Usaco Jan11 Silver

Part of USACO Jan11

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SILVER PROBLEMS
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Three problems numbered 6 through 8
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Problem 6: Profits [Neal Wu, 2007]

The cows have opened a new business, and Farmer John wants to see
how well they are doing. The business has been running for N (1 <=
N <= 100,000) days, and every day i the cows recorded their net
profit P_i (-1,000 <= P_i <= 1,000).

Farmer John wants to find the largest total profit that the cows
have made during any consecutive time period. (Note that a consecutive
time period can range in length from one day through N days.) Help
him by writing a program to calculate the largest sum of consecutive
profits.

PROBLEM NAME: profits

INPUT FORMAT:

* Line 1: A single integer: N

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

SAMPLE INPUT (file profits.in):

7
-3
4
9
-2
-5
8
-3

OUTPUT FORMAT:

* Line 1: A single integer representing the value of the maximum sum
of profits for any consecutive time period.

SAMPLE OUTPUT (file profits.out):

14

OUTPUT DETAILS:

The maximum sum is obtained by taking the sum from the second through
the sixth number (4, 9, -2, -5, 8) => 14.

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Problem 7: Dividing the Gold [Sherry Wu, a classic, 2010]

Bessie and Canmuu found a sack of N (1 <= N <= 250) gold coins
that they wish to divide as evenly as possible. Coin i has value
v_i (1 <= V_i <= 2,000). The cows would like to split the pile as
evenly as they can, but that is not always possible. What is the
smallest difference between the values of the two piles?

In addition, the Bessie and Canmuu have found that there might be
multiple ways to split the piles with that minimum difference. They
would also like to know the number of ways to split the coins as
fairly as possible. If it isn't possible to split the piles evenly, Bessie
will get the higher-valued pile.

By way of example, consider a sack of five coins of values: 2, 1,
8, 4, and 16. Bessie and Canmuu split the coins into two piles, one
pile with one coin worth 16, and the other pile with the remaining
coins worth 1+2+4+8=15. Therefore the difference is 16-15 = 1.  This
is the only way for them to split the coins this way, so the number
of ways to split it evenly is just 1.

Note that same-valued coins can be switched among the piles to
increase the number of ways to perform an optimal split. Thus, the
set of coins {1, 1, 1, 1} has six different ways to split into two
optimal partitions, each with two coins.

PROBLEM NAME: divgold

INPUT FORMAT:

* Line 1: A single integer: N

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

SAMPLE INPUT (file divgold.in):

5
2
1
8
4
16

OUTPUT FORMAT:

* Line 1: A single integer that is the smallest difference of two
partitions.

* Line 2: A single integer that is the number of ways to split the
coins with the minimum difference printed in line 1. Since
this number can get quite large, print the remainder when
divided by 1,000,000.

SAMPLE OUTPUT (file divgold.out):

1
1

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Problem 8: Traffic Lights [Adopted from IOI'99, 2008]

Kenosha, the city nearest Farmer John, has M (1 <= M <= 14,000)
roads conveniently numbered 1..M that connect N (2 <= N <= 300)
junctions which are conveniently numbered 1..N. No two roads connect
the same pair of junctions. No road connects a junction to itself.
The integer travel time T_ij (1 <= T_ij <= 100) between junctions
i and j is the same for both directions (i.e., T_ij = T_ji).

Each junction has a single traffic light with two colors: blue or
purple. The color of each light alternates periodically: blue for
certain duration and then purple for another duration.  Traffic is
permitted to commence travel down the road between any two junctions,
if and only if the lights at both junctions are the same color at
the moment of departing from one junction for the other.  The lights
do not necessarily have to be the same on the whole trip down the

If a vehicle arrives at a junction just at the moment the lights
switch it must consider the new colors of lights. Vehicles are
allowed to wait at the junctions. You are given the city map which
shows:
* The travel times T_ij for all roads
* The durations of the two colors at junction i. (DB_i (1 <=
DB_i <= 100) for the blue light and DP_i (1 <= DP_i <= 100)
for the purple light)
* The initial color C_i of the light at junction i (a letter
'B' or 'P' with the obvious meaning) and the remaining time
R_i (1 <= R_i <= 100) for this color to change

Find the minimum time one needs to get from a given source S (1 <=
S <= N) to a given destination D (1 <= D <= N; D != S).

Consider the map below with four junctions and five roads. FJ wants
to travel from junction 1 to junction 4. The first light is blue;
the rest are purple.
Init  Remg  Blue   Purple
4       76         Junction  Color Time  Cycle  Cycle
>>[1B]===[2P]======          1        B    2     16      99
|   /          \         2        P    6     32      13
40|  /75          \        3        P    2     87       4
| /              \       4        P   38     96      49
[3P]===============[4P]>>
77

The minimum time is 127 utilizing the path 1-2-4.

Initially, the light at junction 1 is blue. Since the light at
junction 2 is purple, vehicle waits at junction 1 for 2 seconds
then travels 4 seconds to junction 2.

At time 6, the light at junction 2 switches to blue whereas the
light at junction 4 has 32 more seconds to switch to blue. However,
after 32 seconds, the light at junction 2 switches to purple and
the light at junction 4 switches to blue at the same time. So the
vehicle needs to wait 13 seconds more for junction 2 to switch to
blue then the lights have the same color and vehicle travels 76
seconds to the destination junction 4.

The total time is 2+4+32+13+76=127 seconds.

Below is a more graphical presentation of this travel plan:
1    1    1    1    1    1
1    1    2    2    3    3    4    4    5    5    6    6    7    7    8    8    9    9    0    0    1    1    2    2
0....5....0....5....0....5....0....5....0....5....0....5....0....5....0....5....0....5....0....5....0....5....0....5....0....5..
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J1 BBBPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPBBBBBBBBBBBBBBBBPPPPPPPPPP
J2 PPPPPPBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBPPPPPPPPPPPPPBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBPPPPPPPPPPPPPBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
J3 PPPBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBPPPPBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
J4 PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
FJ 1..>>>2............................................>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>4

PROBLEM NAME: traffic

INPUT FORMAT:

* Line 1: Two space-separated integers: S and D

* Line 2: Two space-separated integers: N and M

* Lines 3..N+2: Line i+2 line describes junction i with a character
and three integers (all separated by a single space): C_i,
R_i, DB_i, and DP_i

* Lines N+3..N+M+2: Line N+2+k describes road k with three integers:
i, j, and T_ij

SAMPLE INPUT (file traffic.in):

1 4
4 5
B 2 16 99
P 6 32 13
P 2 87 4
P 38 96 49
1 2 4
1 3 40
2 3 75
2 4 76
3 4 77

OUTPUT FORMAT:

* Line 1: One integer: the time taken by a minimum-time path from the
source junction to the destination junction. If there is no
path, output 0.

SAMPLE OUTPUT (file traffic.out):

127

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page revision: 1, last edited: 08 Jul 2011 02:33