Why does QuickSort use O(log(n)) extra space?

问题

I have implemented the below quicksort algorithm. Online I've read that it has a space requirement of O(log(n)). Why is this the case? I'm not creating any extra data structures.

Is it because my recursion will use some extra space on the stack? If this is the case, is it possible to do it with less memory by not having it be recursive (instead making it iterative)?

private static void quickSort (int[] array, int left, int right) {
    int index = partition(array, left, right);

    //Sort left half
    if (left < index - 1)
        quickSort(array, left, index - 1);

    //Sort right half
    if (index < right)
        quickSort(array, index , right);
}

private static int partition (int array[], int left, int right) {
    int pivot = array[(left + right) / 2]; //Pick pivot point
    while (left <= right) {
        //Find element on left that should be on right
        while (array[left] < pivot)
            left++;

        //Find element on right that should be on left
        while (array[right] > pivot)
            right--;

        //Swap elements and move left and right indices
        if (left <= right) {
            int temp = array[left];
            array[left] = array[right];
            array[right] = temp;
            left++;
            right--;
        }
    }
    return left;
}

回答1:

Correct, the extra space are the log(n) stack frames. From the Wikipedia article of Quicksort:

There is a more complex version which uses an in-place partition algorithm and can achieve the complete sort using O(log n) space (not counting the input) on average (for the call stack).

While you could implement quicksort iteratively (i.e., using a loop instead of recursion), you would then need to maintain an auxiliary stack, because Quicksort has two recursive calls and not just one.

Finally, as other answers have pointed out, O(log(n)) is for pretty much all practical applications very, very small. Every constant factor, like the overhead of your data structure, will have a greater impact on memory usage.

回答2:

To get rid of the recursive call you would have to use a stack in your code, and it would still occupy log(n) space.

回答3:

If you read further in the Wikipedia article, you will find a more thorough discussion of space complexity. In particular, they write:

Quicksort with in-place and unstable partitioning uses only constant additional space before making any recursive call. Quicksort must store a constant amount of information for each nested recursive call. Since the best case makes at most O(log n) nested recursive calls, it uses O(log n) space. However, without Sedgewick's trick to limit the recursive calls, in the worst case quicksort could make O(n) nested recursive calls and need O(n) auxiliary space.

Practically speaking, O(log n) memory is nothing. For instance, if you were to sort 1 billion ints, storing them would require 4 GB, but the stack would only require about 30 stack frames, at something like 40 bytes, so about 1200 Bytes in total.

回答4:

Yes, it is because of the stack frames, and yes, it may be possible to convert it to an iterative algorithm by doing something very clever (although nothing is immediately coming to me). But why? O(log(n)) space is almost nothing. For reference, even if you have an array of the maximum size allowed by Java, thats 2^31 elements, which is about 8 GB. Quicksort would require 31 stack frames. Ballpark, maybe 100 bytes per frame? So 3 KB total, which is nothing compared to the memory for the actual array.

In reality, almost any time something is O(log(n)), it's pretty much the same as constant.

回答5:

Sorry for reviving this old question, but I just found an entirely different (but slightly silly) answer to your question, on planetmath.org:

Any sorting algorithm that operates on a contiguous array requires O⁢(log ⁡n) extra space, since this is the number of bite [sic] required to represent an index into the array.

来源：https://stackoverflow.com/questions/12573330/why-does-quicksort-use-ologn-extra-space