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Such problems arise in approximation algorithms; a famous example is the directed Steiner tree problem, for which there is a quasi-polynomial time approximation algorithm achieving an approximation factor of () (n being the number of vertices), but showing the existence of such a polynomial time algorithm is an open problem.
Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n.
For example, the worst-case complexity of solving a linear program using the simplex algorithm is exponential, [2] although the observed number of steps in practice is roughly linear. [3] [4] The simplex algorithm is in fact much faster than the ellipsoid method in practice, although the latter has polynomial-time worst-case complexity.
Therefore, the time complexity, generally called bit complexity in this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a n × n integer matrix is O ( n 3 ) {\displaystyle O(n^{3})} for the usual algorithms ( Gaussian elimination ).
Minimizing the depth/span is important in designing parallel algorithms, because the depth/span determines the shortest possible execution time. [8] Alternatively, the span can be defined as the time T ∞ spent computing using an idealized machine with an infinite number of processors. [9] The cost of the computation is the quantity pT p. This ...
An important application of divide and conquer is in optimization, [example needed] 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 ...
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.
Advantages of this method are: A solution is guaranteed (as long as the puzzle is valid). Solving time is mostly unrelated to degree of difficulty. [dubious – discuss] The algorithm (and therefore the program code) is simpler than other algorithms, especially compared to strong algorithms that ensure a solution to the most difficult puzzles.