Akra–Bazzi method

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In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the well-known master theorem, which assumes that the sub-problems have equal size.

The formula[edit]

The Akra–Bazzi method applies to recurrence formulas of the form

The conditions for usage are:

  • sufficient base cases are provided
  • and are constants for all i
  • for all i
  • for all i
  • , where c is a constant and O notates Big O notation
  • for all i
  • is a constant

The asymptotic behavior of T(x) is found by determining the value of p for which and plugging that value into the equation

(see Θ). Intuitively, represents a small perturbation in the index of T. By noting that and that is always between 0 and 1, can be used to ignore the floor function in the index. Similarly, one can also ignore the ceiling function. For example, and will, as per the Akra–Bazzi theorem, have the same asymptotic behavior.

An example[edit]

Suppose is defined as 1 for integers and for integers . In applying the Akra–Bazzi method, the first step is to find the value of p for which . In this example, p = 2. Then, using the formula, the asymptotic behavior can be determined as follows:

Significance[edit]

The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the runtime of many divide-and-conquer algorithms. For example, in the merge sort, the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as and

for integers , and can thus be computed using the Akra–Bazzi method to be .

References[edit]