Problem A. Dijkstra
Statement
You are to write a program that receives a weighted directed graph and finds
distances from source 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 edges.
Vertices are numbered with integers from 1 to N.
Input file format
First line of input file contains three integers
N M and
S,
where
M is the number of edges. Next
M lines contain three
integers each —
starting vertex number, ending vertex number and weight of some edge respectively.
All weights are positive.
There is at most one edge connecting two vertices in every direction.
Output file format
Output file must contain
N integers — distances from
source vertex to all vertices.
If some vertices are not reachable from
S, corresponding numbers must be −1.
Constraints
1 ≤ N ≤ 1000.
All weights are less or equal than 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. 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 Ith 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 C. FloydWarshall
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
u_{i} v_{i} w_{i} — 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
jth column of
ith 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
