Tasks in which recursion is either the fastest or only way to produce a result

Question

I've just finished studying recursion at university. One thing that stood out for me however was that in both the lectures and in the practical we completed, all the tasks we were asked to do could be performed faster, and in less code, using iterative means.
This was something the lecturer confirmed.

Could somebody please give me some examples of situations when recursion is a better solution than iterative techniques? Additionally, are there any situations in which recursion is the only way to sole a problem?

Answer 1

There are no questions which can only be solved with recursion. This is because they can be solved with Turing machines, which don't use recursion.

The set of problems which can be solved with a TM are exactly the same as the ones which can be solved with recursion (or its formal model, the Lambda Calculus).

In particular, if you want to simulate recursion iteratively, the way to do this is to use a data structure called a stack (which simulates the call stack for functions).

As for algorithms that can be solved better using recursion, there are tons. I'm surprised that your recursive versions were longer, as recursion usually leads to less code. This is one of the reasons haskell is gaining popularity.

Consider the algorithm quicksort, for sorting lists. In rough pseudocode, it's as follows:

function quicksort(list)
    if length(list) <= 1
        return list       
    pivot = first element of list
    lessList = []
    equalList = []
    greaterList = []
    for each element in list:
        if element < pivot, add to lessList
        if element == pivot, add to equalList
        if element > pivot, add to greater list
   sortedLess = quicksort(lessList)
   sortedGreater = quicksort(greaterList)
   return sortedLess ++ equalList ++ sortedGreater

where ++ means concatenation.

The code isn't purely functional, but by dividing the list into different parts, and sorting each sublist recursively, we get a very short $O(n\log n)$ sort.

Recursion is also very useful for recursive data structures. Often times you'll have a traversal on trees, of the following form:

function traverseTree(tree)
    if (tree is a single node)
        do something to that node
    else, for each child of tree:
         traverseTree(child)

Answer 2

Anything that can be implemented through recursion can be implemented through iteration, and vice versa. So there is no task which it is impossible to accomplish without recursion (assuming a programming language that has the usual iterative constructs).

The cost of transforming a recursive program into an equivalent interative one is up to polylogarithmic time in most settings, and close to constant time (depending on the size of the code, but not on the size of the data) in practice. This is because the gist of the transformation from a recursive program to an iterative program is to push a record on a global stack every time a function is called, and pop that record when the function returns. This doesn't make any significant change to running time, other than memory management for the additional stack. So where a recursive program exists, an equally fast equivalent iterative program exists (for reasonable values of “equally”).

For some recursive programs, a transformation to an iterative program can significantly increase the complexity in terms of code maintenance. All that stack management can amount to significant amounts of code, especially to ensure that you have captured the right data in stack records.

A typical example where recursion is natural and avoiding it is cumbersome is tree traversal. Consider a program that manipulates binary trees and must often traverse them. A recursive traversal goes like this:

def traverse(tree):
    do_something(tree.node_payload)
    if tree.left_child != None: traverse(tree.left_child)
    if tree.right_child != None: traverse(tree.right_child)

An iterative traversal requires explicit stack maintenance — and this is a simplified example which doesn't maintain local data as it traverses the tree:

def traverse(tree):
    stack = new_empty_stack()
    stack.push(tree)
    while not stack.is_empty():
       node = stack.pop()
       do_something(tree.node_payload)
        if tree.left_child != None: stack.push(tree.left_child)
        if tree.right_child != None: stack.push(tree.right_child)

Exercise: write a recursive program that sums the elements of a tree of integers multiplied by their depth. Write an iterative program that does the same thing.