Usaco Dec07 Gold

Part of USACO Dec07

                           GOLD PROBLEMS
                  Three problems numbered 1 through 3

Problem 1: Sightseeing Cows [Reid Barton, 2007]

Farmer John has decided to reward his cows for their hard work by
taking them on a tour of the big city! The cows must decide how
best to spend their free time.

Fortunately, they have a detailed city map showing the L (2 <= L
<= 1000) major landmarks (conveniently numbered 1.. L) and the P
(2 <= P <= 5000) unidirectional cow paths that join them. Farmer
John will drive the cows to a starting landmark of their choice,
from which they will walk along the cow paths to a series of other
landmarks, ending back at their starting landmark where Farmer John
will pick them up and take them back to the  farm. Because space
in the city is at a premium, the cow paths are very narrow and so
travel along each cow path is only allowed in one fixed direction.

While the cows may spend as much time as they like in the city,
they do tend to get bored easily. Visiting each new landmark is
fun, but walking between them takes time. The cows know the exact
fun values F_i (1 <= F_i <= 1000) for each landmark i.

The cows also know about the cowpaths. Cowpath i connects landmark
L1_i to L2_i (in the direction L1_i -> L2_i) and requires time T_i
(1 <= T_i <= 1000) to traverse.

In order to have the best possible day off, the cows want to maximize
the average fun value per unit time of their trip.  Of course, the
landmarks are only fun the first time they are visited; the cows
may pass through the landmark more than once, but they do not
perceive its fun value again. Furthermore, Farmer John is making
the cows visit at least two landmarks, so that they get some exercise
during their day off.

Help the cows find the maximum fun value per unit time that they
can achieve.

PROBLEM NAME: sightsee


* Line 1: Two space-separated integers: L and P

* Lines 2..L+1: Line i+1 contains a single one integer: F_i

* Lines L+2..L+P+1: Line L+i+1 describes cow path i with three
        space-separated integers: L1_i, L2_i, and T_i


5 7
1 2 3
2 3 2
3 4 5
3 5 2
4 5 5
5 1 3
5 2 2


* Line 1: A single number given to two decimal places (do not perform
        explicit rounding), the maximum possible average fun per unit
        time, or 0 if the cows cannot plan any trip at all in
        accordance with the above rules.

SAMPLE OUTPUT (file sightsee.out):



The trip 1 -> 2 -> 3 -> 5 -> 1 has a total fun value of 60 and a
length of 10 units for an average fun per unit time of 6. The trip
2 -> 3 -> 5 -> 2 only has an average fun per unit time of 30/6 =
5, and any trip involving landmark 4 has an average fun per unit
time of less than 4.


Problem 2: Gourmet Grazers [Alex Schwendner, 2007]

Like so many others, the cows have developed very haughty tastes
and will no longer graze on just any grass. Instead, Farmer John
must purchase gourmet organic grass at the Green Grass Grocers store
for each of his N (1 <= N <= 100,000) cows.

Each cow_i demands grass of price at least A_i (1 <= A_i <=
1,000,000,000) and with a greenness score at least B_i (1 <= B_i
<= 1,000,000,000). The GGG store has M (1 <= M <= 100,000) different
types of grass available, each with a price C_i (1 <= C_i <=
1,000,000,000) and a greenness score of D_i (1 <= D_i <= 1,000,000,000).
Of course, no cow would sacrifice her individuality, so no two cows
can have the same kind of grass.

Help Farmer John satisfy the cows' expensive gourmet tastes while
spending as little money as is necessary.



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

* Lines 2..N+1: Line i+1 contains two space-separated integers: A_i
        and B_i

* Lines N+2..N+M+1: Line i+N+1 contains two space-separated integers:
        C_i and D_i


4 7
1 1
2 3
1 4
4 2
3 2
2 1
4 3
5 2
5 4
2 6
4 4


* Line 1: A single integer which is the minimum cost to satisfy all
        the cows. If that is not possible, output -1.

SAMPLE OUTPUT (file gourmet.out):



Cow 1 eats grass 2 at cost 2, cow 2 eats grass 3 at cost 4, cow 3 eats
grass 6 at cost 2, and cow 4 eats grass 7 at cost 4, for a total cost of 12.


Problem 3: Best Cow Line, Gold [Christos Tzamos, 2007]

FJ is about to take his N (1 <= N <= 30,000) cows to the annual
"Farmer of the Year" competition. In this contest every farmer
arranges his cows in a line and herds them past the judges.

The contest organizers adopted a new registration scheme this year:
simply register the initial letter of every cow in the order they
will appear (e.g., If FJ takes Bessie, Sylvia, and Dora in that
order, he just registers BSD). After the registration phase ends,
every group is judged in increasing lexicographic order (i.e.,
alphabetical order) according to the string of the initials of the
cows' names.

FJ is very busy this year and has to hurry back to his farm, so he
wants to be judged as early as possible. He decides to rearrange
his cows, who have already lined up, before registering them.

FJ marks a location for a new line of the competing cows. He then
proceeds to marshal the cows from the old line to the new one by
repeatedly sending either the first or last cow in the (remainder
of the) original line to the end of the new line. When he's finished,
FJ takes his cows for registration in this new order.

Given the initial order of his cows, determine the least lexicographic
string of initials he can make this way.



* Line 1: A single integer: N

* Lines 2..N+1: Line i+1 contains a single initial ('A'..'Z') of the
        cow in the ith position in the original line




FJ has 6 cows in this order: ACDBCB


The least lexicographic string he can make. Every line (except perhaps
the last one) contains the initials of 80 cows ('A'..'Z') in the new

SAMPLE OUTPUT (file bclgold.out):



  Step   Original     New
   #1     ACDBCB
   #2      CDBCB     A
   #3      CDBC      AB
   #4      CDB       ABC
   #5      CD        ABCB
   #6       D        ABCBC
   #7                ABCBCD

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License