Problem A. Dijkstra ≡

Author:	StdAlg		Time limit:	1 sec
Input file:	input.txt		Memory limit:	64 Mb
Output file:	output.txt

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 copy	0 5 7 17 -1 copy

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Problem B. Fast Dijkstra ≡

Author:	StdAlg		Time limit:	1 sec
Input file:	input.txt		Memory limit:	64 Mb
Output file:	output.txt

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 copy	0 5 7 17 -1 copy

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Problem C. SST for sparse graph ≡

Author:	StdAlg		Time limit:	1 sec
Input file:	input.txt		Memory limit:	64 Mb
Output file:	output.txt

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 copy	20 copy

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Problem D. Floyd-Warshall ≡

Author:	StdAlg		Time limit:	1 sec
Input file:	input.txt		Memory limit:	64 Mb
Output file:	output.txt

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 NxN. 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 copy	0 5 7 0 2 0 copy

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