Here is a famous story in Chinese history.
"That
was about 2300 years ago. General Tian Ji was a high official in the country Qi.
He likes to play horse racing with the king and others."
"Both of Tian
and the king have three horses in different classes, namely, regular, plus, and
super. The rule is to have three rounds in a match; each of the horses must be
used in one round. The winner of a single round takes two hundred silver dollars
from the loser."
"Being the most powerful man in the country, the king
has so nice horses that in each class his horse is better than Tian's. As a
result, each time the king takes six hundred silver dollars from
Tian."
"Tian Ji was not happy about that, until he met Sun Bin, one of
the most famous generals in Chinese history. Using a little trick due to Sun,
Tian Ji brought home two hundred silver dollars and such a grace in the next
match."
"It was a rather simple trick. Using his regular class horse race
against the super class from the king, they will certainly lose that round. But
then his plus beat the king's regular, and his super beat the king's plus. What
a simple trick. And how do you think of Tian Ji, the high ranked official in
China?"
Were Tian Ji lives in
nowadays, he will certainly laugh at himself. Even more, were he sitting in the
ACM contest right now, he may discover that the horse racing problem can be
simply viewed as finding the maximum matching in a bipartite graph. Draw Tian's
horses on one side, and the king's horses on the other. Whenever one of Tian's
horses can beat one from the king, we draw an edge between them, meaning we wish
to establish this pair. Then, the problem of winning as many rounds as possible
is just to find the maximum matching in this graph. If there are ties, the
problem becomes more complicated, he needs to assign weights 0, 1, or -1 to all
the possible edges, and find a maximum weighted perfect
matching...
However, the horse racing problem is a very special case of
bipartite matching. The graph is decided by the speed of the horses --- a vertex
of higher speed always beat a vertex of lower speed. In this case, the weighted
bipartite matching algorithm is a too advanced tool to deal with the
problem.
In this problem, you are asked to write a program to solve this
special case of matching problem.
