Problem A. Fast Dijkstra
Statement
You are to write a program that receives a weighted directed graph and finds
all distances from fixed vertex S to all other vertices. Distance from S to
some vertex W is the minimal length of path going from S to W. Length of path
is the sum of weights of its arcs.
Input file format
Input file contains two integers N, M and S.
Vertices are numbered with integer numbers from 1 to N. S is the number of
source vertex. M is the number of arcs. Each of next M lines contain three
integers — numbers of starting and ending vertices of some arc and its weight
respectively. All weights are positive. There is at most one arc connecting
two vertices in every direction.
Output file format
Output file must contain N numbers. Each I-th number is the distance from
vertex S to vertex I. If some vertices are not reachable from S, corresponding
numbers must be −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 |
5 3 1
1 2 5
1 3 7
3 4 10
|
0 5 7 17 -1
|
Problem B. 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 C. 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
|
