ACM International Collegiate
Programming Contest, Asia-Amritapuri Site, 2011
Problem D: Wizarding Duel
You are the organizer of a Wizarding Duel tournament at Hogwarts. N
players participated in the tournament and each player played with
every other player exactly once, and each such game resulted in a
winner and a loser (no drawn games). It is now time to declare the
results. Each player's score is the number of games the player won.
However, you realize that something possibly went wrong with the
scoring system, and the scores that you have noted down might be wrong.
In fact, the scores you have noted down could simply not be possible.
For example, suppose there are 3 players and the scores that you noted
are 0,0 and 2. Clearly this is not possible as the game between the
first two players must have had a winner and thus both the players
cannot have score 0.
While you have no way to figure out the correct scores of each player,
you've decided to set the scores right by adjusting the scores of some
players. The new set of scores should be possible to have arisen in the
tournament, otherwise there would be ample proof that the scoring is
wrong. However, making drastic changes might cause suspicion. So you've
decided to adjust the scores so that the sum of the absolute
differences between the old and the new scores of each player is
minimized. In other words, if the original scores which you noted are
a1,..,aN, you must change them to the series of possible scores
b1,...bN such that the sum |ai - bi| is minimized.
Input (STDIN):
The first line contains the number of test cases T. T test cases
follow. Each case contains an integer N on the first line, followed by
the set of scores which you have noted down: a1..aN.
Output (STDOUT):
Output T lines, one for each test case, containing the minimum sum of
absolute values in order to make the scorecard a valid one.
Constraints:
1 <= T <= 20
2 <= N <= 50
0 <= ai <= 100
Time Limit: 3 seconds
Memory Limit: 64 MB
Sample Input:
2
3
0 0 2
5
5 3 2 1 4
Sample Output:
1
5