TADM2E 4.9

1) Sort the sets (which takes $ O(n \log{}n) $) and use the algorithm described in 2) which takes $ O(n) $ (which is also $ O(n \log{}n) $).

2) A and B are sorted (assume in ascending order). The fact that the sets are sorted implies that there is a comparison defined on the elements of A and B (i.e. we can tell whether an element is greater, equal to or smaller than another element).

• let U be the set which will contain the union
• while A and B are not empty:
  • if the first element of A is equal to the last element added to U, remove the first element of A and continue with the next iteration
  • if the first element of B is equal to the last element added to U, remove the first element of B and continue with the next iteration
  • if the first (lowest) element of A is strictly smaller than the first (lowest) element of B, remove the first element of A and add it to U, then continue with the next iteration
  • if the first (lowest) element of A is strictly greater than the first (lowest) element of B, remove the first element of B and add it to U, then continue with the next iteration
  • if the first (lowest) element of A is equal to the first (lowest) element of B, remove the first element from each A and B and add one of them to U, then continue with the next iteration

• After the while loop, either A or B or both are empty. If one of them is non-empty, add its elements to U.