Divide-and-conquer algorithm: Difference between revisions

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{{Short description|Algorithms which recursively solve subproblems}}

In [[computer science]], '''divide and conquer''' is an [[algorithm design paradigm]] based on multi-branched [[recursion]]. A divide-and-conquer [[algorithm]] works[[Recursion by(computer science)|recursively]] breakingbreaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.

ThisThe divide-and-conquer technique is the basis of efficient algorithms for all kinds ofmany 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>{{Citationcite book |last1=Blahut |first1=Richard |author1-link=Richard Blahut |title=Fast Algorithms for Signal Processing needed|date=October14 2017May 2014 |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 [[Recursion (computer science)|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.

Understanding and designing divide-and-conquer algorithms is a complex skill that requires a good understanding of the nature of the underlying problem to be solved. As when proving a [[theorem]] by [[Mathematical induction|induction]], it is often necessary to replace the original problem with a more general or complicated problem in order to initialize the recursion, and there is no systematic method for finding the proper generalization.{{Clarify|date=October 2017}} These divide-and-conquer complications are seen when optimizing the calculation of a [[Fibonacci number#Matrix form|Fibonacci number with efficient double recursion]].{{Why|date=October 2017}}{{Citation needed|date=October 2017}}

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 ==

[[File:Merge sort algorithm diagram.svg|thumb|Divide-and-conquer approach to sort the list (38, 27, 43, 3, 9, 82, 10) in increasing order. ''Upper half:'' splitting into sublists; ''mid:'' a one-element list is trivially sorted; ''lower half:'' composing sorted sublists.]]

The divide-and-conquer [[Programming paradigm|paradigm]] is often used to find an optimal solution of a problem. Its basic idea is to decompose a given problem into two or more similar, but simpler, subproblems, to solve them in turn, and to compose their solutions to solve the given problem. Problems of sufficient simplicity are solved directly.

For example, to sort a given list of ''n'' [[Natural number|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 analoganalogue 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&printsec=frontcover#v=onepage&q=%22Divide-and-conquer%22&f=false|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]].

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

[[Binary search algorithm|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.

An early example of a divide-and-conquer algorithm with multiple subproblems is [[Carl Friedrich Gauss|Gauss]]'s 1805 description of what is now called the [[Cooley–Tukey FFT algorithm|Cooley–Tukey fast Fourier transform]] (FFT) algorithm,<ref name=Heideman84>Heideman, M. T., D. H. Johnson, and C. S. Burrus, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.331.4791&rep=rep1&type=pdf Gauss and the history of the fast Fourier transform]", IEEE ASSP Magazine, 1, (4), 14–21 (1984).</ref> although he did not [[analysis of algorithms|analyze its operation count]] quantitatively, and FFTs did not become widespread until they were rediscovered over a century later.

An early two-subproblem D&C algorithm that was specifically developed for computers and properly analyzed is the [[merge sort]] algorithm, invented by [[John von Neumann]] in 1945.<ref>{{ cite book | last=Knuth | first=Donald | authorlinkauthor-link=Donald Knuth | year=1998 | title=[[The Art of Computer Programming]]: Volume 3 Sorting and Searching | url=https://archive.org/details/artofcomputerpro03knut | url-access=limited | page=[https://archive.org/details/artofcomputerpro03knut/page/159 159] | publisher=Addison-Wesley | isbn=0-201-89685-0 }}</ref>

Another notable example is the [[Karatsuba algorithm|algorithm]] invented by [[Anatolii Alexeevitch Karatsuba|Anatolii A. Karatsuba]] in 1960<ref>{{cite journal| last=Karatsuba | first=Anatolii A. | authorlinkauthor-link=Anatolii Alexeevitch Karatsuba |author2=Yuri P. Ofman |authorlink2author-link2=Yuri Petrovich Ofman | year=1962 | title=Умножение многозначных чисел на автоматах | journal=[[Doklady Akademii Nauk SSSR]] | volume=146 | pages=293–294}} Translated in {{cite journal| title=Multiplication of Multidigit Numbers on Automata | journal=Soviet Physics Doklady | volume=7 | year=1963 | pages=595–596 | bibcode=1963SPhD....7..595K |url={{Google books|MrkOAAAAIAAJ|plainurl=true}} | last1=Karatsuba | first1=A. | last2=Ofman | first2=Yu. }}</ref> that could multiply two ''n''-[[Units of information|digit]] numbers in <math>O(n^{\log_2 3})</math> operations (in [[Big O notation]]). This algorithm disproved [[Andrey Kolmogorov]]'s 1956 conjecture that <math>\Omega(n^2)</math> operations would be required for that task.

As another example of a divide-and-conquer algorithm that did not originally involve computers, [[Donald Knuth]] gives the method a [[post office]] typically uses to route mail: letters are sorted into separate bags for different geographical areas, each of these bags is itself sorted into batches for smaller sub-regions, and so on until they are delivered.<ref name=Knuth3>Donald E. Knuth, ''The Art of Computer Programming: Volume 3, Sorting and Searching'', second edition (Addison-Wesley, 1998).</ref> This is related to a [[radix sort]], described for [[IBM 80 series Card Sorters|punch-card sorting]] machines as early as 1929.<ref name=Knuth3/>

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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''&nbsp;−&nbsp;-1</math>.

=== Algorithm efficiency ===

The divide-and-conquer paradigm often helps in the discovery of efficient algorithms. It was the key, for example, to [[Karatsuba algorithm|Karatsuba]]'s fast multiplication method, the quicksort and mergesort algorithms, the [[Strassen algorithm]] for [[matrix multiplication]], and fast Fourier transforms.

In all these examples, the D&C approach led to an improvement in the [[asymptotic complexity|asymptotic cost]] of the solution. For example, if (a) the [[Recursion (computer science)|base cases]] have constant-bounded size, the work of splitting the problem and combining the partial solutions is proportional to the problem's size <math>n</math>, and (b) there is a bounded number <math>p</math> of sub-problems of size ~ <math>\frac{n}{p}</math> at each stage, then the cost of the divide-and-conquer algorithm will be <math>O(n\log_{p}n)</math>.

For example, if (a) the [[Recursion (computer science)|base cases]] have constant-bounded size, the work of splitting the problem and combining the partial solutions is proportional to the problem's size ''n'', and (b) there is a bounded number ''p'' of subproblems of size ~ ''n''/''p'' at each stage, then the cost of the divide-and-conquer algorithm will be O(''n'' log_''p''''n'').

For other types of divide-and-conquer approaches, running times can also be generalized. For example, when a) the work of splitting the problem and combining the partial solutions take <math>cn</math> time, where <math>n</math> is the input size and <math>c</math> is some constant; b) when <math>n < 2</math>, the algorithm takes time upper-bounded by <math>c</math>, and c) there are <math>q</math> subproblems where each subproblem has size ~ <math>\frac{n}{2}</math>. Then, the running times are as follows:

* if the number of subproblems <math>q > 2</math>, then the divide-and-conquer algorithm's running time is bounded by <math>O(n^{\log_{2}q})</math>.

* if the number of subproblems is exactly one, then the divide-and-conquer algorithm's running time is bounded by <math>O(n)</math>.<ref name="kleinberg&Tardos">{{cite book |last1=Kleinberg |first1=Jon |last2=Tardos |first2=Eva |title=Algorithm Design |date=March 16, 2005 |publisher=[[Pearson Education]] |isbn=9780321295354 |pages=214–220 |edition=1 |url=https://www.pearson.com/en-us/subject-catalog/p/algorithm-design/P200000003259/9780137546350 |access-date=26 January 2025}}</ref>

If, instead, the work of splitting the problem and combining the partial solutions take <math>cn^2</math> time, and there are 2 subproblems where each has size <math>\frac{n}{2}</math>, then the running time of the divide-and-conquer algorithm is bounded by <math>O(n^2)</math>.<ref name="kleinberg&Tardos"/>

=== Parallelism ===

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

=== Memory access ===

Divide-and-conquer algorithms naturally tend to make efficient use of [[memory cache]]s. The reason is that once a sub-problem is small enough, it and all its sub-problems can, in principle, be solved within the [[Cache (computing)|cache]], without accessing the slower [[Computer data storage|main memory]]. An algorithm designed to exploit the cache in this way is called ''[[cache-oblivious algorithm|cache-oblivious]]'', because it does not contain the cache size as an explicit [[Parameter (computer programming)|parameter]].<ref name="cahob">{{cite journalbook | author = M. Frigo |author2=C. E. Leiserson |author3=H. Prokop | title =40th Cache-obliviousAnnual algorithmsSymposium |on journalFoundations =of Proc.Computer 40thScience Symp(Cat. onNo.99CB37039) the|chapter=Cache-oblivious Foundationsalgorithms of Computer Science|pages=285–297 | year = 1999|chapter-url=https://dspace.mit.edu/bitstream/handle/1721.1/80568/43558192-MIT.pdf;sequence=2|doi=10.1109/SFFCS.1999.814600 |isbn=0-7695-0409-4 |s2cid=62758836 }}</ref> Moreover, D&C algorithms can be designed for important algorithms (e.g., sorting, FFTs, and matrix multiplication) to be ''optimal'' cache-oblivious algorithms–they use the cache in a probably optimal way, in an asymptotic sense, regardless of the cache size. In contrast, the traditional approach to exploiting the cache is ''blocking'', as in [[loop nest optimization]], where the problem is explicitly divided into chunks of the appropriate size—this can also use the cache optimally, but only when the algorithm is tuned for the specific cache sizes of a particular machine.

Moreover, D&C algorithms can be designed for important algorithms (e.g., sorting, FFTs, and matrix multiplication) to be ''optimal'' cache-oblivious algorithms–they use the cache in a probably optimal way, in an asymptotic sense, regardless of the cache size. In contrast, the traditional approach to exploiting the cache is ''blocking'', as in [[loop nest optimization]], where the problem is explicitly divided into chunks of the appropriate size—this can also use the cache optimally, but only when the algorithm is tuned for the specific cache size(s) of a particular machine.

The same advantage exists with regards to other hierarchical storage systems, such as [[Non-uniform memory access|NUMA]] or [[virtual memory]], as well as for multiple levels of cache: once a sub-problem is small enough, it can be solved within a given level of the hierarchy, without accessing the higher (slower) levels.

=== 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, ana [[Fast Fourier transform|Fast Fourier FFTTransform]] 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.

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|bibcode=2005IEEEP..93..216F |citeseerx=10.1.1.66.3097 |s2cid=6644892 }}</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>

=== SharingDynamic repeatedprogramming for overlapping 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 which 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 ==

{{Portal|Computer Science}}

{{commonsCommons category|Divide-and-conquer algorithms}}

* [[{{annotated link|Akra–Bazzi method]]}}

* [[{{annotated link|Decomposable aggregation function]]}}

* {{annotated link|Divide and conquer|"Divide and conquer"}}

* [[{{annotated link|Fork–join model]]}}

* [[{{annotated link|Master theorem (analysis of algorithms)]]}}

* [[{{annotated link|Mathematical induction]]}}

* [[MapReduce]]

* {{annotated link|MapReduce}}

* [[{{annotated link|Heuristic (computer science)]]}}

== References ==

{{Reflist}}{{Algorithmic paradigms}}

<references/>

{{Authority control}}

{{DEFAULTSORT:Divide And Conquer Algorithm}}

[[Category:Divide-and-conquer algorithms| ]]

[[Category:Algorithms]]

[[Category:Optimization algorithms and methods]]