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Chapter 2: Problem 16

Suppose that, in a divide-and-conquer algorithm, we always divide an in stance of size \(n\) of a problem into 10 subinstances of size \(n / 3,\) and the dividing and combining steps take a time in \(\Theta\left(n^{2}\right)\). Write a recumence equation for the running time \(T(n),\) and solve the equation for \(T(n)\).

Short Answer

Expert verified
The running time T(n) of the algorithm is \(\Theta(n^{2}log(n))\).

Step by step solution

01

Identify the Dividing and Combining Operation

The problem states that the algorithm is a divide-and-conquer variety, with a problem of size n being divided into 10 subproblems of size n/3. This means we can say the division operation happens with 10 cases each of size n/3. Also, it further asserts that the divide and combine steps take \(\Theta\left(n^{2}\right)\) time.
02

Define the Recurrence Relation

Combining the above information, we formulate the recurrence relation for the run-time T(n) of the algorithm as \(T(n) = 10T(n/3) + \Theta\left(n^{2}\right)\). This equation is made up of two parts, the first term 10T(n/3) represents the time taken to solve the 10 subproblems of size n/3 and the second term \(\Theta\left(n^{2}\right)\) represents time taken to divide and later combine the solutions.
03

Solve the Recurrence Relation

The solution to the recurrence relation is obtained using the Master Theorem. The Master Theorem applies to recurrences of the form \(T(n) = aT(n/b) + f(n)\). In this case, a=10, b=3, and f(n) = \(\Theta\left(n^{2}\right)\). Comparing this with \(n^{log_{b}{a}} = n^{log_{3}{10}}\), it is seen that \(f(n)\) grows faster. Therefore, the time complexity T(n) belongs to \(\Theta(n^{2}log(n))\).

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Most popular questions from this chapter

Write for the following problem a recursive algorithm whose worst-case time complexity is not worse than \(\Theta(n \lg n)\). Given a list of \(n\) distinct positive integers, partition the list into two sublists, cach of size \(n / 2,\) such that the difference between the sums of the integers in the two sublists is maximized. You may assume that \(n\) is a multiple of 2.

Suppose that, even unrealistically, we are to search a list of 700 million items using Binary Search (Algorithm 2.1). What is the maximum number of com parisons that this algorithm must perform before finding a given item or concluding that it is not in the list?

Write an algorithm that sorts a list of \(n\) items by dividing it into three sublists or almost \(n / 3\) items, sorting each sublist recursively and merging the three sorted sublists. Analyze your algorithm, and give the results using order notation.

Use Binary Search (Algorithm 2.1 ) to search for the integer 120 in the following list (array) of integers. Show the actions step by step. $$\begin{array}{lllllllll} 12 & 34 & 37 & 45 & 57 & 82 & 99 & 120 & 134 \end{array}$$

Assuming that Quicksort uses the first item in the list as the pivot item: (a) Give a list of \(n\) items (for example, an array of 10 integers) representing the worst-case scenario. (b) Give a list of \(n\) items (for example, an array of 10 integers) representing the best-case scenario.
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