Q:

Given an array, find out the next smaller element for each element

Given an array find the next smaller element in array for each element without changing the original order of the elements.
For example, suppose the given array is 4,2,1,5,3. 
The resultant array would be 2,1,-1,3,-1.
I was asked this question in an interview, but i couldn't think of a solution better than the trivial O(n^2) solution.
Any approach that I could think of, i.e. making a binary search tree, or sorting the array, will distort the original order of the elements and hence lead to a  wrong result.
Any help would be highly appreciated.

A:

O(N) Algorithm

Initialize output array to all -1s.
Create an empty stack of indexes of items we have visited in the input array but don't yet know the answer for in the output array.
Iterate over each element in the input array:

Is it smaller than the item indexed by the top of the stack?

Yes. It is the first such element to be so. Fill in the corresponding element in our output array, remove the item from the stack, and try again until the stack is empty or the answer is no.
No. Continue to 3.2.

Add this index to the stack. Continue iteration from 3.

Python implementation
def find_next_smaller_elements(xs):
    ys=[-1 for x in xs]
    stack=[]
    for i,x in enumerate(xs):
        while len(stack)>0 and x<xs[stack[-1]]:
           ys[stack.pop()]=x
        stack.append(i)
    return ys

>>> find_next_smaller_elements([4,2,1,5,3])
[2, 1, -1, 3, -1]
>>> find_next_smaller_elements([1,2,3,4,5])
[-1, -1, -1, -1, -1]
>>> find_next_smaller_elements([5,4,3,2,1])
[4, 3, 2, 1, -1]
>>> find_next_smaller_elements([1,3,5,4,2])
[-1, 2, 4, 2, -1]
>>> find_next_smaller_elements([6,4,2])
[4, 2, -1]

Explanation
How it works
This works because whenever we add an item to the stack, we know its value is greater or equal to every element in the stack already. When we visit an element in the array, we know that if it's lower than any item in the stack, it must be lower than the last item in the stack, because the last item must be the largest. So we don't need to do any kind of search on the stack, we can just consider the last item.
Note: You can skip the initialization step so long as you add a final step to empty the stack and use each remaining index to set the corresponding output array element to -1. It's just easier in Python to initialize it to -1s when creating it.
Time complexity
This is O(N). The main loop clearly visits each index once. Each index is added to the stack exactly once and removed at most once.
Solving as an interview question
This kind of question can be pretty intimidating in an interview, but I'd like to point out that (hopefully) an interviewer isn't going to expect the solution to spring from your mind fully-formed. Talk them through your thought process. Mine went something like this:

Is there some relationship between the positions of numbers and their next smaller number in the array? Does knowing some of them constrain what the others might possibly be?
If I were in front of a whiteboard I would probably sketch out the example array and draw lines between the elements. I might also draw them as a 2D bar graph - horizontal axis being position in input array and vertical axis being value.
I had a hunch this would show a pattern, but no paper to hand. I think the diagram would make it obvious. Thinking about it carefully, I could see that the lines would not overlap arbitrarily, but would only nest.
Around this point, it occurred to me that this is incredibly similar to the algorithm Python uses internally to transform indentation into INDENT and DEDENT virtual tokens, which I'd read about before. See "How does the compiler parse the indentation?" on this page: http://www.secnetix.de/olli/Python/block_indentation.hawk However, it wasn't until I actually worked out an algorithm that I followed up on this thought and determined that it was in fact the same, so I don't think it helped too much. Still, if you can see a similarity to some other problem you know, it's probably a good idea to mention it, and say how it's similar and how it's different.
From here the general shape of the stack-based algorithm became apparent, but I still needed to think about it a bit more to be sure it would work okay for those elements that have no subsequent smaller element.

Even if you don't come up with a working algorithm, try to let your interviewer see what you're thinking about. Often it is the thought process more than the answer that they're interested in. For a tough problem, failing to find the best solution but showing insight into the problem can be better than knowing a canned answer but not being able to give it much analysis.

A:

Start making a BST, starting  from the array end. For each value 'v' answer would be the last node  "Right" that you took on your way to inserting 'v', of which you can easily keep track of in recursive or iterative version.
UPDATE: 
Going by your requirements, you can approach this in a linear fashion:
If every next element is smaller than the current element(e.g. 6 5 4 3 2 1) you can process this linearly without requiring any extra memory. Interesting case arises when you start getting jumbled elements(e.g. 4 2 1 5 3), in which case you need to remember their order as long as you dont' get their 'smaller counterparts'.
A simple stack based approach goes like this:
Push the first element (a[0]) in a stack.
For each next element a[i], you peek into the stack and if value ( peek() ) is greater than the one in hand a[i], you got your next smaller number for that stack element (peek()) { and keep on popping the elements as long as peek() > a[i] }. Pop them out and print/store the corresponding value.
else, simply push back your a[i] into the stack.
In the end stack 'll contain those elements which never had a value smaller than them(to their right). You can fill in -1 for them in your outpput.
e.g. A=[4, 2, 1, 5, 3];

stack: 4
a[i] = 2, Pop 4, Push 2 (you got result for 4)
stack: 2
a[i] = 1, Pop 2, Push 1 (you got result for 2)
stack: 1
a[i] = 5
stack: 1 5
a[i] = 3, Pop 5, Push 3 (you got result for 5)
stack: 1 3
1,3 don't have any counterparts for them. so store -1 for them.

