Minimum Adjacent Swaps to Make a Valid Array

Given an array and a validity condition for its arrangement (e.g., pairing/matching elements or grouping certain values), compute the minimum number of adjacent swaps needed to transform the array into a valid one. The core challenge is to choose target positions for the elements that must move and minimize the total adjacent moves — typically solved by mapping current indices to target indices and optimizing the sum of distances (greedy/median or inversion-count patterns).