Externalize The Stack

Intent:

Rewrite a recursive program/algorithm/function into an iterative one which uses an external stack.

The Problem:

All computer scientists are familiar with recursion; many consider it a more "natural" way than iteration to describe algorithms which must repeat some computation. However, recursive algorithms, when naïvely written, can chew up a valuable resource - TheStack - in languages which use stacks to maintain activation records (this includes CeeLanguage, CeePlusPlus, and JavaLanguage; it excludes SmalltalkLanguage). Some recursive algorithms exhibit TailRecursion and can be rewritten in an iterative way that uses bounded space (and some compilers, especially of functional languages, will perform this transformation as an optimization - see TailCallOptimization). However, many recursive algorithms are not tail-recursive.

The most natural way to express recursive algorithms is with recursive function calls; each instance of the function gets its own activation record to maintain its particular state, while TheStack holds the complete set.

The problems with TheStack in many languages (or the stacks, in multithreaded programs) are as follows:

This causes many recursive functions, naïvely implemented, to fail on large datasets. In some languages, you get a StackOverflowException? or some other controlled error. In others, the stack overflow causes something else important to be overwritten, resulting in death, destruction, and UndefinedBehavior. For a well-known example, see JavaSerializationAndTheStack.

The solution:

ExternalizeTheStack.

Figure out what state must be maintained between successive invocations of the function. Define a structure to hold that (and only that) state. Declare a stack (or a linked list, or whatever you have available) (a linked list is one possible implementation of the AbstractDataType stack, not something to be seen as an alternative to a stack) of that structure. Use it to hold the stacked copies of the recursive state. Then rewrite the remainder of the function to be iterative. (In the case of Java serialization, and many other graph/tree traversal algorithms, the only thing that needs to be maintained in recursive fashion is a BackPointer?).

Advantages:

Disadvantages:

Examples:

A Compromise:

Partly externalize the stack, by reallocating as many of the local variables as possible to global memory (off the stack). If the value of a variable must be preserved during a recursive call, then the variable must be local; but if this never happens, it is safe to make the variable global. However, some languages (e.g., Pascal) do not allow a looping index variable to be global. Sometimes, this can partly solve the stack problem while preserving the readability of the code.


Example

This is contrived, since the iterative version does not actually require an array. This whole page is somewhat contrived, since there's no mention of the fact that ExternalizeTheStack is usually a premature optimization, although there's no question but that it is sometimes essential. Also some of the supposed benefits listed above are platform specific, not universals.

  // recursive
  factorial(int n) {
	if (n <= 2) return n;  // Originally "if (n <= 1) return 1;"
                               // I find it amusing that someone applied this
                               // dead-accurate yet PrematureOptimization, here.
                               // Yes, it optimizes. But does it clarify or obfuscate?
                               // And if one quantifies the optimization: clearly premature.
                               // Whoever did that needs to reconsider their habitual
                               // tradeoffs. This was an inappropriate optimization.
                               // See RulesOfOptimization
	return n * factorial(n-1);
  }

// iterative factorial(int n) { int i, array[100]; // integer overflow will happen well before we reach factorial(100) int result; for (i=1; i<=n; i++) array[i] = i; result = 1; for (i=1; i<=n; i++) result *= array[i]; return result; }

I understand both the recursive and iterative approaches to computing the factorial. This example seems to demonstrate eliminating recursion altogether. I was hoping to see an example of "an iterative [function] which uses an external stack" -- I don't see an external stack in above example. Not to pick on the example too much, but many compilers will preallocate "array" with room for 100 ints -- assuring that the iterative example consumes MORE stack space than the factorial for typical values of n. How does the iterative approach optimize anything (even if prematurely)? The example isn't a great choice, because computing factorial is TailRecursive and thus it doesn't need ExternalizeTheStack. An example of an algorithm which ISN'T tail recursive - depth-first traversal of a tree - is given below. First, our definition of a tree (the trees in this example may have zero, one, or two children - no balancing is assumed - and both leaf and non-leaf nodes have data). This is written in Java/C++-esque pseudocode; with many issues, such as proper encapsulation, elided.

First, the common preliminaries

 class TreeNode? {
	Object data;  // our data
	TreeNode? left, right;  // children
 };

class TreeNodeVisitor? { abstract void visit (TreeNode?); // do something interesting on a tree node };

Second, a depth-first traversal using recursion. Nice and easy.

 void recursiveDepthFirstTraversal (TreeNode? root, TreeNodeVisitor? visitor)
 {
	// do the left branch
	if (root.left) recursiveDepthFirstTraversal (root.left, visitor);

// visit this node visitor.visit (root);

// do the right branch if (root.right) recursiveDepthFirstTraversal (root.right, visitor); }
And third, a version using ExternalizeTheStack. Note that we need to define an auxillary class, TraversalState?, which essentially captures the continuation that would otherwise be saved on the stack. Lots of possible error conditions go unhandled.

 class TraversalState? {
	TreeNode? node;  // node being visited
	bool left_visited; // true if we've already visited the left node
	TraversalState? (TreeNode? n) {
	  node = n; left_visited = false;
	}
 };

void iterativeDepthFirstTraversal (TreeNode? root, TreeNodeVisitor? visitor) { Cons <TraversalState?> stack = new Cons<TraversalState?> (new TraversalState? (root), null); // our external stack, implemented using conses

while (stack != null) { if (!stack.car.left_visited) { stack.car.left_visited = true; if (stack.car.node.left) { stack = new Cons<TraversalState?> (new TraversalState? (stack.car.node.left), stack); } } else { visitor.visit (stack.car.node); if (stack.car.node.right) { stack = new Cons<TravseralState?> (new TraveralState? (stack.car.node.right), stack); } stack = stack.cdr; // done with branch; pop up to parent } } }
As you can see, the externalized and iterative version is considerably more complicated than the recursive version. And since it was typed in by hand (no attempt to run it), I'm not even sure it works. ExternalizeTheStack is definitely one of those things you should only do when you have to - such as when you are short on memory (and need to minimize the size of the saved continuation) or are likely to run into a StackOverflow.


Do not ExternalizeTheStack in languages, that have continuations (see CategoryContinuation), or rather where control flow is realized with continuations like in SML/NJ. In these cases there is no problem with TheStack, because there is only the TheHeap anyway.

Another alternative, if you know that there is a bound on the recursion but the default stack size is not sufficient, is to allocate enough stack space for the calculation in the first case. Often setting the stack size as a compiler option/OS option/interpreter option is the simplest fix on machines with 32 bit or greater virtual memory spaces. -- PeteKirkham


See DesignPatterns See also http://wiki.cs.uiuc.edu/PatternStories/DesignPatterns

CategoryStructuralPatterns

EditText of this page (last edited January 19, 2010) or FindPage with title or text search