man wnsp (Fonctions bibliothèques) - shortest path problem

NAME

wn_shortest_path - shortest path problem

SYNOPSIS

#include <wn/wnspmat.h>
#include <wn/wnsp.h>

wn_shortest_path(&code,&len,&result,length_mat,start_node,fin_node)
int code;
double len;
wn_sll result; /* list of edges */
wn_sparse_matrix length_mat;
int start_node,fin_node;

DESCRIPTION

This package solves the "shortest path" problem easily and efficiently using Dijkstra's algorithm [1]. length_mat is treated as a DIRECTED GRAPH; thus it must be square. The matrix entry length_mat[i][j] gives the length of the directed edge from node i to node j. Negative edge lengths are not allowed. result is the list of edges in the shortest path, ordered starting from start_node. len is set to the total length of the solution.

The shortest path problem is the following optimization problem:

Choose the path in the graph length_mat from start_node to fin_node which is the shortest possible path.

Difficult optimization problems from many fields can be put into this form, and thus solved efficiently with this package.

For an introduction to the shortest path problem, consult [1].

RESOURCES

Solving a shortest path problem requires

WORST CASE:

time = e+n*log(n)

stack memory = 1

dynamic memory = n

AVERAGE CASE:

time = depends heavily on problem structure stack memory = 1 dynamic memory = depends heavily on problem structure

where e is the number of matrix entries, and n is the number of nodes in the graph represented by length_mat. (n == len_i == len_j).

If the shortest path involves only a small fraction of the graph nodes, run time and memory usage are usually very much better than the worst case.

Note: the shortest path problem becomes NP-complete if negative edges are allowed [2].

DIAGNOSTICS

code == WN_SUCCESS for successful solution.

code == WN_INFEASIBLE if no path from start_node to fin_node exists.

len_i != len_j causes a crash.

negative edge lengths cause a crash.

SEE ALSO

wncp, wnlp, wnsplx, wnmst

REFERENCES

[1] A. Aho, J. Hopcroft, and J. Ullman: Data Stuctures and Algorithms. Addison-Wesley Publishing.

AUTHOR

Will Naylor

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