Usaco Feb08 Gold

Part of USACO Feb08

                           GOLD PROBLEMS
                  Three problems numbered 1 through 3

Problem 1: Making the Grade [Brian Dean, 2008]

A straight dirt road connects two fields on FJ's farm, but it changes
elevation more than FJ would like. His cows do not mind climbing
up or down a single slope, but they are not fond of an alternating
succession of hills and valleys. FJ would like to add and remove
dirt from the road so that it becomes one monotonic slope (either
sloping up or down).

You are given N integers A_1, . . . , A_N (1 <= N <= 2,000) describing
the elevation (0 <= A_i <= 1,000,000,000) at each of N equally-spaced
positions along the road, starting at the first field and ending
at the other. FJ would like to adjust these elevations to a new
sequence B_1, . . . , B_N that is either nonincreasing or nondecreasing.
Since it costs the same amount of money to add or remove dirt at
any position along the road, the total cost of modifying the road

         |A_1 - B_1| + |A_2 - B_2| + ... + |A_N - B_N|

Please compute the minimum cost of grading his road so it becomes
a continuous slope. FJ happily informs you that signed 32-bit
integers can certainly be used to compute the answer.



* Line 1: A single integer: N

* Lines 2..N+1: Line i+1 contains a single integer elevation: A_i




* Line 1: A single integer that is the minimum cost for FJ to grade
        his dirt road so it becomes nonincreasing or nondecreasing in

SAMPLE OUTPUT (file grading.out):



By changing the first 3 to 2 and the second 3 to 5 for a total cost of
|2-3|+|5-3| = 3 we get the nondecreasing sequence 1,2,2,4,5,5,9.


Problem 2: Cow Game [Shravas Rao, 2007]

The cows have just discovered a new game to play with Farmer John.

Each of the N (2 <= N <= 50,000) cows is standing on a lattice point
of a grid; Cow 1 is standing at the point (0,0). The cows then tell
John the distance in the X-coordinate (0 <= X_i <= 1,000,000) and
Y-coordinate (0 <= Y_i <= 1,000,000) from Cow 1 to Cow 2.

For example, if the distance from Cow 1 to Cow 2 was given as (3,
2), then Cow 2 could be at the points (3, 2), (-3, 2), (3, -2), or
(-3, -2), since Cow 1 is at the point (0, 0). This continues
successively from cow to cow until the cows report the distance
from Cow N-1 to Cow N.

Farmer John wants your help to minimize the smallest possible sum
of the two numbers in the distance from Cow N to Cow 1.

This is an output-only problem. For each case, it is possible for
Cow N to be on the same spot with Cow 1. Should you produce an
answer where the total distance from Cow 1 to Cow N is x, you will
receive a integer version of (1000-x)/1000 of the points for that
test case. Each test case begins with CASE (1 <= CASE <= 25), the
test case ID number.

Input data are in a zip file.



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

* Lines 2..N: Line i+1 contains two space-separated integers that give
        the distance from Cow i to Cow i+1: X_i and Y_i


1 5
2 6
3 1
1 4
4 3


There are 5 cows. The distance from Cow 1 to Cow 2 is (2, 6), the
distance from Cow 2 to Cow 3 is (3, 1), the distance from Cow 3 to
Cow 4 is (1, 4) and the distance from Cow 4 to Cow 5 is (4, 3).


* Line 1: A line like this where ID is the case number: #FILE cgame ID

* Lines 2..N+1: The locations of each of the cows to minimize this
        sum. Line i+1 should contain the x and y coordinates of cow i
        in that order.

SAMPLE OUTPUT (file cgame.out):

#FILE cgame 1
0 0
2 -6
5 -7
4 -3
0 0


This case receives full points since it gives a valid assignment
and the locations of cows 1 and N overlap.


Problem 3: Hotel [Zhou Yuan, 2007]

The cows are journeying north to Thunder Bay in Canada to gain
cultural enrichment and enjoy a vacation on the sunny shores of
Lake Superior. Bessie, ever the competent travel agent, has named
the Bullmoose Hotel on famed Cumberland Street as their vacation
residence. This immense hotel has N (1 <= N <= 50,000) rooms all
located on the same side of an extremely long hallway (all the
better to see the lake, of course).

The cows and other visitors arrive in groups of size D_i (1 <= D_i
<= N) and approach the front desk to check in. Each group i requests
a set of D_i contiguous rooms from Canmuu, the moose staffing the
counter.  He assigns them some set of consecutive room numbers
r..r+D_i-1 if they are available or, if no contiguous set of rooms
is available, politely suggests alternate lodging. Canmuu always
chooses the value of r to be the smallest possible.

Visitors also depart the hotel from groups of contiguous rooms.
Checkout i has the parameters X_i and D_i which specify the vacating
of rooms X_i..X_i+D_i-1 (1 <= X_i <= N-D_i+1). Some (or all) of
those rooms might be empty before the checkout.

Your job is to assist Canmuu by processing M (1 <= M < 50,000)
checkin/checkout requests. The hotel is initially unoccupied.



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

* Lines 2..M+1: Line i+1 contains request expressed as one of two
        possible formats: (a) Two space separated integers
        representing a check-in request: 1 and D_i (b) Three
        space-separated integers representing a check-out: 2, X_i, and


10 6
1 3
1 3
1 3
1 3
2 5 5
1 6


* Lines 1.....: For each check-in request, output a single line with a
        single integer r, the first room in the contiguous sequence of
        rooms to be occupied. If the request cannot be satisfied,
        output 0.

SAMPLE OUTPUT (file hotel.out):


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