Measures of Sortedness

Often it useful to talk about measures of list sortedness. Such a distance measure should assign a score of zero to a sorted list, and any unsorted list should score a positive amount.

Some measures:

• Unsortedp: Is the list completely sorted? Score 1 if no.
• Items out of place: Score 1 for each position that contains an element other than what would be there in a sorted list. Maximum distance is n.
• Inversions: Score 1 for every pair that is out of order. Maximum distance is n(n-1)/2.
• Adjacent inversions: Score 1 for every adjacent pair that is out of order. Maximum distance is n-1.
• Insertion index: Find the longest increasing subsequence (LIS). Score n minus the length of the LIS. The same as asking, what's the smallest number of keys I can remove and the reinsert to get a sorted list? Maximum distance is n-1.

Example:

• 2 1 6 3 5 4 7
• unsortedp: 1
• items out of place: 5.
• inversions: 1+0+3+0+1=5.
• adjacent inversions: 3.
• insertion index: 7-4=3.