BubbleSort is a popular but inefficient sorting algorithm. It works by repeatedly swapping adjacent elements that...

 

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BubbleSort is a popular but inefficient sorting algorithm. It works by repeatedly swapping adjacent elements that are out of order, and doing so enough times that there can be no more elements out of order. 1: BUBBLESORT (A[1..n]) 2: for i=1 to n - 1 3: for jn downto i +1 if A[j] <A[j-1] 5: exchange A[j] with A[j - 1] Since BubbleSort never removes or adds items to the array - it merely swaps them in line 5-in order to prove that BUBBLESORT correctly sorts the input array, we only need to show that at the end of the algorithm A[1] ≤ A[2] ≤ A[3] ≤ ≤ A[n-1] ≤ A[n]. You will do this using loop invariants. (a) (10 pts) State precisely the loop invariant for the inner for loop (line 3), and prove that this loop invariant holds. Your proof should use the structure of the loop invariant proof presented in Chapter 2 of CLRS. Hint: To come up with a good loop invariant, think about what exactly the algorithm is doing. (b) (10 pts) Using the termination condition of the loop invariant proved in part (a), state the loop invariant for the outer for loop (line 2) that will allow you to prove the inequality A[1] ≤ A[2] < A[3]... <A[n - 1] A[n]. Again, your proof should use the structure of the loop invariant proof presented in Chapter 2 of CLRS. (c) (10 pts) What is the worst-case running time of BUBBLESORT? Show your work (using summations). How does it compare asymptotically to the running time of InsertionSort? BubbleSort is a popular but inefficient sorting algorithm. It works by repeatedly swapping adjacent elements that are out of order, and doing so enough times that there can be no more elements out of order. 1: BUBBLESORT (A[1..n]) 2: for i=1 to n - 1 3: for jn downto i +1 if A[j] <A[j-1] 5: exchange A[j] with A[j - 1] Since BubbleSort never removes or adds items to the array - it merely swaps them in line 5-in order to prove that BUBBLESORT correctly sorts the input array, we only need to show that at the end of the algorithm A[1] ≤ A[2] ≤ A[3] ≤ ≤ A[n-1] ≤ A[n]. You will do this using loop invariants. (a) (10 pts) State precisely the loop invariant for the inner for loop (line 3), and prove that this loop invariant holds. Your proof should use the structure of the loop invariant proof presented in Chapter 2 of CLRS. Hint: To come up with a good loop invariant, think about what exactly the algorithm is doing. (b) (10 pts) Using the termination condition of the loop invariant proved in part (a), state the loop invariant for the outer for loop (line 2) that will allow you to prove the inequality A[1] ≤ A[2] < A[3]... <A[n - 1] A[n]. Again, your proof should use the structure of the loop invariant proof presented in Chapter 2 of CLRS. (c) (10 pts) What is the worst-case running time of BUBBLESORT? Show your work (using summations). How does it compare asymptotically to the running time of InsertionSort?

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Part a Loop invariant At the start of each iteration of the inner for loop the subarray Ajn consists ... View the full answer