Question

Fill in the blanks in each of the following statements: a. A selection sort application would take approximately ________ times as long to run on a 128-element vector as on a 32-element vector. b. The efficiency of merge sort is ________.

Short answer

a. 16 times; b. \( O(n \, \log n) \).

Step-by-step solution

Understanding Selection Sort Complexity

Selection sort is a simple sorting algorithm with a time complexity of \( O(n^2) \), where \( n \) is the number of elements in the array or vector. This means that the running time increases quadratically with the number of elements.

Determine the Factor of Increase

If a 32-element vector requires \( k \) time to sort, then a 128-element vector would require \( k \cdot (128/32)^2 = k \cdot 16 \) times the time due to the quadratic nature of selection sort. Therefore, a selection sort would take approximately 16 times as long to run on a 128-element vector as on a 32-element vector.

Understanding Merge Sort Efficiency

Merge sort is a more efficient sorting algorithm with a time complexity of \( O(n \, \log n) \). This means the algorithm has a logarithmic factor in its growth rate, making it more efficient for larger datasets compared to quadratically scaling algorithms like selection sort.

Key concepts

Selection Sort

Selection sort is a straightforward sorting algorithm that is often used for teaching and learning due to its simplicity. Yet, its efficiency is not optimal for large datasets. Here’s why:Selection sort operates by dividing the list or array into two parts: a sorted subsection and an unsorted subsection. Initially, the sorted subsection is empty:

• The algorithm repeatedly selects the smallest (or largest, depending on order) element from the unsorted portion and swaps it with the first unsorted element.
• This process is repeated, moving gradually through the array until all elements are sorted.

The time complexity of this sorting technique is quadratic, specifically, it is represented by the big O notation as \( O(n^2) \), where \( n \) is the number of items being sorted. For a 32-element vector, the time taken is proportional to \( 32^2 \). However, if you increase the vector to 128 elements, the time needed becomes \( 128^2 \), which is 16 times longer due to the nature of quadratic growth.

Merge Sort

Unlike selection sort, merge sort is known for its efficiency, especially with larger datasets. It is a divide-and-conquer algorithm that:

• Splits the array into two halves.
• Recursively sorts each half.
• Merges the sorted halves back together into a fully sorted array.

The power of merge sort lies in its time complexity, which is \( O(n \log n) \). This complexity arises because the list is divided in half at each step (the \( \log n \) part) and each of the \( n \) elements needs to be considered in the merging process. Thus, it balances the merging and dividing steps efficiently compared to simpler methods. By maintaining a logarithmic growth with each split and merge step, merge sort tends to outperform algorithms like selection sort in scenarios involving large numbers of elements. This makes it a favorite for handling more significant sorting tasks in computer science.

Algorithm Efficiency

Algorithm efficiency is crucial when selecting which sorting technique to implement, particularly in the context of resource limitations and performance requirements. Efficiency is generally measured by the time and space complexity of the algorithm:

• Time complexity refers to the rate at which the computational steps increase as the input size grows. For instance, an algorithm with \( O(n^2) \) time complexity will experience a quadratic increase in steps as more data is added.
• Space complexity considers how much memory the algorithm requires relative to the input size.

In practical terms, more efficient algorithms (like merge sort with \( O(n \log n) \)) will handle larger datasets more effectively and quickly than less efficient ones (such as selection sort). Understanding these efficiencies helps developers make informed decisions about which algorithms will best suit their needs, balancing factors like the size of the input data, the processing power available, and the required speed of the operation.