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Chapter 7: Problem 48

Suppose we are to find the \(k\) smallest elements in a list of \(n\) elements, and we are not interested in their relative order. Can a linear-time algorithm be found when \(k\) is a constant? Justify your answer.

Short Answer

Expert verified
Yes, a linear-time algorithm can be found when \(k\) is constant to find \(k\) smallest elements in a list of \(n\) elements. This is because the constant \(k\) does not change the time complexity from \(O(n)\).

Step by step solution

01

Understanding the problem and time complexity

We need to determine if we can find the \(k\) smallest elements in a list of \(n\) elements in linear time. Linear time means that the time it takes to complete an operation, scales at most linearly with the size of the input. This is usually represented as \(O(n)\), where \(n\) is the size of the input.
02

Considering the value of k

In this case, \(k\) represents the number of smallest elements we want to find, and it is stated that \(k\) is a constant. This means that the value of \(k\) doesn't grow with the input size \(n\), and doesn't really affect the time complexity when we consider large input sizes.
03

Evaluating the impact of k on the time complexity

Even though we are searching for \(k\) elements, the fact that \(k\) is a constant plays an important role. Because \(k\) is constant, the time complexity wouldn't change to \(O(nk)\), but remain \(O(n)\) because we are effectively still traversing the list once or a constant number \(k\) of times, and this constant is ignored when calculating time complexity.
04

Concluding

Therefore, even when we are searching for multiple (\(k\)) elements in a list, if \(k\) is a constant, it's possible to find them with a linear time algorithm. The time complexity remains \(O(n)\), as we're simply traversing the list a constant number of times.

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Most popular questions from this chapter

Write an algorithm that checks if an essentially complete binary tree is a heap. Analyze your algorithm, and show the results using order notation.

Among Selection Sort, Insertion Sort, Mergesort, Quicksort, and Heapsort, which algorithm would you choose in each list-sorting situation below? Justify your answers. (a) The list has several hundred records. The records are quite long, but the keys are very short. (b) The list has about 45,000 records. It is necessary that the sort be completed reasonably quickly in all cases. There is barely enough memory to hold the 45,000 records. (c) The list has about 45,000 records, but it starts off only slightly out of order. (d) The list has about 25,000 records. It is desirable to complete the sort as quickly as possible on the average, but it is not critical that the sort be completed quickly in every single case.

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