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Chapter 10: Problem 11

Write a scquential 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 works by successively pairing elements and picking the highest, thereby eventually finding the maximal element in the array. A python code for this algorithm is provided. However, in terms of efficiency, it does not surpasses the standard sequential algorithm for finding the maximal key. Both methods require roughly \(n-1\) comparisons.

Step by step solution

01

Define the tournament method

Tournament algorithm involves a series of comparisons where each element competes with its neighbor. The larger number progresses to the next round of comparisons until one number is left which is the largest. For an array \(A\) of \(n\) elements, firstly pair up the elements and compare. Repeat the process with the 'winners' of the previous round until one is left. In the end, the single remaining number is the largest.
02

Code the algorithm

Here is a python version of this algorithm.\n\n```\ndef Tournament(A):\n n = len(A)\n while n > 1:\n if n % 2 != 0:\n A.append(A[-1])\n A = [max(A[i], A[i+1]) for i in range(0, len(A), 2)]\n n = len(A)\n return A[0]\n```\nThis function gets the input list, does the comparisons until there is one number left in the list and return this number as the largest key in the array.
03

Comparison with standard sequential algorithm

In the standard sequential search for the largest number, we would simply iterate through each number in the list once. It will take \(n-1\) comparisons where \(n\) is the length of the list. For the Tournament method, consider the total number of comparisons: in each round of tournament, about half of the numbers are eliminated. This gives us roughly \(n-1\) comparisons too. So, everywhere one looks at it, the Tournament method does not have fewer steps.

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