Usaco Nov07 Gold

Part of USACO Nov07

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GOLD PROBLEMS
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Three problems numbered 1 through 3
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Problem 1: Telephone Wire [Jeffrey Wang, 2007]

Farmer John's cows are getting restless about their poor telephone
service; they want FJ to replace the old telephone wire with new,
more efficient wire. The new wiring will utilize N (2 <= N <=
100,000) already-installed telephone poles, each with some height_i
meters (1 <= height_i <= 100). The new wire will connect the tops
of each pair of adjacent poles and will incur a penalty cost C *
the two poles' height difference for each section of wire where the
poles are of different heights (1 <= C <= 100). The poles, of course,
are in a certain sequence and can not be moved.

Farmer John figures that if he makes some poles taller he can reduce
an integer X number of meters to a pole at a cost of X^2.

Help Farmer John determine the cheapest combination of growing pole
heights and connecting wire so that the cows can get their new and
improved service.

PROBLEM NAME: telewire

INPUT FORMAT:

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

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

SAMPLE INPUT (file telewire.in):

5 2
2
3
5
1
4

INPUT DETAILS:

There are 5 telephone poles, and the vertical distance penalty is
\$2/meter. The poles initially have heights of 2, 3, 5, 1, and 4,
respectively.

OUTPUT FORMAT:

* Line 1: The minimum total amount of money that it will cost Farmer
John to attach the new telephone wire.

SAMPLE OUTPUT (file telewire.out):

15

OUTPUT DETAILS:

The best way is for Farmer John to raise the first pole by 1 unit and the
fourth pole by 2 units, making the heights (in order) 3, 3, 5, 3, and 4.
This costs \$5. The remaining wiring will cost \$2*(0+2+2+1) = \$10, for a
total of \$15.

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Problem 2: Cow Relays [Erik Bernhardsson, 2003]

For their physical fitness program, N (2 <= N <= 1,000,000) cows
have decided to run a relay race using the T (2 <= T <= 100) cow
trails throughout the pasture.

Each trail connects two different intersections (1 <= I1_i <= 1,000;
1 <= I2_i <= 1,000), each of which is the termination for at least
two trails. The cows know the length_i of each trail (1 <= length_i
<= 1,000), the two intersections the trail connects, and they know
that no two intersections are directly connected by two different
trails. The trails form a structure known mathematically as a graph.

To run the relay, the N cows position themselves at various
intersections (some intersections might have more than one cow).
They must position themselves properly so that they can hand off
the baton cow-by-cow and end up at the proper finishing place.

Write a program to help position the cows. Find the shortest path
that connects the starting intersection (S) and the ending intersection
(E) and traverses exactly N cow trails.

PROBLEM NAME: relays

INPUT FORMAT:

* Line 1: Four space-separated integers: N, T, S, and E

* Lines 2..T+1: Line i+1 describes trail i with three space-separated
integers: length_i, I1_i, and I2_i

SAMPLE INPUT (file relays.in):

2 6 6 4
11 4 6
4 4 8
8 4 9
6 6 8
2 6 9
3 8 9

OUTPUT FORMAT:

* Line 1: A single integer that is the shortest distance from
intersection S to intersection E that traverses exactly N cow
trails.

SAMPLE OUTPUT (file relays.out):

10

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Problem 3: Sunscreen [Russ Cox, 2001]

To avoid unsightly burns while tanning, each of the C (1 <= C <=
2500) cows must cover her hide with sunscreen when they're at the
beach. Cow i has a minimum and maximum SPF rating (1 <= minSPF_i
<= 1,000; minSPF_i <= maxSPF_i <= 1,000) that will work. If the SPF
rating is too low, the cow suffers sunburn; if the SPF rating is
too high, the cow doesn't tan at all.

The cows have a picnic basket with L (1 <= L <= 2500) bottles of
sunscreen lotion, each bottle i with an SPF rating SPF_i (1 <= SPF_i
<= 1,000). Lotion bottle i can cover cover_i cows with lotion. A
cow may lotion from only one bottle.

What is the maximum number of cows that can protect themselves
while tanning given the available lotions?

PROBLEM NAME: tanning

INPUT FORMAT:

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

* Lines 2..C+1: Line i describes cow i's lotion requires with two
integers: minSPF_i and maxSPF_i

* Lines C+2..C+L+1: Line i+C+1 describes a sunscreen lotion bottle i
with space-separated integers: SPF_i and cover_i

SAMPLE INPUT (file tanning.in):

3 2
3 10
2 5
1 5
6 2
4 1

INPUT DETAILS:

3 cows; 2 lotions.  Cows want SPF ratings of 3..10, 2..5, and 1..5. Lotions
available: 6 (for two cows), 4 (for 1 cow).  Cow 1 can use the SPF 6 lotion.
Either cow 2 or cow 3 can use the SPF 4 lotion.  Only 2 cows can be covered.

OUTPUT FORMAT:

A single line with an integer that is the maximum number of cows that
can be protected while tanning

SAMPLE OUTPUT (file tanning.out):

2

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page revision: 7, last edited: 08 Jul 2011 02:49