Divide and conquer algorithm

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In computer science, divide and conquer (D&C) is an important algorithm design paradigm. It works by recursively breaking 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.

This technique is the basis of efficient algorithms for all kinds of problems, such as sorting (quicksort, merge sort) and the discrete Fourier transform (FFTs).

Implementation

Divide-and-conquer algorithms are naturally implemented as recursive procedures. In that case, the partial sub-problems leading to the one currently being solved are implicitly stored in the procedure call stack.

However, D&C solutions can also be implemented by a non-recursive algorithm that stores the partial sub-problems in some explicit data structure, such as a stack, 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 and the branch and bound method for function optimization.

Variations

One variation of divide and conquer is called decrease and conquer, where a solution of problem depends on only one subproblem. There are two advantages of treating this variant separately. Some problems do not need to solve all subproblems, and have a simpler conquer strategy. They can be generally solved with tail recursion. Analysis of these problems is simpler than divide and conquer.

For finding the largest element in a list of numbers:

Algorithm LargestNumber  
  Input: A non-empty list of numbers L.
  Output: The largest number in the list L.
  Comment:   Divide and Conquer
  if L.size == 1 then
     return L.front
  largest1 ←  LargestNumber(L.front...L.mid)
  largest2 ←  LargestNumber(L.mid....L.back)
  if largest1 >  largest2, then
    largest ← largest1
  else
     largest ← largest2
  return largest

Algorithm LargestNumber 
  Input: A non-empty list of numbers L.
  Output: The largest number in the list L.
  Comment:  Decrease and Conquer 
 if L.size == 1 then
     return L.front
  last = L.size - 1;
  largest ←  LargestNumber(L.front...L.last)
  if largest <  L.back then
    largest ← L.back
  return largest

Finding the maximum is solved by decreasing the problem by one element on each pass. More popular example of decrease and conquer is binary search algorithm, where the problem size will be cut down by half in each decrease phase and so completes faster.

Advantages

Solving difficult problems

Divide and conquer is a powerful tool for solving conceptually difficult problems, such as the classic Tower of Hanoi puzzle: 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. Dividing the problem into sub-problems so that the sub-problems can be combined again is often the major difficulty in designing a new algorithm. Indeed, for many such problems the paradigm offers the only simple solution.

Algorithm efficiency

Moreover, divide and conquer often provides a natural way to design efficient algorithms.

For example, if the work of splitting the problem and combining the partial solutions is proportional to the problem's size n, there are a bounded number p of subproblems of size ~ n/p at each stage, and the base cases require O(1) (constant-bounded) time, then the divide-and-conquer algorithm will have O(n log n) complexity. This is used for problems such as sorting and FFTs to reduce the complexity from O(n²), although in general there may also be other approaches to designing efficient algorithms.

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

Divide-and-conquer algorithms naturally tend to make efficient use of memory caches. 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, without accessing the slower main memory. An algorithm designed to exploit the cache in this way is called cache oblivious, because it does not contain the cache size(s) as an explicit parameter.

Moreover, D&C algorithms can be designed for many important algorithms, such as sorting, FFTs, and matrix multiplication, in such a way as to be optimal cache oblivious algorithms—they use the cache in a provably optimal way, in an asymptotic sense, regardless of the cache size. In contrast, the traditional approach to exploiting the cache is blocking, 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 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.

However, the kind of asymptotic optimality described here, analogous to big O notation, ignores constant factors, and additional machine/cache-specific tuning is in general required to achieve optimality in an absolute sense.

Disadvantages

One commonly argued disadvantage of a divide-and-conquer approach is that recursion is slow: the overhead of the repeated subroutine calls, along with that of storing the call stack (the state at each point in the recursion), can outweigh any advantages of the approach. This, however, depends upon the implementation style: with large enough recursive base cases, the overhead of recursion can become negligible for many problems.

Another problem of a divide-and-conquer approach is that, for simple problems, it may be more complicated than an iterative approach, especially if large base cases are to be implemented for performance reasons. For example, to add N numbers, a simple loop to add them up in sequence is much easier to code than a divide-and-conquer approach that breaks the set of numbers into two halves, adds them recursively, and then adds the sums. (On the other hand, the latter approach turns out to be much more accurate in finite-precision floating-point arithmetic.)

Alternatively, an algorithm can be designed by a divide-and-conquer approach and then implemented in an iterative style; of course, any recursive algorithm can be simulated in an iterative style, but by reordering the computation it is sometimes possible to avoid simulating a recursive call stack altogether. The Cooley-Tukey FFT algorithm is a prime example of this: it was initially derived in a divide-and-conquer style, but implementations were traditionally implemented iteratively in a re-ordered fashion that avoided call-stack-like storage (although some recent implementations have returned to a recursive style).

See also

• Mathematical induction
• The Master theorem
• The Akra-Bazzi method
• Divide and rule (politics and sociology)

References

• M. Frigo, C. E. Leiserson, and H. Prokop, "Cache-oblivious algorithms", Proc. 40th Symp. on the Foundations of Computer Science (1999).
• Nicholas J. Higham, "The accuracy of floating point summation", SIAM J. Scientific Computing 14 (4), 783–799 (1993).de:Teile und herrsche (Informatik)

es:Algoritmo divide y vencerás fr:Diviser pour régner (informatique) he:אלגוריתם הפרד ומשול it:Divide et impera (informatica) ja:分割統治法 pl:Dziel i rządź sl:Deli in vladaj (računalništvo) zh:分治法