Question

I learnt about quicksort and how it can be implemented in both Recursive and Iterative method.

In Iterative method:

0. Push the range (0...n) into the stack
1. Partition the given array with a pivot
2. Pop the top element.
3. Push the partitions (index range) onto a stack if the range has more than one element
4. Do the above 3 steps, till the stack is empty

And the recursive version is the normal one defined in wiki.

I learnt that recursive algorithms are always slower than their iterative counterpart.

So, Which method is preferred in terms of time complexity (memory is not a concern)?
Which one is fast enough to use in Programming contest?
Is C++ STL sort() using a recursive approach?

Answer 1

In terms of (asymptotic) time complexity – they are both the same.

"Recursive is slower then iterative" – the rational behind this statement is because of the overhead of the recursive stack (saving and restoring the environment between calls).

However -these are constant number of ops, while not changing the number of "iterations".

Both recursive and iterative quicksort are O(nlogn) average case and O(n^2) worst case.

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EDIT:

just for the fun of it I ran a benchmark with the (java) code attached to the post , and then I ran wilcoxon statistic test, to check what is the probability that the running times are indeed distinct

The results may be conclusive (P_VALUE=2.6e-34, https://en.wikipedia.org/wiki/P-value. Remember that the P_VALUE is P(T >= t | H) where T is the test statistic and H is the null hypothesis). But the answer is not what you expected.

The average of the iterative solution was 408.86 ms while of recursive was 236.81 ms

(Note – I used Integer and not int as argument to recursiveQsort() – otherwise the recursive would have achieved much better, because it doesn’t have to box a lot of integers, which is also time consuming – I did it because the iterative solution has no choice but doing so.

Thus – your assumption is not true, the recursive solution is faster (for my machine and java for the very least) than the iterative one with P_VALUE=2.6e-34.

public static void recursiveQsort(int[] arr,Integer start, Integer end) { 
    if (end - start < 2) return; //stop clause
    int p = start + ((end-start)/2);
    p = partition(arr,p,start,end);
    recursiveQsort(arr, start, p);
    recursiveQsort(arr, p+1, end);

}

public static void iterativeQsort(int[] arr) { 
    Stack<Integer> stack = new Stack<Integer>();
    stack.push(0);
    stack.push(arr.length);
    while (!stack.isEmpty()) {
        int end = stack.pop();
        int start = stack.pop();
        if (end - start < 2) continue;
        int p = start + ((end-start)/2);
        p = partition(arr,p,start,end);

        stack.push(p+1);
        stack.push(end);

        stack.push(start);
        stack.push(p);

    }
}

private static int partition(int[] arr, int p, int start, int end) {
    int l = start;
    int h = end - 2;
    int piv = arr[p];
    swap(arr,p,end-1);

    while (l < h) {
        if (arr[l] < piv) {
            l++;
        } else if (arr[h] >= piv) { 
            h--;
        } else { 
            swap(arr,l,h);
        }
    }
    int idx = h;
    if (arr[h] < piv) idx++;
    swap(arr,end-1,idx);
    return idx;
}
private static void swap(int[] arr, int i, int j) { 
    int temp = arr[i];
    arr[i] = arr[j];
    arr[j] = temp;
}

public static void main(String... args) throws Exception {
    Random r = new Random(1);
    int SIZE = 1000000;
    int N = 100;
    int[] arr = new int[SIZE];
    int[] millisRecursive = new int[N];
    int[] millisIterative = new int[N];
    for (int t = 0; t < N; t++) { 
        for (int i = 0; i < SIZE; i++) { 
            arr[i] = r.nextInt(SIZE);
        }
        int[] tempArr = Arrays.copyOf(arr, arr.length);
        
        long start = System.currentTimeMillis();
        iterativeQsort(tempArr);
        millisIterative[t] = (int)(System.currentTimeMillis()-start);
        
        tempArr = Arrays.copyOf(arr, arr.length);
        
        start = System.currentTimeMillis();
        recursvieQsort(tempArr,0,arr.length);
        millisRecursive[t] = (int)(System.currentTimeMillis()-start);
    }
    int sum = 0;
    for (int x : millisRecursive) {
        System.out.println(x);
        sum += x;
    }
    System.out.println("end of recursive. AVG = " + ((double)sum)/millisRecursive.length);
    sum = 0;
    for (int x : millisIterative) {
        System.out.println(x);
        sum += x;
    }
    System.out.println("end of iterative. AVG = " + ((double)sum)/millisIterative.length);
}