Usaco Open11 Gold

Part of USACO Open11

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GOLD PROBLEMS
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Three problems numbered 1 through 3
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Problem 1: Mowing the Lawn [Neal Wu, 2008]

After winning the annual town competition for best lawn a year ago,
Farmer John has grown lazy; he has not mowed the lawn since then
and thus his lawn has become unruly. However, the competition is
once again coming soon, and FJ would like to get his lawn into
tiptop shape so that he can claim the title.

Unfortunately, FJ has realized that his lawn is so unkempt that he
will need to get some of his N (1 <= N <= 100,000) cows, who are
lined up in a row and conveniently numbered 1..N, to help him. Some
cows are more efficient than others at mowing the lawn; cow i has
efficiency E_i (0 <= E_i <= 1,000,000,000).

FJ has noticed that cows near each other in line often know each
other well; he has also discovered that if he chooses more than K
(1 <= K <= N) consecutive (adjacent) cows to help him, they will
ignore the lawn and start a party instead. Thus, FJ needs you to
assist him: determine the largest total cow efficiency FJ can obtain
without choosing more than K consecutive cows.

PROBLEM NAME: mowlawn

INPUT FORMAT:

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

* Lines 2..N+1: Line i+1 contains the single integer: E_i

SAMPLE INPUT (file mowlawn.in):

5 2
1
2
3
4
5

INPUT DETAILS:

FJ has 5 cows whose efficiencies are 1, 2, 3, 4, and 5, in that
order. He wants to choose some of the cows such that their total
efficiency is maximized, but he cannot choose more than 2 consecutive
cows.

OUTPUT FORMAT:

* Line 1: A single integer that is the best total efficiency FJ can
obtain.

SAMPLE OUTPUT (file mowlawn.out):

12

OUTPUT DETAILS:

FJ chooses all cows but the third. The total efficiency of the cows is thus
1 + 2 + 4 + 5 = 12.

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Problem 2: Odd degrees [Traditional, 2011]

The cows are being invaded! Their republic comprises N (1 <= N <=
50,000) towns that are connected by M (1 <= M <= 100,000) undirected
paths between two towns A_i and B_i (1 <= A_i <= N; 1 <= B_i <= N;
A_i != B_i; no duplicate paths will occur). However the republic
is not necessarily connected--there may be pairs of towns that are
unable to reach each other through the paths.

The cows know their invaders plan to conduct an inventory of every
path within their republic, so they are willing to shut down various
paths to make it as difficult as possible for their invaders to do
so.

such that each town has an odd number of remaining paths connected
to it, or determine if no such subset exists.

For example, consider the following cow republic:

1---2
\ /
3---4

If we keep the paths 1-3, 2-3, and 3-4, and remove the path 1-2, then towns
1, 2, and 4 will be an endpoint of exactly one path, whereas town 3 will be
an endpoint of three paths:

1   2
\ /
3---4

PROBLEM NAME: oddd

INPUT FORMAT:

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

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

SAMPLE INPUT (file oddd.in):

4 4
1 2
2 3
3 1
3 4

OUTPUT FORMAT:

* Line 1: A single integer that is the number of paths to keep. If no
subset exists output only a single line with the integer -1.

* Lines 2..K+1: Each line contains an index of an path to keep, in the
range 1..M. These indices must be pairwise distinct.

SAMPLE OUTPUT (file oddd.out):

3
2
3
4

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Problem 3: Soldering [Michael Cohen, 2011]

The cows are playing with wires! They have learned a technique
called soldering, in which they connect two pieces of wire together
by attaching the endpoint of one wire to a location along the length
of the other. (Soldering endpoint to endpoint is not allowed.) There
can be multiple solder junctions at the same point.

The cows have a plan for an Amazing Structure they would like to
build. It is in the form of a graph with N (1 <= N <= 50,000) nodes
and N-1 edges of unit length so that each pair of nodes is connected.
Each edge is described by a pair of integers, A and B (1 <= A <=
N; 1 <= B <= N; A != B).

The cows are able to buy wire from a local store; however longer
wire is more expensive. In particular the cows can buy a wire of
length L with cost L*L, but they cannot cut wires or join wires
together.

Given the plan, the cows would like solder wires together to build
cost!

Test data worth at least 50% of the points will have N <= 2,000.

Partial feedback will be provided on your first 50 submissions to this problem.

TIME LIMIT: 2 seconds
MEMORY LIMIT: 64 MB

PROBLEM NAME: solder

INPUT FORMAT:

* Line 1: A single integer: N

* Lines 2..N: Two space-separated integers describing an edge: A and B

SAMPLE INPUT (file solder.in):

6
1 2
1 3
1 4
1 5
1 6

OUTPUT FORMAT:

* Line 1: A single integer, the cost of soldering the tree together.
Note that this number may not always fit in a 32-bit integer.

SAMPLE OUTPUT (file solder.out):

7

OUTPUT DETAILS:

Since all nodes in the structure are connected to node 1, we only need to
buy one wire of length 2 and three of length 1, for a total cost of 2 * 2 +
1 * 1 + 1 * 1 + 1 * 1 = 7.

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page revision: 1, last edited: 08 Jul 2011 02:35