Usaco Jan12 Gold

Part of USACO Jan12

                           GOLD PROBLEMS
                  Three problems numbered 1 through 3

Problem 1: Video Game Combos [Neal Wu, 2012]

Bessie is playing a video game! In the game, the three letters 'A', 'B',
and 'C' are the only valid buttons. Bessie may press the buttons in any
order she likes; however, there are only N distinct combos possible (1 <= N
<= 20). Combo i is represented as a string S_i which has a length between 1
and 15 and contains only the letters 'A', 'B', and 'C'.

Whenever Bessie presses a combination of letters that matches with a combo,
she gets one point for the combo. Combos may overlap with each other or
even finish at the same time! For example if N = 3 and the three possible
combos are "ABA", "CB", and "ABACB", and Bessie presses "ABACB", she will
end with 3 points. Bessie may score points for a single combo more than once.

Bessie of course wants to earn points as quickly as possible. If she
presses exactly K buttons (1 <= K <= 1,000), what is the maximum number of
points she can earn?



* Line 1: Two space-separated integers: N and K.

* Lines 2..N+1: Line i+1 contains only the string S_i, representing
        combo i.


3 7


* Line 1: A single integer, the maximum number of points Bessie can

SAMPLE OUTPUT (file combos.out):



The optimal sequence of buttons in this case is ABACBCB, which gives 4
points--1 from ABA, 1 from ABACB, and 2 from CB.


Problem 2: Cow Run [Mark Gordon, 2011]

Farmer John and Bessie have devised a new exercise game for the cows.  The
cows are running on a circular track of length M (2 <= M <= 1,000,000,000)
starting from the same position.  The game proceeds in N (1 <= N <= 14)
rounds using a deck of 8N cards each with a number X_i (0 <= X_i < M)
written on it.

Each round FJ moves the top 8 cards into a separate pile and selects either
the top 4 or bottom 4 cards for Bessie to play with.  Bessie then chooses
either the top 2 cards or bottom 2 cards of the 4 cards FJ selected.  After
this FJ calls out the number on the top card, X_top, and the cows run a
distance of R * X_top, where R is the total distance the cows have run so
far.  Bessie then calls out the number on the bottom card, X_bottom, and the
cows run a distance of X_bottom.

FJ is concerned that after the exercise the cows will be too tired to get
back to the beginning of the track if they end up too far away.  He
believes if the cows end up more than a distance of K (0 <= K <=
floor(M/2)) from their starting position they won't be able to get back

It is guaranteed that if FJ plays correctly, he will always be able to 
ensure the cows can come home, irrespective of the moves made by Bessie!
For each round, your task is to determine which half of the cards FJ should
choose such that no matter what Bessie does from that point on, FJ can
always get the cows home.  Bessie will then make the move provided in the
input and you can then continue onto the next round.  Note that even though
Bessie's moves are provided to you in the input, you are to specify moves
for FJ that would have worked no matter what Bessie chooses (so it is
effectively as if FJ does not really know what Bessie will do during her



* Line 1: Three space-separated integers N, M, K

* Line 2: A string N characters.  If the ith character is 'T' it means
        Bessie will select the top 2 cards in the ith round.  Otherwise
        the ith character will be 'B' to indicate Bessie will select
        the bottom 2 cards in the ith round.

* Lines 3..2+N: Each line contains eight integers representing the 8
        cards to be used that round from top to bottom.


2 2 0
1 0 0 0 0 0 0 1
0 1 1 1 0 0 1 0


* Line 1: A string of N characters where the ith character is 'T' if
        FJ should choose the top 4 cards or 'B' if FJ should choose
        the bottom 4 cards in the ith round.  If there are multiple
        ways to get the cows home, choose the lexicographically first
        (that is, the string that is alphabetically smallest).

SAMPLE OUTPUT (file cowrun.out):



The cows must end up exactly where they started to be able to come home.  
Note that FJ is not aware of what choices Bessie is going to make
beforehand.  If FJ did know, he could have chosen the bottom half each time.


Problem 3: Bovine Alliance [Mark Gordon, 2011]

Bessie and her bovine pals from nearby farms have finally decided that they
are going to start connecting their farms together by trails in an effort
to form an alliance against the farmers.  The cows in each of the N (1 <= N
<= 100,000) farms were initially instructed to build a trail to exactly one
other farm, for a total of N trails.  However months into the project only
M (1 <= M < N) of these trails had actually been built.

Arguments between the farms over which farms already built a trail now
threaten to split apart the cow alliance.  To ease tension, Bessie wishes
to calculate how many ways the M trails that exist so far could have been
built.  For example, if there is a trail connecting farms 3 and 4, then one
possibility is that farm 3 built the trail, and the other possibility is
that farm 4 built the trail.  Help Bessie by calculating the number of
different assignments of trails to the farms that built them, modulo
1,000,000,007.  Two assignments are considered different if there is at
least one trail built by a different farm in each assignment.

PROBLEM NAME: alliance


* Line 1: Two space-separated integers N and M

* Lines 2..1+M: Line i+1 describes the ith trail.  Each line contains
        two space-separated integers u_i and v_i (1 <= u_i, v_i <= N,
        u_i != v_i) describing the pair of farms connected by the


5 4
1 2
3 2
4 5
4 5


Note that there can be two trails between the same pair of farms.


* Line 1: A single line containing the number of assignments of trails
        to farms, taken modulo 1,000,000,007.  If no assignment
        satisfies the above conditions output 0.

SAMPLE OUTPUT (file alliance.out):



There are 6 possible assignments.  Letting {a,b,c,d} mean that farm 1
builds trail a, farm 2 builds trail b, farm 3 builds trail c, and farm 4
builds trail d, the assignments are:
{2, 3, 4, 5}
{2, 3, 5, 4}
{1, 3, 4, 5}
{1, 3, 5, 4}
{1, 2, 4, 5}
{1, 2, 5, 4}

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