Question

I have an array A[] with 4 element A={
8 1 2 4 }. How to sort it with minimized cost. Criteria is defined as follows-

a. It is possible to swap any 2 element.

b. The cost of any swap is sum of the element value , Like if i swap 8 and 4 the cost is 12 an resultant array is look like A={4 1 2 8}, which is still unsorted so more swap needed.

c. Need to find a way to sort the array with minimum cost.

From my observation greedy will not work, like in each step place any element to its sorted position in array with minimum cost. So a DP solution needed.
Can any one help??

Answer 1

Swap 2 and 1, and then 1 and 4, and then 1 and 8? Or is it a general question?

For a more general approach you could try:

1. Swapping every pair of 2 elements (with the highest sum) if they are perfect swaps (i.e. swapping them will put them both at their right spot). Th

2. Use the lowest element as a pivot for swaps (by swapping the element whose spot it occupies), until it reaches its final spot

3. Then, you have two possibilities:

   1. Repeat step 2: use the lowest element not in its final spot as a pivot until it reaches its final spot, then go back to step 3

   2. Or swap the lowest element not in its final spot (l2) with the lowest element (l1), repeat step 2 until l1 reaches the final spot of l2. Then:

      1. Either swap l1 and l2 again, go to step 3.1
      2. Or go to step 3.2 again, with the next lowest element not in its final spot being used.

When all this is done, if some opposite swaps are performed one next to another (for example it could happen from going to step 2. to step 3.2.), remove them.

There are still some things to watch out for, but this is already a pretty good approximation. Step one and two should always work though, step three would be the one to improve in some borderline cases.

Example of the algorithm being used:

With {8 4 5 3 2 7}: (target array {2 3 4 5 7 8})

• Step 2: 2 <> 7, 2 <> 8

• Array is now {2, 4, 5, 3, 7, 8}

• Choice between 3.1 and 3.2:

  • 3.1 gives 3 <> 5, 3 <> 4

  • 3.2 gives 2 <> 3, 2 <> 5, 2 <> 4, 2 <> 3

  • 3 <> 5, 3 <> 4 is the better result

Conclusion: 2 <> 7, 2 <> 8, 3 <> 5, 3 <> 4 is the best answer.

With {1 8 9 7 6} (resulting array {1 6 7 8 9})

• You’re beginning at step three already

• Choice between 3.1 and 3.2:

  • 3.1 gives 6 <> 9, 6 <> 7, 6 <> 8 (total: 42)

  • 3.2 gives 1 <> 6, 1 <> 9, 1 <> 7, 1 <> 8, 1 <> 6 (total: 41)

So 1 <> 6, 1 <> 9, 1 <> 7, 1 <> 8, 1 <> 6 is the best result