Divide-and-conquer algorithm: Difference between revisions

Revision as of 23:56, 9 December 2018

Revision as of 00:06, 10 December 2018

Line 58:

=== Explicit stack ===

Divide -and -conquer algorithms can also be implemented by a non-recursive program that stores the partial sub-problems in some explicit data structure, such as a [[stack (data structure)|stack]], [[queue (data structure)|queue]], or [[priority queue]]. This approach allows more freedom in the choice of the sub-problem that is to be solved next, a feature that is important in some applications — e.g. in [[breadth first recursion|breadth-first recursion]] and the [[branch -and -bound]] method for function optimization. This approach is also the standard solution in programming languages that do not provide support for recursive procedures.

=== Stack size ===

In recursive implementations of D&C algorithms, one must make sure that there is sufficient memory allocated for the recursion stack, otherwise the execution may fail because of [[stack overflow]]. D&C algorithms that are time-efficient often have relatively small recursion depth. For example, the quicksort algorithm can be implemented so that it never requires more than <math>\log_2 n</math> nested recursive calls to sort <math>n</math> items.

Stack overflow may be difficult to avoid when using recursive procedures, since many compilers assume that the recursion stack is a contiguous area of memory, and some allocate a fixed amount of space for it. Compilers may also save more information in the recursion stack than is strictly necessary, such as return address, unchanging parameters, and the internal variables of the procedure. Thus, the risk of stack overflow can be reduced by minimizing the parameters and internal variables of the recursive procedure, or by using an explicit stack structure.

=== Choosing the base cases ===

In any recursive algorithm, there is considerable freedom in the choice of the ''base cases'', the small subproblems that are solved directly in order to terminate the recursion.

Choosing the smallest or simplest possible base cases is more elegant and usually leads to simpler programs, because there are fewer cases to consider and they are easier to solve. For example, an FFT algorithm could stop the recursion when the input is a single sample, and the quicksort list-sorting algorithm could stop when the input is the empty list; in both examples there is only one base case to consider, and it requires no processing.

On the other hand, efficiency often improves if the recursion is stopped at relatively large base cases, and these are solved non-recursively, resulting in a [[hybrid algorithm]]. This strategy avoids the overhead of recursive calls that do little or no work, and may also allow the use of specialized non-recursive algorithms that, for those base cases, are more efficient than explicit recursion. A general procedure for a simple hybrid recursive algorithm is ''short-circuiting the base case,'', also known as ''[[arm's-length recursion]].''. In this case whether the next step will result in the base case is checked before the function call, avoiding an unnecessary function call. For example, in a tree, rather than recursing to a child node and then checking ifwhether it is null, checking null before recursing; this avoids half the function calls in some algorithms on binary trees. Since a D&C algorithm eventually reduces each problem or sub-problem instance to a large number of base instances, these often dominate the overall cost of the algorithm, especially when the splitting/joining overhead is low. Note that these considerations do not depend on whether recursion is implemented by the compiler or by an explicit stack.

Thus, for example, many library implementations of quicksort will switch to a simple loop-based [[insertion sort]] (or similar) algorithm once the number of items to be sorted is sufficiently small. Note that, if the empty list were the only base case, sorting a list with ''n'' entries would entail maximally ''n'' quicksort calls that would do nothing but return immediately. Increasing the base cases to lists of size 2 or less will eliminate most of those do-nothing calls, and more generally a base case larger than 2 is typically used to reduce the fraction of time spent in function-call overhead or stack manipulation.

Alternatively, one can employ large base cases that still use a divide-and-conquer algorithm, but implement the algorithm for predetermined set of fixed sizes where the algorithm can be completely [[loop unwinding|unrolled]] into code that has no recursion, loops, or [[Conditional (programming)|conditionals]] (related to the technique of [[partial evaluation]]). For example, this approach is used in some efficient FFT implementations, where the base cases are unrolled implementations of divide-and-conquer FFT algorithms for a set of fixed sizes.<ref name="fftw">{{cite journal | author = Frigo, M. |author2=Johnson, S. G. | url = http://www.fftw.org/fftw-paper-ieee.pdf | title = The design and implementation of FFTW3 | journal = Proceedings of the IEEE | volume = 93 | issue = 2 |date=February 2005 | pages = 216–231 | doi = 10.1109/JPROC.2004.840301}}</ref> [[Source -code generation]] methods may be used to produce the large number of separate base cases desirable to implement this strategy efficiently.<ref name="fftw"/>

The generalized version of this idea is known as recursion "unrolling" or "coarsening", and various techniques have been proposed for automating the procedure of enlarging the base case.<ref>Radu Rugina and Martin Rinard, "[http://people.csail.mit.edu/rinard/paper/lcpc00.pdf Recursion unrolling for divide and conquer programs]," in ''Languages and Compilers for Parallel Computing'', chapter 3, pp. 34–48. ''Lecture Notes in Computer Science'' vol. 2017 (Berlin: Springer, 2001).</ref>

=== Sharing repeated subproblems ===

For some problems, the branched recursion may end up evaluating the same sub-problem many times over. In such cases it may be worth identifying and saving the solutions to these overlapping subproblems, a technique commonly known as [[memoization]]. Followed to the limit, it leads to [[bottom-up design|bottom-up]] divide-and-conquer algorithms such as [[dynamic programming]] and [[chart parsing]].

== See also ==