Inversions from Sorting with Distance-Based Errors

We study the number of inversions after running the Insertion Sort or Quicksort algorithm, when errors in the comparisons occur with some probability. We investigate the case in which probabilities depend on the difference between the two numbers to be compared and only differences up to some threshold τ are prone to errors. We give upper bounds for this model and show that for constant τ , the expected number of inversions is linear in the number of elements to be sorted. For Insertion Sort, we also yield an upper bound on the expected number of runs, i.e., the number of consecutive increasing subsequences.