Integer, Intent (In)	::	n, ns, ne, nnz, icol(nnz)
Integer, Intent (Inout)	::	irow(nnz), ifail
Integer, Intent (Out)	::	path(n), iwork(3*n+1)
Real (Kind=nag_wp), Intent (In)	::	d(nnz)
Real (Kind=nag_wp), Intent (Out)	::	splen, work(2*n)
Logical, Intent (In)	::	direct

C Header Interface

#include <nag.h>

void	h03adf_ (const Integer n, const Integer ns, const Integer ne, const logical direct, const Integer nnz, const double d[], Integer irow[], const Integer icol[], double splen, Integer path[], Integer iwork[], double work[], Integer *ifail)

The routine may be called by the names h03adf or nagf_mip_shortestpath.

3 Description

h03adf attempts to find the shortest path through a directed or undirected acyclic network, which consists of a set of points called vertices and a set of curves called arcs that connect certain pairs of distinct vertices. An acyclic network is one in which there are no paths connecting a vertex to itself. An arc whose origin vertex is

i

and whose destination vertex is

j

can be written as

i \to j

. In an undirected network the arcs

i \to j

and

j \to i

are equivalent (i.e.,

i \leftrightarrow j

), whereas in a directed network they are different. Note that the shortest path may not be unique and in some cases may not even exist (e.g., if the network is disconnected).

The network is assumed to consist of

n

vertices which are labelled by the integers

1, 2, \dots, n

. The lengths of the arcs between the vertices are defined by the

n \times n

distance matrix

d

, in which the element

d_{i j}

gives the length of the arc

i \to j

;

d_{i j} = 0

if there is no arc connecting vertices

i

and

j

(as is the case for an acyclic network when

i = j

). Thus the matrix

D

is usually sparse. For example, if

n = 4

and the network is directed, then

d = (\begin{matrix} 0 & d_{12} & d_{13} & d_{14} \\ d_{21} & 0 & d_{23} & d_{24} \\ d_{31} & d_{32} & 0 & d_{34} \\ d_{41} & d_{42} & d_{43} & 0 \end{matrix}) .

If the network is undirected,

d

is symmetric since

d_{i j} = d_{j i}

(i.e., the length of the arc

i \to j \equiv

the length of the arc

j \to i

The method used by h03adf is described in detail in Section 9.

4 References

Dijkstra E W (1959) A note on two problems in connection with graphs Numer. Math. 1 269–271

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of vertices.

Constraint:

n \geq 2

2: $ns$ – Integer Input

3: $ne$ – Integer Input

On entry:

n_{s}

and

n_{e}

, the labels of the first and last vertices, respectively, between which the shortest path is sought.

Constraints:

$1 \leq ns \leq n$ ;
$1 \leq ne \leq n$ ;
$ns \neq ne$ .

4: $direct$ – Logical Input

On entry: indicates whether the network is directed or undirected.

$direct = .TRUE.$: The network is directed.
$direct = .FALSE.$: The network is undirected.

5: $nnz$ – Integer Input

On entry: the number of nonzero elements in the distance matrix

D

Constraints:

if $direct = .TRUE.$ , $1 \leq nnz \leq n \times (n - 1)$ ;
if $direct = .FALSE.$ , $1 \leq nnz \leq n \times (n - 1) / 2$ .

6: $d (nnz)$ – Real (Kind=nag_wp) array Input

On entry: the nonzero elements of the distance matrix

D

, ordered by increasing row index and increasing column index within each row. More precisely,

d (k)

must contain the value of the nonzero element with indices (

irow (k), icol (k)

); this is the length of the arc from the vertex with label

irow (k)

to the vertex with label

icol (k)

. Elements with the same row and column indices are not allowed. If

direct = .FALSE.

, then only those nonzero elements in the strict upper triangle of

d

need be supplied since

d_{i j} = d_{j i}

. (f11zaf may be used to sort the elements of an arbitrarily ordered matrix into the required form.)

Constraint:

d (k) > 0.0

, for

k = 1, 2, \dots, nnz

7: $irow (nnz)$ – Integer array Input

8: $icol (nnz)$ – Integer array Input

On entry:

irow (k)

and

icol (k)

must contain the row and column indices, respectively, for the nonzero element stored in

d (k)

Constraints:

irow and icol must satisfy the following constraints (which may be imposed by a call to f11zaf):

$irow (k - 1) < irow (k)$ ;
$irow (k - 1) = irow (k)$ and $icol (k - 1) < icol (k)$ , for $k = 2, 3, \dots, nnz$ .

In addition, if

direct = .TRUE.

1 \leq irow (k) \leq n

1 \leq icol (k) \leq n

and

irow (k) \neq icol (k)

;

if $direct = .FALSE.$ , $1 \leq irow (k) < icol (k) \leq n$ .

9: $splen$ – Real (Kind=nag_wp) Output

On exit: the length of the shortest path between the specified vertices

n_{s}

and

n_{e}

10: $path (n)$ – Integer array Output

On exit: contains details of the shortest path between the specified vertices

n_{s}

and

n_{e}

. More precisely,

ns = path (1) \to path (2) \to \dots \to path (p) = ne

for some

p \leq n

. The remaining

(n - p)

elements are set to zero.

11: $iwork (3 \times n + 1)$ – Integer array Workspace

12: $work (2 \times n)$ – Real (Kind=nag_wp) array Workspace

13: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

On entry, $ne = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq ne \leq n$ .

On entry, $ns = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq ns \leq n$ .

On entry, $ns = ⟨ value ⟩$ and $ne = ⟨ value ⟩$ .
Constraint: $ns \neq ne$ .

$ifail = 2$: On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $direct = .FALSE.$ , $1 \leq nnz \leq n \times (n - 1) / 2$ .

On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $direct = .TRUE.$ , $1 \leq nnz \leq n \times (n - 1)$ .

$ifail = 3$: On entry, $k = ⟨ value ⟩$ , $irow (k) = ⟨ value ⟩$ , $icol (k) = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq irow (k) \leq n$ , $1 \leq icol (k) \leq n$ ; $icol (k) \neq irow (k)$ when $direct = .TRUE.$ .

$ifail = 4$: On entry, $k = ⟨ value ⟩$ , $irow (k) = ⟨ value ⟩$ , $icol (k) = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq irow (k) < icol (k) \leq n$ when $direct = .FALSE.$ .

$ifail = 5$: On entry, $k = ⟨ value ⟩$ , $d (k) = ⟨ value ⟩$ .
Constraint: $d (k) > 0.0$ .

$ifail = 6$: On entry, $k = ⟨ value ⟩$ , $irow (k) = ⟨ value ⟩$ , $irow (k - 1) = ⟨ value ⟩$ , $icol (k) = ⟨ value ⟩$ , $icol (k - 1) = ⟨ value ⟩$ .
Constraints: $irow (k - 1) < irow (k)$ or $irow (k - 1) = irow (k)$ and $icol (k - 1) < icol (k)$ .

$ifail = 7$: On entry, $k = ⟨ value ⟩$ $irow (k) = ⟨ value ⟩$ , $irow (k - 1) = ⟨ value ⟩$ $icol (k) = ⟨ value ⟩$ , $icol (k - 1) = ⟨ value ⟩$ .
Constraint: $irow (k) \neq irow (k - 1)$ or $icol (k) \neq icol (k - 1)$ .

$ifail = 8$: On entry, $ns = ⟨ value ⟩$ and $ne = ⟨ value ⟩$ .
No connected network exists between vertices ns and ne.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The results are exact, except for the obvious rounding errors in summing the distances in the length of the shortest path.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

h03adf is not threaded in any implementation.

9 Further Comments

h03adf is based upon Dijkstra's algorithm (see Dijkstra (1959)), which attempts to find a path

n_{s} \to n_{e}

between two specified vertices

n_{s}

and

n_{e}

of shortest length

d (n_{s}, n_{e})

The algorithm proceeds by assigning labels to each vertex, which may be temporary or permanent. A temporary label can be changed, whereas a permanent one cannot. For example, if vertex

p

has a permanent label

(q, r)

, then

r

is the distance

d (n_{s}, r)

and

q

is the previous vertex on a shortest length

n_{s} \to p

path. If the label is temporary, then it has the same meaning but it refers only to the shortest

n_{s} \to p

path found so far. A shorter one may be found later, in which case the label may become permanent.

The algorithm consists of the following steps.

1.Assign the permanent label $(-, 0)$ to vertex $n_{s}$ and temporary labels $(-, \infty)$ to every other vertex. Set $k = n_{s}$ and go to 2.
2.Consider each vertex $y$ adjacent to vertex $k$ with a temporary label in turn. Let the label at $k$ be $(p, q)$ and at $y (r, s)$ . If $q + d_{k y} < s$ , then a new temporary label $(k, q + d_{k y})$ is assigned to vertex $y$ ; otherwise no change is made in the label of $y$ . When all vertices $y$ with temporary labels adjacent to $k$ have been considered, go to 3.
3.From the set of temporary labels, select the one with the smallest second component and declare that label to be permanent. The vertex it is attached to becomes the new vertex $k$ . If $k = n_{e}$ go to 4. Otherwise go to 2 unless no new vertex can be found (e.g., when the set of temporary labels is ‘empty’ but $k \neq n_{e}$ , in which case no connected network exists between vertices $n_{s}$ and $n_{e}$ ).
4.To find the shortest path, let $(y, z)$ denote the label of vertex $n_{e}$ . The column label ( $z$ ) gives $d (n_{s}, n_{e})$ while the row label ( $y$ ) then links back to the previous vertex on a shortest length $n_{s} \to n_{e}$ path. Go to vertex $y$ . Suppose that the (permanent) label of vertex $y$ is $(w, x)$ , then the next previous vertex is $w$ on a shortest length $n_{s} \to y$ path. This process continues until vertex $n_{s}$ is reached. Hence the shortest path is

$n_{s} \to \dots \to w \to y \to n_{e},$

which has length $d (n_{s}, n_{e})$ .

10 Example

This example finds the shortest path between vertices

1

and

11

for the undirected network

Figure 1

h03ad: FL CL CPP AD PY MB

NAG FL Interfaceh03adf (shortestpath)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
h03adf (shortestpath)