Question

Suppose that, in a divide-and-conquer algorithm, we always divide an in stance of size \(n\) of a problem into \(n\) subinstances of size \(n / 3,\) and the dividing and combining steps take lincar time. Write a recurrence equation for the running time \(T(n),\) and solve this recurrence equation for \(T(n) .\) Show your solution in order notation.

Short answer

The exact solution cannot be calculated using common methods such as the Master's theorem due to the unique nature of the equation as it both creates an exponential number of recursive calls while also shrinking the problem size exponentially. It is approximately \(O(n \log n)\).

Step-by-step solution

Write the Recurrence Relation

Based on the problem, since \(n\) subinstances of size \(n / 3\) are created, and the division and combination steps take linear time, the recurrence relation for the running time \(T(n)\) can be written as: \(T(n) = n * T(n / 3) + n\).

Solve the Recurrence

Determine the solution using the Master method. The Master's theorem would be applicable if the equation was in the form, \(T(n) = a * T(n / b) + f(n)\) where \(a \geq 1\) and \(b > 1\). However the equation \(T(n) = n * T(n / 3) + n\) does not conform to the Master's theorem because there is \(n\) sub-problems, instead the size reduces to \(n / 3\) for each one. Due to the unique nature of this equation, there is no immediate direct solution using standard methods such as the Master method.

Determining Running Time and Final Notation

Given that traditional methods such as the Master theorem don't apply in this case, you wouldn't be able to derive an exact solution. Instead, let's determine approximately what the time complexity would be. The function seems to both create an exponential number of recursive calls while also shrinking the problem size exponentially, all while taking linear time to divide/combine. This seems to suggest a possible running time complexity of \(O(n \log n)\). Remember, however, that this is just an approximation, the exact Big O notation cannot be determined without a more sophisticated approach.

Key concepts

Recurrence Relation

A recurrence relation in the context of divide-and-conquer algorithms is a mathematical equation that is used to define the running time of an algorithm by expressing it in terms of the running time of smaller instances of the problem. In our exercise, given that the problem of size n is divided into n subinstances of size n/3, and the dividing and combining steps are said to take linear time, we come up with a recurrence relation:

• \(T(n) = n * T(n / 3) + n\)

The first term, n * T(n / 3), represents the time taken for all subinstances, assuming that the time to solve each subinstance is T(n / 3). The second term, n, accounts for the linear time taken for dividing the problem and combining the results of the subproblems. To improve the understanding of this relation, it's helpful to visualize the recursion tree that this formula implies, where each node at depth d has n / 3^d subinstances. The understanding of such recursion trees is crucial in identifying patterns in the algorithm's behavior, which can eventually lead to a solution for the recurrence relation.

Running Time Complexity

Running time complexity is a measure of the time an algorithm takes to run as a function of the length of the input. In symbolic terms, we generally look to express the complexity of an algorithm using Big O notation, such as O(n), O(n log n), or O(n^2).
For the divide-and-conquer problem referenced, determining the exact running time complexity is non-trivial due to the unusual nature of the recurrence relation. The 'n' recursive calls at each level, combined with the 'divide by three' creates a complexity that doesn't fit standard categories. Following a rough analysis of the recurrence, our initial intuition might lead us to an approximate complexity of O(n log n), considering that the problem size is reduced exponentially while the number of subproblems grows linearly. However, without a more nuanced method, such as an iterated recursion tree or the Akra-Bazzi theorem, we can't pin down the exact complexity. This highlights how critical the understanding of big O concepts is for computer scientists, as it enables them to predict algorithm efficiency in a wide range of scenarios.

Master Theorem

The Master theorem offers a shortcut to finding closed-form solutions to certain kinds of recurrence relations that are common in divide-and-conquer algorithms. The standard form it deals with is:
\(T(n) = a * T(\frac{n}{b}) + f(n)\)
where a is the number of subproblems at each level, b is the factor by which the subproblem size is reduced, and f(n) signifies the extra work done outside the recursive calls (e.g., dividing and combining results).
Unfortunately, our exercise presented a problem where a is equal to n, which breaks the assumptions of the Master theorem, rendering it non-applicable. This exception is an excellent example of how the Master theorem is powerful yet has its limitations. Being familiar with it can vastly ease the problem-solving process, but it also requires an understanding of when and how it is applicable. When dealing with recurrences that don't fit the Master theorem mold, alternative methods and deeper analysis are required.