Larry has been given a permutation of a sequence of natural numbers incrementing from
1 as an array. He must determine whether the array can be sorted using the following operation any number of times:
- Choose any consecutive indices and rotate their elements in such a way that
ABC -> BCA -> CAB -> ABC.
For example, if :
A rotate [1,6,5,2,4,3] [6,5,2] [1,5,2,6,4,3] [5,2,6] [1,2,6,5,4,3] [5,4,3] [1,2,6,3,5,4] [6,3,5] [1,2,3,5,6,4] [5,6,4] [1,2,3,4,5,6] YES
On a new line for each test case, print
A can be fully sorted. Otherwise, print
I’m providing the solution for Python and JS, please leave on the comments if you found a better way.
That I could think of, there are 2 approaches to solve this problem. The first one being the obvious, doing all the permutations and evaluating at the end when 2 items are left if they are in order.
The other way, which happens I knew it already, has to do with inversion . Inversions are particularly interesting for this problem. Now if you don’t know about inversions, is very unlikely to find this solution on your own.
By definition, The inversion number is the cardinality of inversion set. It is a common measure of the sortedness of a permutation or sequence.
What’s an interesting property of the inversion number is that the order of the permutation won’t actually affect its even/odd nature for the entire array.
Here is that demonstrated with a few examples:
Example 1: 312 inversion(312) = 2 312 -> 123 = sorted! Example 2: 231 inversion(231) = 2 231 -> 312 -> 123 = sorted! Example 3: 213 inversion(231) = 1 213 -> 132 -> 321 -> 213 != can't be sorted
So we can say that only those combinations with even inversion number can be sorted, and that’s what we are going for.