bzoj#P2183. Pku3137 Enjoyable Commutation
Pku3137 Enjoyable Commutation
题目描述
Isaac is tired of his daily trip to his office, using the same shortest route everyday. Although this saves his time, he must see the same scenery again and again. He cannot stand such a boring commutation any more.
One day, he decided to improve the situation. He would change his route everyday at least slightly. His new scheme is as follows. On the first day, he uses the shortest route. On the second day, he uses the second shortest route, namely the shortest except one used on the first day. In general, on the -th day, the -th shortest route is chosen. Visiting the same place twice on a route should be avoided, of course.
You are invited to help Isaac, by writing a program which finds his route on the -th day. The problem is easily modeled using terms in the graph theory. Your program should find the -th shortest path in the given directed graph.
结点有向简单图中的简单 短路(结点不能重复经过)。路径长度相同时按照字典序排列。
输入格式
The input consists of multiple datasets, each in the following format.
Every input item in a dataset is a non-negative integer. Two or more input items in a line are separated by a space.
is the number of nodes in the graph. You can assume the inequality . is the number of (directed) edges. is the start node, and is the goal node. They are between and , inclusive. You are required to find the k-th shortest path from to . You can assume and .
The -th edge is from the node to with the length . Both and are between and , inclusive. is between and , inclusive. You can directly go from to , but not from to unless an edge from to is explicitly given. The edge connecting the same pair of nodes is unique, if any, that is, if , it is never the case that equals and equals . Edges are not connecting a node to itself, that is, never equals . Thus the inequality holds.
Note that the given graph may be quite unrealistic as a road network. Both the cases and are included in the judges’ data.
The last dataset is followed by a line containing five zeros (separated by a space).
输出格式
For each dataset in the input, one line should be output as specified below. An output line should not contain extra characters such as spaces.
If the number of distinct paths from to is less than , the string None should be printed. Note that the first letter of None
is in uppercase, while the other letters are in lowercase.
If the number of distinct paths from to is or more, the node numbers visited in the -th shortest path should be printed in the visited order, separated by a hyphen (minus sign). Note that must be the first, and must be the last in the printed line.
In this problem the term shorter (thus shortest also) has a special meaning. A path is defined to be shorter than , if and only if one of the following conditions holds.
- The length of is less than the length of . The length of a path is defined to be the sum of lengths of edges on the path.
- The length of is equal to the length of , and ’s sequence of node numbers comes earlier than ’s in the dictionary order. Let’s specify the latter condition more precisely. Denote ’s sequence of node numbers by , and ’s by . and should be observed. The sequence comes earlier than in the dictionary order, if for some and , , and ( is numerically smaller than ).
A path visiting the same node twice or more is not allowed.
样例输入
5 20 10 1 5
1 2 1
1 3 2
1 4 1
1 5 3
2 1 1
2 3 1
2 4 2
2 5 2
3 1 1
3 2 2
3 4 1
3 5 1
4 1 1
4 2 1
4 3 1
4 5 2
5 1 1
5 2 1
5 3 1
5 4 1
4 6 1 1 4
2 4 2
1 3 2
1 2 1
1 4 3
2 3 1
3 4 1
3 3 5 1 3
1 2 1
2 3 1
1 3 1
0 0 0 0 0
样例输出
1-2-4-3-5
1-2-3-4
None
样例说明
In the case of the first dataset, there are paths from the node to . They are ordered as follows (The number in parentheses is the length of the path).
数据规模与约定
对于 的数据,,。
题目来源
Japan 2006 K短路