Score : 900 points
Diverta City is a new city consisting of N towns numbered 1, 2, ..., N.
The mayor Ringo is planning to connect every pair of two different towns with a bidirectional road. The length of each road is undecided.
A Hamiltonian path is a path that starts at one of the towns and visits each of the other towns exactly once. The reversal of a Hamiltonian path is considered the same as the original Hamiltonian path.
There are N! / 2 Hamiltonian paths. Ringo wants all these paths to have distinct total lengths (the sum of the lengths of the roads on a path), to make the city diverse.
Find one such set of the lengths of the roads, under the following conditions:
Input is given from Standard Input in the following format:
N
Print a set of the lengths of the roads that meets the objective, in the following format:
w_{1, 1} \ w_{1, 2} \ w_{1, 3} \ ... \ w_{1, N} w_{2, 1} \ w_{2, 2} \ w_{2, 3} \ ... \ w_{2, N} : : : w_{N, 1} \ w_{N, 2} \ w_{N, 3} \ ... \ w_{N, N}
where w_{i, j} is the length of the road connecting Town i and Town j, which must satisfy the following conditions:
If there are multiple sets of lengths of the roads that meet the objective, any of them will be accepted.
3
0 6 15 6 0 21 15 21 0
There are three Hamiltonian paths. The total lengths of these paths are as follows:
They are distinct, so the objective is met.
4
0 111 157 193 111 0 224 239 157 224 0 258 193 239 258 0
There are 12 Hamiltonian paths, with distinct total lengths.