Describe how the number of comparisons used in the worst...

Describe how the number of comparisons used in the worst case changes when the size of the list to be sorted doubles from $n$ to $2 n,$ where $n$ is a positive integer when these sorting algorithms are used. a) bubble sort $\quad$ b) insertion sort c) selection sort (described in the preamble to Exercise 43 in Section 3.1$)$ d) binary insertion sort (described in the preamble to Exercise 49 in Section 3.1 )

Describe how the number of comparisons used in the worst case changes when the size of the list to be sorted doubles from $n$ to $2 n,$ where $n$ is a positive integer when these sorting algorithms are used.
a) bubble sort $\quad$ b) insertion sort
c) selection sort (described in the preamble to Exercise 43 in Section 3.1$)$
d) binary insertion sort (described in the preamble to Exercise 49 in Section 3.1 )

Key Concepts

Scaling Behavior with Input Doubling

Scaling behavior with input doubling involves understanding how the performance of an algorithm changes when the size of the input is multiplied by a constant factor, such as two. This concept is crucial when comparing algorithms because it shows that for quadratic algorithms the cost increases by roughly four times, while for algorithms with n log n complexity the cost increases by a factor that is closer to twice the original, possibly slightly more due to the logarithmic component.

Logarithmic Growth and n log n Complexity

Logarithmic growth, as seen in expressions like O(log n) or more generally O(n log n), describes scenarios where the number of operations increases more slowly compared to quadratic growth. For algorithms that combine linear and logarithmic components, such as binary insertion sort, doubling the input size increases the total operations roughly by a factor of 2 times a term that grows logarithmically, highlighting a more efficient scaling compared to quadratic algorithms.

Time Complexity Analysis

Time complexity analysis is the study of how the running time or number of operations of an algorithm grows as a function of the input size. This concept helps in predicting the efficiency of an algorithm, particularly in worst-case scenarios, by providing theoretical bounds on performance through mathematical expressions such as Big O notation.

Quadratic Time Complexity

Quadratic time complexity, represented as O(n²), describes algorithms for which the number of comparisons (or assignments) grows proportional to the square of the input size. When the size of the input doubles, the number of operations increases by roughly a factor of four. This concept is central to algorithms like bubble sort, insertion sort, and selection sort in their worst-case scenario.

Transcript

00:01 It's clarissa enumerate.

00:02 So for a, we have bubble sort, which successfully compares two adjacent elements and sorts them in decreasing order.

00:10 So let's say the list contains n elements.

00:15 So the first pass, we compare n minus one pairs of adjacent elements.

00:23 So there's n minus one comparisons.

00:25 For the second one, we make n minus two pairs of adjacent elements.

00:29 So there's n minus two comparisons.

00:31 And for the third pass, we compare n minus three pairs of adjacent elements.

00:35 So there's n minus three comparisons.

00:38 So the bubble sort algorithm stops if we make one comparison in some pass, which is as known as the last pass.

00:46 So let's say, like we said, there's going to be n elements.

00:52 So the bubble sort algorithm will stop when it's n minus one elements.

00:57 So the number of comparisons is equal to i'll put number of c for short is equal to n minus 1 plus n minus 2 plus n minus 3 all the way till plus 1, which is then equal to the sum notation of n minus 1, i equals 1 for i.

01:33 And since the sum notation for n minus 1 to n for i is equal to n times n plus 1 over 2, we know the number of comparisons is going to be equal to n minus 1 times n over 2, which is equal to n square minus 2 over 2.

02:04 So if we would then have a list of 2n elements instead, we would have 2n square minus 2 comparisons.

02:14 So we note that the number of comparisons more than doubled and less than quadruples when the size of the list increases from n to 2n.

02:26 So for part b, we have our insertion sort, which compares the first, the second element, with the first and correctly sorts the two numbers.

02:40 So it takes the next element afterwards and correctly inserts it in the part of the string that we know is correctly sorted and so on...

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