Problem A. Ford-Bellman
Statement
You are to write a program that finds shortest distances from vertex S
to all other vertices in a given directed weighted graph.
Graph consists of
N vertices, numbered from 1 to
N, and
M edges.
Input file format
Input file contains two integers
N M S, followed my
M triplets of integers
ui vi wi — starting vertex, ending vertex and weight or the edge.
There is at most one edge connecting any two vertices in every direction. There are no cycles of negative weight.
Output file format
Output file must contain a vector of
N integers — distances from vertex
S to other vertices.
The distance from any vertex to itself is considered to be 0.
If some vertex is not reachable from
S, corresponding cell of the vector must contain empty space.
Constraints
0 ≤
N,
M ≤ 1000.
All weights are less than 1000 by absolute value.
Sample tests
| No. |
Input file (input.txt) |
Output file (output.txt) |
|---|
| 1 |
3 3 1
1 2 5
1 3 10
2 3 2
|
0 5 7
|
Problem B. Floyd-Warshall
Statement
You are to write a program that finds shortest distances between all pairs
of vertices in a directed weighted graph.
Graph consists of
N vertices, numbered from 1 to
N, and
M edges.
Input file format
Input file contains two integers
N and
M, followed my
M triplets of integers
ui vi wi — starting vertex, ending vertex and weight or the edge.
There is at most one edge connecting two vertices in every direction. There are no cycles of negative weight.
Output file format
Output file must contain a matrix of size
Nx
N.
Element in the
j-th column of
i-th row mush be the shortest distance between
vertices
i and
j.
The distance from the vertex to itself is considered to be 0.
If some vertex is not reachable from some other,
there must be empty space in corresponding cell of matrix.
Constraints
0 ≤
N ≤ 100.
All weights are less than 1000 by absolute value.
Sample tests
| No. |
Input file (input.txt) |
Output file (output.txt) |
|---|
| 1 |
3 3
1 2 5
1 3 10
2 3 2
|
0 5 7
0 2
0
|
Problem C. SST for sparse graph
Statement
You are to write a program that receives a weighted undirected graph and finds length of its shortest spanning tree.
Input file format
Input file contains two integers N, M.
Vertices are numbered with integer numbers from 1 to N. M is the number of edges. Each of next M lines contain three
integers describing an edge — numbers of vertices, connected by an edge and its weight respectively. All weights are
positive. There is at most one edge connecting two vertices.
Output file format
Output file must contain a signle integer number — length of the SST. If it is impossible to construct spanning tree,
output file must contain −1.
Constraints
1 ≤ N, M ≤ 100000
All weights are less or equal 1000.
Sample tests
| No. |
Input file (input.txt) |
Output file (output.txt) |
|---|
| 1 |
3 3
1 2 10
2 3 10
3 1 10
|
20
|
Problem D. Shortest Spanning Tree
Statement
You are to write a program that receives a weighted undirected graph and finds length of its shortest spanning tree.
Input file format
Input file contains two integers N, M.
Vertices are numbered with integer numbers from 1 to N. M is the number of edges. Each of next M lines contain three
integers describing an edge — numbers of vertices, connected by an edge and its weight respectively. All weights are
positive. There is at most one edge connecting two vertices.
Output file format
Output file must contain a signle integer number — length of the SST. If it is impossible to construct spanning tree,
output file must contain −1.
Constraints
1 ≤ N ≤ 1000
All weights are less or equal 1000.
Sample tests
| No. |
Input file (input.txt) |
Output file (output.txt) |
|---|
| 1 |
3 3
1 2 10
2 3 10
3 1 10
|
20
|
