OI题库

Surround the Islands [Fatih Gelgi, 2008]

Farmer John has bought property in the Caribbean and is going to
try to raise dairy cows on a big farm composed of islands. Set in
his ways, he wants to surround all the islands with fence.

Each island in the farm has the shape of a polygon. He fences the
islands one side at a time (between a consecutive pair of vertices)
and proceeds clockwise around a given island with his fencing
operations. Since he wants to fence all the islands, he must at
some point travel to any other islands using a boat.

He can start fencing at any vertex and, at any vertex he encounters,
travel to some vertex on another island, fence all the way around it, and
then IMMEDIATELY return back to the same vertex on the original island
using the same path he traveled before. Each boat trip has a cost defined
by a supplied matrix.

The islands are described by a set of N (3 <= N <= 500) pairs of
vertices V1,V2 (1 <= V1 <= N; 1 <= V2 <= N) although you must figure
out how to assemble them into islands. The vertices are conveniently
numbered 1..N.

The cost of traveling by boat between each pair of vertices is given
by a symmetric cost matrix whose elements fall in the range 0..1000.

What is the minimum cost of surrounding the islands with the fence?

PROBLEM NAME: surround

INPUT FORMAT:

* Line 1: A single integer: N

* Lines 2..N+1: Each line describes an island's border with two
        space-separated integers: V1 and V2

* Lines N+2..2*N+1: Line i-N-1 contains N integers that describe row i
        of the cost matrix: Row_i

SAMPLE INPUT (file surround.in):

12
1 7
7 3
3 6
6 10
10 1
2 12
2 9
8 9
8 12
11 5
5 4
11 4
0 15 9 20 25 8 10 13 17 8 8 7
15 0 12 12 10 10 8 15 15 8 8 9
9 12 0 25 20 18 16 14 13 7 12 12
20 12 25 0 8 13 14 15 15 10 10 10
25 10 20 8 0 16 20 18 17 18 9 11
8 10 18 13 16 0 10 9 11 10 8 12
10 8 16 14 20 10 0 18 20 6 16 15
13 15 14 15 18 9 18 0 5 12 12 13
17 15 13 15 17 11 20 5 0 22 8 10
8 8 7 10 18 10 6 12 22 0 11 12
8 8 12 10 9 8 16 12 8 11 0 9
7 9 12 10 11 12 15 13 10 12 9 0

INPUT DETAILS:

  1        10            4
    xxxxxxx              x
   xxxxxxxxx            xxxx
7 xxxxxxxxxxx 6        xxxxxxx
 xxxxxxxxxxx       11 xxxxxxxxxx 5
  xxxxxxx
   xxx
  3         12 xxxxxxx 2
              xxxxxxxx
              xxxxxxxx
             xxxxxxxxx
             xxxxxxxxx
            xxxxxxxxxx
            xxxxxxxxxx
          8 xxxxxxxxxx 9

The example describes three islands: {1,7,3,6,10}, {4,5,11} and
{2,9,8,12}. The travel costs are provided as a matrix. For example,
the travel cost from vertex 1 to 2 is 15.

OUTPUT FORMAT:

* Line 1: A single integer that specifies the minimum cost of building
        the fence

SAMPLE OUTPUT (file surround.out):

30

OUTPUT DETAILS:

There is more than one solution. One is: FJ starts from vertex 3
then 7 and stops at 1, travels to 11 followed by 4,5,11. He then
returns back to 1, and travels to 12 followed by 2,9,8,12. Finally,
he returns back to 1 and continues with 10,6,3,7. The costs are 8
* 2 = 16 for traveling from 1 to 11 and returning back, and 7 * 2
= 14 for traveling from 1 to 12 and back -- a total cost of 30.