Analysis of Bubble Sort and Loop Invariant N1

Analysis of Bubble Sort and Loop Invariant

N-1 Iteration 1 42 2 35 3 12 4 77 5 5 6 101 1 35 2 12 3 42 4 5 5 77 6 101 1 12 3 5 3 35 4 42 5 77 6 101 1 12 2 35 2 5 4 42 5 77 6 101 1 5 2 12 3 35 4 42 5 77 6 101

Bubble Sort Algorithm Bubble-Sort( A ): To do N-1 iterations for i = 1 … (N – 1): Python Code in http: //www. geekviewpoint. com/python/sorting/bubblesort Outer loop for k = 1 … (N – 1): if A[k] > A[k+1]: Swap( A, k, k+1 ) Inner loop To bubble a value

Bubble Sort Time Complexity • Best-Case Time Complexity – The scenario under which the algorithm will do the least amount of work (finish the fastest) • Worst-Case Time Complexity – The scenario under which the algorithm will do the largest amount of work (finish the slowest)

Bubble Sort Time Complexity Called Linear Time O(N 2) Order-of-N-square • Best-Case Time Complexity – Array is already sorted – (N-1) + (N-2) + (N-3) + …. + (1) = (N-1)* N / 2 comparisons Called Quadratic Time O(N 2) Order-of-N-square • Worst-Case Time Complexity – Need N-1 iterations – (N-1) + (N-2) + (N-3) + …. + (1) = (N-1)* N / 2

Loop Invariant • It is a condition or property that is guaranteed to be correct with each iteration in the loop • Usually used to prove the correctness of the algorithm To do N-1 iterations for i = 1 … (N – 1): Outer loop for k = 1 … (N – 1): if A[k] > A[k+1]: Swap( A, k, k+1 ) Inner loop To bubble a value

Loop Invariant for Bubble Sort To do N-1 iterations for i = 1 … (N – 1): • By the end of iteration i the right-most i items (largest) are sorted and in place Outer loop for k = 1 … (N – 1): if A[k] > A[k+1]: Swap( A, k, k+1 ) Inner loop To bubble a value

N-1 Iterations 1 42 2 35 3 12 4 77 5 5 6 101 1 st 1 35 2 12 3 42 4 5 5 77 6 101 2 nd 1 12 3 5 3 35 4 42 5 77 6 101 3 rd 1 12 2 35 2 5 4 42 5 77 6 101 4 th 1 5 2 12 3 35 4 42 5 77 6 101 5 th

Correctness of Bubble Sort (using Loop Invariant) • Bubble sort has N-1 Iterations • Invariant: By the end of iteration i the right-most i items (largest) are sorted and in place • Then: After the N-1 iterations The right-most N-1 items are sorted – This implies that all the N items are sorted