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Chapter 2: Problem 7

Use the divide-and-conquer approach to write an algorithm that finds the largest item in a list of \(n\) items. Analyze your algorithm, and show the results in order notation.

Short Answer

Expert verified
The algorithm employs a divide-and-conquer strategy to find the maximum number in a list. It recursively divides the list until one element – which is the maximum of a list of one – remains. It then combines results by comparing and returning the larger of the two elements. The time complexity, evaluated using the Master's theorem, is \(O(n)\), where \(n\) represents the list size.

Step by step solution

01

Understand and Apply the Divide-and-Conquer Approach

Begin by dividing the list into two halves. This is the 'divide' part of the approach. We continue dividing the sublists into two until there is only one element left in each sublist. At this point, it can be stated that a single item is in itself the largest item of a 'list' of one.
02

Recursive Function

Create a recursive function, typically called 'find_max()', that uses the divide-and-conquer approach. The function 'find_max()' takes two parameters: the list and the size of the list. If the size is 1, it returns the single element as the maximum. Else, it calls itself, each time with one half of the list.
03

Conquer

Now, we 'conquer' by combining the solutions in this case. We compare the largest elements of the two sublists (returned by our 'find_max()' function), and return the greater one. This process continues recursively till we get the maximum of the entire list.
04

Time Complexity Analysis

The time complexity of this divide-and-conquer algorithm can be represented by the recurrence relation \(T(n) = 2T(n/2) + 1\). From the Master's theorem, we know that the solution to such a recurrence relation is \(O(n)\), where \(n\) represents the list size.

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Most popular questions from this chapter

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 \(T(n)\).

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. $$\begin{array}{lllllllll} 12 & 34 & 37 & 45 & 57 & 82 & 99 & 120 & 134 \end{array}$$

How many multiplications would be performed in finding the product of two \(64 \times 64\) matrices using the standard algorithm?

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.

Write algorithms that perform the operations \(u \times 10^{m}\) \(u\) divide \(10^{n}\) \(u\) rem \(10 "\) where \(u\) represents a large integer, \(m\) is a nonnegative integer, divide returns the quotient in integer division, and rem returns the remainder. Analyze your algorithms, and show that these operations can be done in linear time.
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