
     = n (c + (15/16)a)
     <= an

    provided a > 16c.

    This gives T(n) = O(n). It's clearly Omega(n), so we get T(n) = Theta(n).


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    A quick Google on that ('kth largest element array') returned this: http://discuss.joelonsoftware.com/default.asp?interview.11.509587.17

    "Make one pass through tracking the three largest values so far." 
    

    (it was specifically for 3d largest)

    and this answer:

    Build a heap/priority queue.  O(n)
    Pop top element.  O(log n)
    Pop top element.  O(log n)
    Pop top element.  O(log n)
    
    Total = O(n) + 3 O(log n) = O(n)
    

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    You do like quicksort. Pick an element at random and shove everything either higher or lower. At this point you'll know which element you actually picked, and if it is the kth element you're done, otherwise you repeat with the bin (higher or lower), that the kth element would fall in. Statistically speaking, the time it takes to find the kth element grows with n, O(n).


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    A Programmer's Companion to Algorithm Analysis gives a version that is O(n), although the author states that the constant factor is so high, you'd probably prefer the naive sort-the-list-then-select method.

    I answered the letter of your question :)


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    The C++ standard library has almost exactly that function call nth_element, although it does modify your data. It has expected linear run-time, O(N), and it also does a partial sort.

    const int N = ...;
    double a[N];
    // ... 
    const int m = ...; // m < N
    nth_element (a, a + m, a + N);
    // a[m] contains the mth element in a
    

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    Although not very sure about O(n) complexity, but it will be sure to be between O(n) and nLog(n). Also sure to be closer to O(n) than nLog(n). Function is written in Java

    public int quickSelect(ArrayList<Integer>list, int nthSmallest){
        //Choose random number in range of 0 to array length
        Random random =  new Random();
        //This will give random number which is not greater than length - 1
        int pivotIndex = random.nextInt(list.size() - 1); 
    
        int pivot = list.get(pivotIndex);
    
        ArrayList<Integer> smallerNumberList = new ArrayList<Integer>();
        ArrayList<Integer> greaterNumberList = new ArrayList<Integer>();
    
        //Split list into two. 
        //Value smaller than pivot should go to smallerNumberList
        //Value greater than pivot should go to greaterNumberList
        //Do nothing for value which is equal to pivot
        for(int i=0; i<list.size(); i++){
            if(list.get(i)<pivot){
                smallerNumberList.add(list.get(i));
            }
            else if(list.get(i)>pivot){
                greaterNumberList.add(list.get(i));
            }
            else{
                //Do nothing
            }
        }
    
        //If smallerNumberList size is greater than nthSmallest value, nthSmallest number must be in this list 
        if(nthSmallest < smallerNumberList.size()){
            return quickSelect(smallerNumberList, nthSmallest);
        }
        //If nthSmallest is greater than [ list.size() - greaterNumberList.size() ], nthSmallest number must be in this list
        //The step is bit tricky. If confusing, please see the above loop once again for clarification.
        else if(nthSmallest > (list.size() - greaterNumberList.size())){
            //nthSmallest will have to be changed here. [ list.size() - greaterNumberList.size() ] elements are already in 
            //smallerNumberList
            nthSmallest = nthSmallest - (list.size() - greaterNumberList.size());
            return quickSelect(greaterNumberList,nthSmallest);
        }
        else{
            return pivot;
        }
    }
    

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    I implemented finding kth minimimum in n unsorted elements using dynamic programming, specifically tournament method. The execution time is O(n + klog(n)). The mechanism used is listed as one of methods on Wikipedia page about Selection Algorithm (as indicated in one of the posting above). You can read about the algorithm and also find code (java) on my blog page Finding Kth Minimum. In addition the logic can do partial ordering of the list - return first K min (or max) in O(klog(n)) time.

    Though the code provided result kth minimum, similar logic can be employed to find kth maximum in O(klog(n)), ignoring the pre-work done to create tournament tree.


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    You can do it in O(n + kn) = O(n) (for constant k) for time and O(k) for space, by keeping track of the k largest elements you've seen.

    For each element in the array you can scan the list of k largest and replace the smallest element with the new one if it is bigger.

    Warren's priority heap solution is neater though.


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    Sexy quickselect in Python

    def quickselect(arr, k):
        '''
         k = 1 returns first element in ascending order.
         can be easily modified to return first element in descending order
        '''
    
        r = random.randrange(0, len(arr))
    
        a1 = [i for i in arr if i < arr[r]] '''partition'''
        a2 = [i for i in arr if i > arr[r]]
    
        if k <= len(a1):
            return quickselect(a1, k)
        elif k > len(arr)-len(a2):
            return quickselect(a2, k - (len(arr) - len(a2)))
        else:
            return arr[r]
    

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    Find the median of the array in linear time, then use partition procedure exactly as in quicksort to divide the array in two parts, values to the left of the median lesser( < ) than than median and to the right greater than ( > ) median, that too can be done in lineat time, now, go to that part of the array where kth element lies, Now recurrence becomes: T(n) = T(n/2) + cn which gives me O (n) overal.


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    Below is the link to full implementation with quite an extensive explanation how the algorithm for finding Kth element in an unsorted algorithm works. Basic idea is to partition the array like in QuickSort. But in order to avoid extreme cases (e.g. when smallest element is chosen as pivot in every step, so that algorithm degenerates into O(n^2) running time), special pivot selection is applied, called median-of-medians algorithm. The whole solution runs in O(n) time in worst and in average case.

    Here is link to the full article (it is about finding Kth smallest element, but the principle is the same for finding Kth largest):

    Finding Kth Smallest Element in an Unsorted Array


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    As per this paper Finding the Kth largest item in a list of n items the following algorithm will take O(n) time in worst case.

    1. Divide the array in to n/5 lists of 5 elements each.
    2. Find the median in each sub array of 5 elements.
    3. Recursively find the median of all the medians, lets call it M
    4. Partition the array in to two sub array 1st sub-array contains the elements larger than M , lets say this sub-array is a1 , while other sub-array contains the elements smaller then M., lets call this sub-array a2.
    5. If k <= |a1|, return selection (a1,k).
    6. If k− 1 = |a1|, return M.
    7. If k> |a1| + 1, return selection(a2,k −a1 − 1).

    Analysis: As suggested in the original paper:

    We use the median to partition the list into two halves(the first half, if k <= n/2 , and the second half otherwise). This algorithm takes time cn at the first level of recursion for some constant c, cn/2 at the next level (since we recurse in a list of size n/2), cn/4 at the third level, and so on. The total time taken is cn + cn/2 + cn/4 + .... = 2cn = o(n).

    Why partition size is taken 5 and not 3?

    As mentioned in original paper:

    Dividing the list by 5 assures a worst-case split of 70 − 30. Atleast half of the medians greater than the median-of-medians, hence atleast half of the n/5 blocks have atleast 3 elements and this gives a 3n/10 split, which means the other partition is 7n/10 in worst case. That gives T(n) = T(n/5)+T(7n/10)+O(n). Since n/5+7n/10 < 1, the worst-case running time isO(n).

    Now I have tried to implement the above algorithm as:

    public static int findKthLargestUsingMedian(Integer[] array, int k) {
            // Step 1: Divide the list into n/5 lists of 5 element each.
            int noOfRequiredLists = (int) Math.ceil(array.length / 5.0);
            // Step 2: Find pivotal element aka median of medians.
            int medianOfMedian =  findMedianOfMedians(array, noOfRequiredLists);
            //Now we need two lists split using medianOfMedian as pivot. All elements in list listOne will be grater than medianOfMedian and listTwo will have elements lesser than medianOfMedian.
            List<Integer> listWithGreaterNumbers = new ArrayList<>(); // elements greater than medianOfMedian
            List<Integer> listWithSmallerNumbers = new ArrayList<>(); // elements less than medianOfMedian
            for (Integer element : array) {
                if (element < medianOfMedian) {
                    listWithSmallerNumbers.add(element);
                } else if (element > medianOfMedian) {
                    listWithGreaterNumbers.add(element);
                }
            }
            // Next step.
            if (k <= listWithGreaterNumbers.size()) return findKthLargestUsingMedian((Integer[]) listWithGreaterNumbers.toArray(new Integer[listWithGreaterNumbers.size()]), k);
            else if ((k - 1) == listWithGreaterNumbers.size()) return medianOfMedian;
            else if (k > (listWithGreaterNumbers.size() + 1)) return findKthLargestUsingMedian((Integer[]) listWithSmallerNumbers.toArray(new Integer[listWithSmallerNumbers.size()]), k-listWithGreaterNumbers.size()-1);
            return -1;
        }
    
        public static int findMedianOfMedians(Integer[] mainList, int noOfRequiredLists) {
            int[] medians = new int[noOfRequiredLists];
            for (int count = 0; count < noOfRequiredLists; count++) {
                int startOfPartialArray = 5 * count;
                int endOfPartialArray = startOfPartialArray + 5;
                Integer[] partialArray = Arrays.copyOfRange((Integer[]) mainList, startOfPartialArray, endOfPartialArray);
                // Step 2: Find median of each of these sublists.
                int medianIndex = partialArray.length/2;
                medians[count] = partialArray[medianIndex];
            }
            // Step 3: Find median of the medians.
            return medians[medians.length / 2];
        }
    

    Just for sake of completion, another algorithm makes use of Priority Queue and takes time O(nlogn).

    public static int findKthLargestUsingPriorityQueue(Integer[] nums, int k) {
            int p = 0;
            int numElements = nums.length;
            // create priority queue where all the elements of nums will be stored
            PriorityQueue<Integer> pq = new PriorityQueue<Integer>();
    
            // place all the elements of the array to this priority queue
            for (int n : nums) {
                pq.add(n);
            }
    
            // extract the kth largest element
            while (numElements - k + 1 > 0) {
                p = pq.poll();
                k++;
            }
    
            return p;
        }
    

    Both of these algorithms can be tested as:

    public static void main(String[] args) throws IOException {
            Integer[] numbers = new Integer[]{2, 3, 5, 4, 1, 12, 11, 13, 16, 7, 8, 6, 10, 9, 17, 15, 19, 20, 18, 23, 21, 22, 25, 24, 14};
            System.out.println(findKthLargestUsingMedian(numbers, 8));
            System.out.println(findKthLargestUsingPriorityQueue(numbers, 8));
        }
    

    As expected output is: 18 18


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    How about this kinda approach

    Maintain a buffer of length k and a tmp_max, getting tmp_max is O(k) and is done n times so something like O(kn)

    enter image description here

    Is it right or am i missing something ?

    Although it doesn't beat average case of quickselect and worst case of median statistics method but its pretty easy to understand and implement.


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    iterate through the list. if the current value is larger than the stored largest value, store it as the largest value and bump the 1-4 down and 5 drops off the list. If not,compare it to number 2 and do the same thing. Repeat, checking it against all 5 stored values. this should do it in O(n)


  • Clara Lee
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    i would like to suggest one answer

    if we take the first k elements and sort them into a linked list of k values

    now for every other value even for the worst case if we do insertion sort for rest n-k values even in the worst case number of comparisons will be k*(n-k) and for prev k values to be sorted let it be k*(k-1) so it comes out to be (nk-k) which is o(n)

    cheers


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    Explanation of the median - of - medians algorithm to find the k-th largest integer out of n can be found here: http://cs.indstate.edu/~spitla/presentation.pdf

    Implementation in c++ is below:

    #include <iostream>
    #include <vector>
    #include <algorithm>
    using namespace std;
    
    int findMedian(vector<int> vec){
    //    Find median of a vector
        int median;
        size_t size = vec.size();
        median = vec[(size/2)];
        return median;
    }
    
    int findMedianOfMedians(vector<vector<int> > values){
        vector<int> medians;
    
        for (int i = 0; i < values.size(); i++) {
            int m = findMedian(values[i]);
            medians.push_back(m);
        }
    
        return findMedian(medians);
    }
    
    void selectionByMedianOfMedians(const vector<int> values, int k){
    //    Divide the list into n/5 lists of 5 elements each
        vector<vector<int> > vec2D;
    
        int count = 0;
        while (count != values.size()) {
            int countRow = 0;
            vector<int> row;
    
            while ((countRow < 5) && (count < values.size())) {
                row.push_back(values[count]);
                count++;
                countRow++;
            }
            vec2D.push_back(row);
        }
    
        cout<<endl<<endl<<"Printing 2D vector : "<<endl;
        for (int i = 0; i < vec2D.size(); i++) {
            for (int j = 0; j < vec2D[i].size(); j++) {
                cout<<vec2D[i][j]<<" ";
            }
            cout<<endl;
        }
        cout<<endl;
    
    //    Calculating a new pivot for making splits
        int m = findMedianOfMedians(vec2D);
        cout<<"Median of medians is : "<<m<<endl;
    
    //    Partition the list into unique elements larger than 'm' (call this sublist L1) and
    //    those smaller them 'm' (call this sublist L2)
        vector<int> L1, L2;
    
        for (int i = 0; i < vec2D.size(); i++) {
            for (int j = 0; j < vec2D[i].size(); j++) {
                if (vec2D[i][j] > m) {
                    L1.push_back(vec2D[i][j]);
                }else if (vec2D[i][j] < m){
                    L2.push_back(vec2D[i][j]);
                }
            }
        }
    
    //    Checking the splits as per the new pivot 'm'
        cout<<endl<<"Printing L1 : "<<endl;
        for (int i = 0; i < L1.size(); i++) {
            cout<<L1[i]<<" ";
        }
    
        cout<<endl<<endl<<"Printing L2 : "<<endl;
        for (int i = 0; i < L2.size(); i++) {
            cout<<L2[i]<<" ";
        }
    
    //    Recursive calls
        if ((k - 1) == L1.size()) {
            cout<<endl<<endl<<"Answer :"<<m;
        }else if (k <= L1.size()) {
            return selectionByMedianOfMedians(L1, k);
        }else if (k > (L1.size() + 1)){
            return selectionByMedianOfMedians(L2, k-((int)L1.size())-1);
        }
    
    }
    
    int main()
    {
        int values[] = {2, 3, 5, 4, 1, 12, 11, 13, 16, 7, 8, 6, 10, 9, 17, 15, 19, 20, 18, 23, 21, 22, 25, 24, 14};
    
        vector<int> vec(values, values + 25);
    
        cout<<"The given array is : "<<endl;
        for (int i = 0; i < vec.size(); i++) {
            cout<<vec[i]<<" ";
        }
    
        selectionByMedianOfMedians(vec, 8);
    
        return 0;
    }
    

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    There is also Wirth's selection algorithm, which has a simpler implementation than QuickSelect. Wirth's selection algorithm is slower than QuickSelect, but with some improvements it becomes faster.

    In more detail. Using Vladimir Zabrodsky's MODIFIND optimization and the median-of-3 pivot selection and paying some attention to the final steps of the partitioning part of the algorithm, i've came up with the following algorithm (imaginably named "LefSelect"):

    #define F_SWAP(a,b) { float temp=(a);(a)=(b);(b)=temp; }
    
    # Note: The code needs more than 2 elements to work
    float lefselect(float a[], const int n, const int k) {
        int l=0, m = n-1, i=l, j=m;
        float x;
    
        while (l<m) {
            if( a[k] < a[i] ) F_SWAP(a[i],a[k]);
            if( a[j] < a[i] ) F_SWAP(a[i],a[j]);
            if( a[j] < a[k] ) F_SWAP(a[k],a[j]);
    
            x=a[k];
            while (j>k & i<k) {
                do i++; while (a[i]<x);
                do j--; while (a[j]>x);
    
                F_SWAP(a[i],a[j]);
            }
            i++; j--;
    
            if (j<k) {
                while (a[i]<x) i++;
                l=i; j=m;
            }
            if (k<i) {
                while (x<a[j]) j--;
                m=j; i=l;
            }
        }
        return a[k];
    }
    

    In benchmarks that i did here, LefSelect is 20-30% faster than QuickSelect.


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    Haskell Solution:

    kthElem index list = sort list !! index
    
    withShape ~[]     []     = []
    withShape ~(x:xs) (y:ys) = x : withShape xs ys
    
    sort []     = []
    sort (x:xs) = (sort ls `withShape` ls) ++ [x] ++ (sort rs `withShape` rs)
      where
       ls = filter (<  x)
       rs = filter (>= x)
    

    This implements the median of median solutions by using the withShape method to discover the size of a partition without actually computing it.


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    Here is a C++ implementation of Randomized QuickSelect. The idea is to randomly pick a pivot element. To implement randomized partition, we use a random function, rand() to generate index between l and r, swap the element at randomly generated index with the last element, and finally call the standard partition process which uses last element as pivot.

    #include<iostream>
    #include<climits>
    #include<cstdlib>
    using namespace std;
    
    int randomPartition(int arr[], int l, int r);
    
    // This function returns k'th smallest element in arr[l..r] using
    // QuickSort based method.  ASSUMPTION: ALL ELEMENTS IN ARR[] ARE DISTINCT
    int kthSmallest(int arr[], int l, int r, int k)
    {
        // If k is smaller than number of elements in array
        if (k > 0 && k <= r - l + 1)
        {
            // Partition the array around a random element and
            // get position of pivot element in sorted array
            int pos = randomPartition(arr, l, r);
    
            // If position is same as k
            if (pos-l == k-1)
                return arr[pos];
            if (pos-l > k-1)  // If position is more, recur for left subarray
                return kthSmallest(arr, l, pos-1, k);
    
            // Else recur for right subarray
            return kthSmallest(arr, pos+1, r, k-pos+l-1);
        }
    
        // If k is more than number of elements in array
        return INT_MAX;
    }
    
    void swap(int *a, int *b)
    {
        int temp = *a;
        *a = *b;
        *b = temp;
    }
    
    // Standard partition process of QuickSort().  It considers the last
    // element as pivot and moves all smaller element to left of it and
    // greater elements to right. This function is used by randomPartition()
    int partition(int arr[], int l, int r)
    {
        int x = arr[r], i = l;
        for (int j = l; j <= r - 1; j++)
        {
            if (arr[j] <= x) //arr[i] is bigger than arr[j] so swap them
            {
                swap(&arr[i], &arr[j]);
                i++;
            }
        }
        swap(&arr[i], &arr[r]); // swap the pivot
        return i;
    }
    
    // Picks a random pivot element between l and r and partitions
    // arr[l..r] around the randomly picked element using partition()
    int randomPartition(int arr[], int l, int r)
    {
        int n = r-l+1;
        int pivot = rand() % n;
        swap(&arr[l + pivot], &arr[r]);
        return partition(arr, l, r);
    }
    
    // Driver program to test above methods
    int main()
    {
        int arr[] = {12, 3, 5, 7, 4, 19, 26};
        int n = sizeof(arr)/sizeof(arr[0]), k = 3;
        cout << "K'th smallest element is " << kthSmallest(arr, 0, n-1, k);
        return 0;
    }
    

    The worst case time complexity of the above solution is still O(n2).In worst case, the randomized function may always pick a corner element. The expected time complexity of above randomized QuickSelect is Θ(n)


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    1. Have Priority queue created.
    2. Insert all the elements into heap.
    3. Call poll() k times.

      public static int getKthLargestElements(int[] arr)
      {
          PriorityQueue<Integer> pq =  new PriorityQueue<>((x , y) -> (y-x));
          //insert all the elements into heap
          for(int ele : arr)
             pq.offer(ele);
          // call poll() k times
          int i=0;
          while(i&lt;k)
           {
             int result = pq.poll();
           } 
         return result;        
      }
      

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    This is an implementation in Javascript.

    If you release the constraint that you cannot modify the array, you can prevent the use of extra memory using two indexes to identify the "current partition" (in classic quicksort style - http://www.nczonline.net/blog/2012/11/27/computer-science-in-javascript-quicksort/).

    function kthMax(a, k){
        var size = a.length;
    
        var pivot = a[ parseInt(Math.random()*size) ]; //Another choice could have been (size / 2) 
    
        //Create an array with all element lower than the pivot and an array with all element higher than the pivot
        var i, lowerArray = [], upperArray = [];
        for (i = 0; i  < size; i++){
            var current = a[i];
    
            if (current < pivot) {
                lowerArray.push(current);
            } else if (current > pivot) {
                upperArray.push(current);
            }
        }
    
        //Which one should I continue with?
        if(k <= upperArray.length) {
            //Upper
            return kthMax(upperArray, k);
        } else {
            var newK = k - (size - lowerArray.length);
    
            if (newK > 0) {
                ///Lower
                return kthMax(lowerArray, newK);
            } else {
                //None ... it's the current pivot!
                return pivot;
            }   
        }
    }  
    

    If you want to test how it perform, you can use this variation:

        function kthMax (a, k, logging) {
             var comparisonCount = 0; //Number of comparison that the algorithm uses
             var memoryCount = 0;     //Number of integers in memory that the algorithm uses
             var _log = logging;
    
             if(k < 0 || k >= a.length) {
                if (_log) console.log ("k is out of range"); 
                return false;
             }      
    
             function _kthmax(a, k){
                 var size = a.length;
                 var pivot = a[parseInt(Math.random()*size)];
                 if(_log) console.log("Inputs:", a,  "size="+size, "k="+k, "pivot="+pivot);
    
                 // This should never happen. Just a nice check in this exercise
                 // if you are playing with the code to avoid never ending recursion            
                 if(typeof pivot === "undefined") {
                     if (_log) console.log ("Ops..."); 
                     return false;
                 }
    
                 var i, lowerArray = [], upperArray = [];
                 for (i = 0; i  < size; i++){
                     var current = a[i];
                     if (current < pivot) {
                         comparisonCount += 1;
                         memoryCount++;
                         lowerArray.push(current);
                     } else if (current > pivot) {
                         comparisonCount += 2;
                         memoryCount++;
                         upperArray.push(current);
                     }
                 }
                 if(_log) console.log("Pivoting:",lowerArray, "*"+pivot+"*", upperArray);
    
                 if(k <= upperArray.length) {
                     comparisonCount += 1;
                     return _kthmax(upperArray, k);
                 } else if (k > size - lowerArray.length) {
                     comparisonCount += 2;
                     return _kthmax(lowerArray, k - (size - lowerArray.length));
                 } else {
                     comparisonCount += 2;
                     return pivot;
                 }
         /* 
          * BTW, this is the logic for kthMin if we want to implement that... ;-)
          * 
    
                 if(k <= lowerArray.length) {
                     return kthMin(lowerArray, k);
                 } else if (k > size - upperArray.length) {
                     return kthMin(upperArray, k - (size - upperArray.length));
                 } else 
                     return pivot;
         */            
             }
    
             var result = _kthmax(a, k);
             return {result: result, iterations: comparisonCount, memory: memoryCount};
         }
    

    The rest of the code is just to create some playground:

        function getRandomArray (n){
            var ar = [];
            for (var i = 0, l = n; i < l; i++) {
                ar.push(Math.round(Math.random() * l))
            }
    
            return ar;
        }
    
        //Create a random array of 50 numbers
        var ar = getRandomArray (50);   
    

    Now, run you tests a few time. Because of the Math.random() it will produce every time different results:

        kthMax(ar, 2, true);
        kthMax(ar, 2);
        kthMax(ar, 2);
        kthMax(ar, 2);
        kthMax(ar, 2);
        kthMax(ar, 2);
        kthMax(ar, 34, true);
        kthMax(ar, 34);
        kthMax(ar, 34);
        kthMax(ar, 34);
        kthMax(ar, 34);
        kthMax(ar, 34);
    

    If you test it a few times you can see even empirically that the number of iterations is, on average, O(n) ~= constant * n and the value of k does not affect the algorithm.


  • Translate

    I came up with this algorithm and seems to be O(n):

    Let's say k=3 and we want to find the 3rd largest item in the array. I would create three variables and compare each item of the array with the minimum of these three variables. If array item is greater than our minimum, we would replace the min variable with the item value. We continue the same thing until end of the array. The minimum of our three variables is the 3rd largest item in the array.

    define variables a=0, b=0, c=0
    iterate through the array items
        find minimum a,b,c
        if item > min then replace the min variable with item value
        continue until end of array
    the minimum of a,b,c is our answer
    

    And, to find Kth largest item we need K variables.

    Example: (k=3)

    [1,2,4,1,7,3,9,5,6,2,9,8]
    
    Final variable values:
    
    a=7 (answer)
    b=8
    c=9
    

    Can someone please review this and let me know what I am missing?


  • Translate

    Here is the implementation of the algorithm eladv suggested(I also put here the implementation with random pivot):

    public class Median {
    
        public static void main(String[] s) {
    
            int[] test = {4,18,20,3,7,13,5,8,2,1,15,17,25,30,16};
            System.out.println(selectK(test,8));
    
            /*
            int n = 100000000;
            int[] test = new int[n];
            for(int i=0; i<test.length; i++)
                test[i] = (int)(Math.random()*test.length);
    
            long start = System.currentTimeMillis();
            random_selectK(test, test.length/2);
            long end = System.currentTimeMillis();
            System.out.println(end - start);
            */
        }
    
        public static int random_selectK(int[] a, int k) {
            if(a.length <= 1)
                return a[0];
    
            int r = (int)(Math.random() * a.length);
            int p = a[r];
    
            int small = 0, equal = 0, big = 0;
            for(int i=0; i<a.length; i++) {
                if(a[i] < p) small++;
                else if(a[i] == p) equal++;
                else if(a[i] > p) big++;
            }
    
            if(k <= small) {
                int[] temp = new int[small];
                for(int i=0, j=0; i<a.length; i++)
                    if(a[i] < p)
                        temp[j++] = a[i];
                return random_selectK(temp, k);
            }
    
            else if (k <= small+equal)
                return p;
    
            else {
                int[] temp = new int[big];
                for(int i=0, j=0; i<a.length; i++)
                    if(a[i] > p)
                        temp[j++] = a[i];
                return random_selectK(temp,k-small-equal);
            }
        }
    
        public static int selectK(int[] a, int k) {
            if(a.length <= 5) {
                Arrays.sort(a);
                return a[k-1];
            }
    
            int p = median_of_medians(a);
    
            int small = 0, equal = 0, big = 0;
            for(int i=0; i<a.length; i++) {
                if(a[i] < p) small++;
                else if(a[i] == p) equal++;
                else if(a[i] > p) big++;
            }
    
            if(k <= small) {
                int[] temp = new int[small];
                for(int i=0, j=0; i<a.length; i++)
                    if(a[i] < p)
                        temp[j++] = a[i];
                return selectK(temp, k);
            }
    
            else if (k <= small+equal)
                return p;
    
            else {
                int[] temp = new int[big];
                for(int i=0, j=0; i<a.length; i++)
                    if(a[i] > p)
                        temp[j++] = a[i];
                return selectK(temp,k-small-equal);
            }
        }
    
        private static int median_of_medians(int[] a) {
            int[] b = new int[a.length/5];
            int[] temp = new int[5];
            for(int i=0; i<b.length; i++) {
                for(int j=0; j<5; j++)
                    temp[j] = a[5*i + j];
                Arrays.sort(temp);
                b[i] = temp[2];
            }
    
            return selectK(b, b.length/2 + 1);
        }
    }
    

  • Translate

    it is similar to the quickSort strategy, where we pick an arbitrary pivot, and bring the smaller elements to its left, and the larger to the right

        public static int kthElInUnsortedList(List<int> list, int k)
        {
            if (list.Count == 1)
                return list[0];
    
            List<int> left = new List<int>();
            List<int> right = new List<int>();
    
            int pivotIndex = list.Count / 2;
            int pivot = list[pivotIndex]; //arbitrary
    
            for (int i = 0; i < list.Count && i != pivotIndex; i++)
            {
                int currentEl = list[i];
                if (currentEl < pivot)
                    left.Add(currentEl);
                else
                    right.Add(currentEl);
            }
    
            if (k == left.Count + 1)
                return pivot;
    
            if (left.Count < k)
                return kthElInUnsortedList(right, k - left.Count - 1);
            else
                return kthElInUnsortedList(left, k);
        }
    


  • Translate

    You can find the kth smallest element in O(n) time and constant space. If we consider the array is only for integers.

    The approach is to do a binary search on the range of Array values. If we have a min_value and a max_value both in integer range, we can do a binary search on that range. We can write a comparator function which will tell us if any value is the kth-smallest or smaller than kth-smallest or bigger than kth-smallest. Do the binary search until you reach the kth-smallest number

    Here is the code for that

    class Solution:

    def _iskthsmallest(self, A, val, k):
        less_count, equal_count = 0, 0
        for i in range(len(A)):
            if A[i] == val: equal_count += 1
            if A[i] < val: less_count += 1
    
        if less_count >= k: return 1
        if less_count + equal_count < k: return -1
        return 0
    
    def kthsmallest_binary(self, A, min_val, max_val, k):
        if min_val == max_val:
            return min_val
        mid = (min_val + max_val)/2
        iskthsmallest = self._iskthsmallest(A, mid, k)
        if iskthsmallest == 0: return mid
        if iskthsmallest > 0: return self.kthsmallest_binary(A, min_val, mid, k)
        return self.kthsmallest_binary(A, mid+1, max_val, k)
    
    # @param A : tuple of integers
    # @param B : integer
    # @return an integer
    def kthsmallest(self, A, k):
        if not A: return 0
        if k > len(A): return 0
        min_val, max_val = min(A), max(A)
        return self.kthsmallest_binary(A, min_val, max_val, k)
    

  • Translate

    There is also one algorithm, that outperforms quickselect algorithm. It's called Floyd-Rivets (FR) algorithm.

    Original article: https://doi.org/10.1145/360680.360694

    Downloadable version: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.309.7108&rep=rep1&type=pdf

    Wikipedia article https://en.wikipedia.org/wiki/Floyd%E2%80%93Rivest_algorithm

    I tried to implement quickselect and FR algorithm in C++. Also I compared them to the standard C++ library implementations std::nth_element (which is basically introselect hybrid of quickselect and heapselect). The result was quickselect and nth_element ran comparably on average, but FR algorithm ran approx. twice as fast compared to them.

    Sample code that I used for FR algorithm:

    template <typename T>
    T FRselect(std::vector<T>& data, const size_t& n)
    {
        if (n == 0)
            return *(std::min_element(data.begin(), data.end()));
        else if (n == data.size() - 1)
            return *(std::max_element(data.begin(), data.end()));
        else
            return _FRselect(data, 0, data.size() - 1, n);
    }
    
    template <typename T>
    T _FRselect(std::vector<T>& data, const size_t& left, const size_t& right, const size_t& n)
    {
        size_t leftIdx = left;
        size_t rightIdx = right;
    
        while (rightIdx > leftIdx)
        {
            if (rightIdx - leftIdx > 600)
            {
                size_t range = rightIdx - leftIdx + 1;
                long long i = n - (long long)leftIdx + 1;
                long long z = log(range);
                long long s = 0.5 * exp(2 * z / 3);
                long long sd = 0.5 * sqrt(z * s * (range - s) / range) * sgn(i - (long long)range / 2);
    
                size_t newLeft = fmax(leftIdx, n - i * s / range + sd);
                size_t newRight = fmin(rightIdx, n + (range - i) * s / range + sd);
    
                _FRselect(data, newLeft, newRight, n);
            }
            T t = data[n];
            size_t i = leftIdx;
            size_t j = rightIdx;
            // arrange pivot and right index
            std::swap(data[leftIdx], data[n]);
            if (data[rightIdx] > t)
                std::swap(data[rightIdx], data[leftIdx]);
    
            while (i < j)
            {
                std::swap(data[i], data[j]);
                ++i; --j;
                while (data[i] < t) ++i;
                while (data[j] > t) --j;
            }
    
            if (data[leftIdx] == t)
                std::swap(data[leftIdx], data[j]);
            else
            {
                ++j;
                std::swap(data[j], data[rightIdx]);
            }
            // adjust left and right towards the boundaries of the subset
            // containing the (k - left + 1)th smallest element
            if (j <= n)
                leftIdx = j + 1;
            if (n <= j)
                rightIdx = j - 1;
        }
    
        return data[leftIdx];
    }
    
    template <typename T>
    int sgn(T val) {
        return (T(0) < val) - (val < T(0));
    }
    

  • Translate

    What I would do is this:

    initialize empty doubly linked list l
    for each element e in array
        if e larger than head(l)
            make e the new head of l
            if size(l) > k
                remove last element from l
    
    the last element of l should now be the kth largest element
    

    You can simply store pointers to the first and last element in the linked list. They only change when updates to the list are made.

    Update:

    initialize empty sorted tree l
    for each element e in array
        if e between head(l) and tail(l)
            insert e into l // O(log k)
            if size(l) > k
                remove last element from l
    
    the last element of l should now be the kth largest element
    

  • Translate

    First we can build a BST from unsorted array which takes O(n) time and from the BST we can find the kth smallest element in O(log(n)) which over all counts to an order of O(n).