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Chapter 11: Problem 11

Write a sequential algorithm that implements the Tournament method to find the largest key in an array of \(n\) keys. Show that this algorithm is no more efficient than the standard sequential algorithm.

Short Answer

Expert verified
The Tournament method for finding the largest key in an array is not more efficient than the standard sequential search. Both have a time complexity of \ (O(n)\).

Step by step solution

01

Understanding the Tournament Method

The tournament method is a method used for finding the maximum (or minimum) value in an array. It pits pairs of elements against each other in a tournament-style setting, such that in each 'round', the larger elements from each pair advance to the next round. This continues until the 'final round' where the largest element becomes the 'winner'. This is implemented by splitting the array into pairs of elements, comparing each pair, and sending the largest value of each pair to the next round.
02

Implementing the Tournament Algorithm

To implement the Tournament algorithm, start by comparing each set of two elements from the array. Store the larger value from each pair in a new array. Repeat this process with the new array until only one element, the maximum key, remains. This is the winner or the largest key in the array. The pseudocode for this algorithm would look something like this: pseudocode: 1. initialize an empty array to hold the 'winners' 2. while the array has more than 1 element: - create a temporary array - for each pair in the array: - find the larger value - add the larger value to the temporary array - replace the original array with the temporary array
03

Comparing Efficiency with Sequential Search

Sequential search will scan each element in the array once, thus it has a time complexity of \ (O(n)\). The Tournament method, however, although involves multiple rounds, still compares each pair only once. Therefore, the Tournament method also has a time complexity of \ (O(n)\). Thus, the efficiency of both methods in terms of time complexity is equal.

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Most popular questions from this chapter

Consider the problem of adding \(n\) numbers in a list. If it takes \(t_{a}\) time for one person to add two numbers and it takes no time to pass the result of an addition from one person to another, how many people do we need to minimize the total time spent to get the final answer? What will be the minimum amount of time needed to find the answer, if we assume we have enough people? Justify your answer.

If we have two people add \(n\) numbers in a list and it takes \(t_{a}\) time for one person to add two numbers, how long will it take the two people to add all \(n\) numbers in the list, if we consider the operation of addition as the basic operation and include \(t_{p}\) time for passing the result of an addition from one person to the other? Justify your answer.

Write a CREW PRAM algorithm for adding all \(n\) numbers in a list in \(\theta(\lg n)\) time.

Consider the proble of adding two \(n \times n\) matrices. If it takes \(t_{a}\) time for one person to add two numbers, how many people do we need to minimize the total time spent to get the final answer? What will be the minimum amount of time needed to find the answer, if we assume that we have enough people? Justify your answers.

Write a CREW PRAM algorithm for determining for each element of an \(n\) -element linked list if it is the middle ( \([n / 2]\) th ) element in \(\theta\) ( \(\lg n\) ) time.
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