Question

Write an algorithm that sorts a list of $n$ elements in nonincreasing order by finding the largest and smallest elements and exchanges those elements with the elements in the first and last positions. Then the size of the list is reduced by $2,$ excluding the two elements that are already in the proper positions, and the process is repeated on the remaining part of the list until the entire list is sorted. Analyze your algorithm and show the results using order notation.

Short answer

The algorithm sorts the list in non-increasing order by finding and swapping the smallest and largest elements in the unsorted part of the list. It continues to do so by reducing the unsorted list size by two. The time complexity of this algorithm is $O(n^2)$.

Step-by-step solution

Write the Algorithm

Begin with a list of $n$ elements. The algorithm is as follows:\n\n1. For each unsorted part from $i=0$ till $(n-i-1)$, find both the largest and smallest elements.\n2. Swap the smallest element with the first element in the unsorted part, from the left side. Swap the largest element with the first element in the unsorted part, from the right side.\n3. Move the boundaries of the unsorted part from both sides, and repeat steps 1 and 2 until the entire list is sorted.

Calculate Time Complexity

To analyze the time complexity:\n\n1. The outer loop will run $\frac{n}{2}$ times for each unsorted sub-list.\n2. The inner loop will run in $O(n)$ for finding both smallest and largest elements.\n\nHence, the total time complexity of the algorithm will be $O(\frac{n}{2}\ast n)$ = $O(n^2)$.

Represent in Order Notation

Time complexity in order notation will be $O(n^2)$.

Key concepts

Nonincreasing Order

Sorting a list in nonincreasing order means arranging the elements from largest to smallest. In the context of our discussed algorithm, this involves finding both the largest and the smallest elements in any segment of the list and placing them into their correct positions incrementally.
This process helps organize the list such that as we proceed, the largest elements get pushed towards the start, and the list continues to order itself from there.

• "Nonincreasing" implies that each successive element is less than or equal to the previous one.
• In our algorithm, the recurring task is to continually adjust the list, pulling the highest numbers forward until the entire list meets the nonincreasing criterion.
• This technique ensures that each step brings the partial list closer to completion, ultimately achieving a fully sorted list.

Understanding nonincreasing order is crucial because it influences how sorting algorithms are constructed and what intermediate steps they take to reach the desired outcome.

Time Complexity

Time complexity measures how the runtime of an algorithm scales with the size of the input data. It provides insights into the efficiency of the algorithm.
In the algorithm provided, time complexity is concerned with how many operations are necessary to sort a list of size $n$.

• The main work of the algorithm happens in a loop which executes for every unsorted section of the list.
• The outer loop runs approximately $\frac{n}{2}$ times as it processes two elements (largest and smallest) with each iteration, gradually reducing the list size.
• Within each iteration of this loop, there is another loop that searches through the remaining unsorted elements linearly, leading to $O(n)$ operations per iteration.

When combined, this results in a quadratic time complexity of $O(n^2)$, indicating that the time required grows exponentially with the number of elements.

Order Notation

Order notation, often denoted as "Big O notation", describes an algorithm's performance in terms of the input size $n$. It focuses on the upper limit, or the worst-case scenario, of an algorithm’s execution time.
In our case, the algorithm's time complexity is expressed as $O(n^2)$, which indicates it has a quadratic growth pattern.

• When we express time complexity using order notation, we ignore constants and lower-order terms, concentrating on the dominant behavior as $n$ becomes very large.
• An $O(n^2)$ algorithm suggests that doubling the number of elements results in a fourfold increase in the work done.
• This notation aids in comparing algorithms regardless of variations in hardware or implementation subtleties, as it standardizes the analysis focusing on growth rates.

Thus, order notation is invaluable for predicting scalability and performance.

Algorithm Analysis

Algorithm analysis is the study of algorithms to determine their efficiency and correctness. It involves evaluating an algorithm’s complexity, stability, and resource usage, like time or memory.
The given sorting algorithm must be analyzed for both its correctness in producing a nonincreasing order and its efficiency through measured time complexity.

• Correctness is established by ensuring that every step consistently transforms the list towards being fully sorted.
• Ensuring that the largest and smallest elements are appropriately swapped at each iteration reduces the portion of the unsorted list incrementally, thus confirming correct functionality.
• By analyzing time complexity, we understand that the algorithm is not efficient for very large data sets, emphasizing its quadratic nature $O(n^2)$.

Through analysis, developers can choose the most appropriate algorithm based on situational needs, balancing performance against other constraints.