Divide-and-conquer algorithm: Difference between revisions

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The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as [[sorting algorithm|sorting]] (e.g., [[quicksort]], [[merge sort]]), [[multiplication algorithm|multiplying large numbers]] (e.g., the [[Karatsuba algorithm]]), finding the [[Closest pair of points problem|closest pair of points]], [[syntactic analysis]] (e.g., [[top-down parser]]s), and computing the [[discrete Fourier transform]] ([[fast Fourier transform|FFT]]).<ref>{{cite book |last1=Blahut |first1=Richard |title=Fast Algorithms for Signal Processing |publisher=Cambridge University Press |isbn=978-0-511-77637-3 |pages=139–143}}</ref>

Designing efficient divide-and-conquer algorithms can be difficult. As in [[mathematical induction]], it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cost is often determined by solving [[recurrence relation]]s.

== Divide and conquer ==

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For example, to sort a given list of ''n'' natural numbers, split it into two lists of about ''n''/2 numbers each, sort each of them in turn, and interleave both results appropriately to obtain the sorted version of the given list (see the picture). This approach is known as the [[merge sort]] algorithm.

The name "divide and conquer" is sometimes applied to algorithms that reduce each problem to only one sub-problem, such as the [[binary search]] algorithm for finding a record in a sorted list (or its analog in [[numerical algorithm|numerical computing]], the [[bisection algorithm]] for [[root-finding algorithm|root finding]]).<ref name="CormenLeiserson2009">{{cite book|author1=Thomas H. Cormen|author2=Charles E. Leiserson|author3=Ronald L. Rivest|author4=Clifford Stein|title=Introduction to Algorithms|url=https://books.google.com/books?id=aefUBQAAQBAJ&q=%22Divide-and-conquer%22|date=31 July 2009|publisher=MIT Press|isbn=978-0-262-53305-8}}</ref> These algorithms can be implemented more efficiently than general divide-and-conquer algorithms; in particular, if they use [[tail recursion]], they can be converted into simple [[loop (computing)|loops]]. Under this broad definition, however, every algorithm that uses recursion or loops could be regarded as a "divide-and-conquer algorithm". Therefore, some authors consider that the name "divide and conquer" should be used only when each problem may generate two or more subproblems.<ref>Brassard, G., and Bratley, P. Fundamental of Algorithmics, Prentice-Hall, 1996.</ref> The name '''decrease and conquer''' has been proposed instead for the single-subproblem class.<ref>Anany V. Levitin, ''Introduction to the Design and Analysis of Algorithms'' (Addison Wesley, 2002).</ref>

An important application of divide and conquer is in optimization,{{Examples|date=October 2017}} where if the search space is reduced ("pruned") by a constant factor at each step, the overall algorithm has the same asymptotic complexity as the pruning step, with the constant depending on the pruning factor (by summing the [[geometric series]]); this is known as [[prune and search]].

== Early historical examples ==

Early examples of these algorithms are primarily decreasedecreased and conquer – the original problem is successively broken down into ''single'' subproblems, and indeed can be solved iteratively.

Binary search, a decrease-and-conquer algorithm where the subproblems are of roughly half the original size, has a long history. While a clear description of the algorithm on computers appeared in 1946 in an article by [[John Mauchly]], the idea of using a sorted list of items to facilitate searching dates back at least as far as [[Babylonia]] in 200&nbsp;BC.<ref name=Knuth3/> Another ancient decrease-and-conquer algorithm is the [[Euclidean algorithm]] to compute the [[greatest common divisor]] of two numbers by reducing the numbers to smaller and smaller equivalent subproblems, which dates to several centuries BC.

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=== Solving difficult problems ===

Divide and conquer is a powerful tool for solving conceptually difficult problems: all it requires is a way of breaking the problem into sub-problems, of solving the trivial cases, and of combining sub-problems to the original problem. Similarly, decrease and conquer only requires reducing the problem to a single smaller problem, such as the classic [[Tower of Hanoi]] puzzle, which reduces moving a tower of height <math>n</math> to movingmove a tower of height <math>n-1</math>.

=== Algorithm efficiency ===

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=== Parallelism ===

Divide-and-conquer algorithms are naturally adapted for execution in multi-processor machines, especially [[shared-memory]] systems where the communication of data between processors does not need to be planned in advance, because distinct sub-problems can be executed on different processors.

=== Memory access ===

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=== Roundoff control ===

In computations with rounded arithmetic, e.g. with [[floating-point]] numbers, a divide-and-conquer algorithm may yield more accurate results than a superficially equivalent iterative method. For example, one can add ''N'' numbers either by a simple loop that adds each datum to a single variable, or by a D&C algorithm called [[pairwise summation]] that breaks the data set into two halves, recursively computes the sum of each half, and then adds the two sums. While the second method performs the same number of additions as the first, and pays the overhead of the recursive calls, it is usually more accurate.<ref>Nicholas J. Higham, "[https://pdfs.semanticscholar.org/5c17/9d447a27c40a54b2bf8b1b2d6819e63c1a69.pdf The accuracy of floating -point summation]", ''SIAM J. Scientific Computing'' '''14''' (4), 783–799 (1993).</ref>

== Implementation issues ==

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=== 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 whether 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 <math>n</math> entries would entail maximally <math>n</math> 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.

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=== 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 is 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 ==