Report ESL-R-683
August, 1976
SHORTEST ROUTE ALGORITHMS FOR
SPARSELY CONNECTED NETWORKS
by
Joe E. Defenderfer
Electronic Systems Laboratory
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
-2-
ABSTRACT
This report studies the shortest route problem for networks that
are less than fully connected.
Two algorithms are presented which
exploit the absence of arcs in solving the shortest route problem.
The first, which is designated the NXN algorithm, would tend to be
the more applicable to networks typically encountered in practice.
The second, which is an improvement on Hu's decomposition shortest
route algorithm, is more efficient for a small class of networks;
however, it generally requires less memory to hold the required
decomposition information in the computer than does the NXN algorithm.
-3-
Acknowledgements
I wish to thank Professors John M. Wozencraft and Robert G. Gallager
who provided encouragement and many helpful discussions.
This work was supported by a Sloan Research Traineeship and the
Advanced Research Projects Agency under Contract ONR/N00014-75-C-1183.
-4-
Table of Contents
Page
Abstract
2
Acknowledgements
3
Section I
Introduction
7
II
The NXN Algorithm
9
III
Decomposing the Network for the NXN
Algorithm
12
IV
The IHU Algorithm
15
V
Decomposing the Network for the IHU
Algorithm
20
Some Examples Using the IHU and NXN
Algorithms
21
VI
Appendix
Section IA
Introduction
27
IIA
Bookkeeping for Shortest Routes
28
IIIA
The Main Program
29
IVA
Subroutine DIJKST
32
VA
Subroutines DECIHU and IHU
33
VIA
Subroutines DECNXN and NXN
37
VIIA
Program Listing
42
Biblography
71
Distribution List
72
-5-
Figures
Figure 1
An N node network with an NXN decomposition.
2
The network of Figure 1 with a distinct NXN
decomposition.
3
The form of the D matrix for the IHU algorithm.
4
The topology of ARPA network with decomposition
information.
5
A 47 node symmetric network with decomposition
information.
6
A 64 node network with decomposition information.
7
An example network with seven nodes and thirteen arcs.
8
Node renumbering and partitioning by subroutine DECNXN
for the network of Figure 7.
9
The topology of Figure 7 with new node numbers as assigned
by DECNXN.
-6-
Tables
Table 1
Comparative performance of three different shortest
route algorithms on the three sample networks.
2
Topology cards for the network of Figure 7.
3
Cards punched by subroutine DECIHU for the network
of Figure 7.
4
Values of variables in common block IHUSTF deduced
from Table 3.
5
Cards punched by subroutine DECNXN for the network of
Figure 7.
6
A Fortran Program which interprets the cards punched
by DECNXN.
-7-
Section I
Introduction
The problem of finding all the shortest routes in a directed network
has an extensive literature
[3,9] due to the number of network
problems to which shortest route algorithms are applied.
This paper
presents two new shortest route algorithms which can significantly reduce
the required computation time when the network is less than fully connected.
The first is based on original decomposition ideas and is called the nodeby-node decomposition
(NXN) algorithm. $ The second is based on Hu's de-
composition algorithm [5,6,11] and is designated the improved Hu (IHU)
algorithm.
The shortest route problem is formulated as a shortest distance problem
where D =
[dij] is a given matrix.
The number dij represents the length
of the directed arc from node i to node j, and thus it is assumed di. = 0.
A path P from i to j is an ordered sequence i = ko,kl,..., k
l,k m
j,
m
and the length of the path, L(P), is defined as L(P) =
dk
k
If P
r=l
r-l r
is any closed path, then it is assumed L(P) > 0 so that the shortest distance
problem is well defined.
d.j
= min L(P)
Then the problem is to find D* = [d]I where
for P ranging over all paths from i to j.
Knowing D* alone
does not specify the shortest routes, but is a well documented fact that
by appropriate bookkeeping as one calculates D*, the shortest routes can also
be established.
tAfter completion of this paper, the equivalence of the NXN algorithm with
previous work done at Network Analysis Corporation under ARPA Order No. 1523
[8] was discovered.
Typically D* is calculated as a series of refinements on D.
algorithm
Floyd's
[4] is cited for an N node network:
For every i E{1,2,...,N}, do step a:
a)
For every j,k c{1,2,...,N},
b)
where " -"
do step b:
djk + min (djkl dji + dik)
means "is replaced by".
The algorithm requires N 3 additions
and N 3 comparisons, and it is generally assumed additions and comparisons
take about the same amount of time so that one says Floyd's algorithms
requires 2N3 operations.
D.
At the conclusion of the algorithm D* has replaced
Proof of the algorithm is found elsewhere [6], but the interested reader
can easily convince himself that when i has been stepped from 1 though i
0
then the current value of djk is the minimal distance over all paths from
j to k under the condition that the intermediate nodes are elements of the
set {1,2,..., i }.
No algorithm which solves the shortest route algorithm could be any
simpler to encode, but there are a variety of faster algorithms in terms
of number of operations
[7,10].
The standard against which the new
decomposition algorithms will be measured is Yen's implementation of
Dijkstra's algorithm [2,11]
3 3
requiring -N
operations.
The algorithms
claiming even less operations are not significantly faster, theoretically,
for networks of the size for which computational experience is cited in
this paper; furthermore, some of the apparent gains of the theoretically
faster algorithms would be offset by their additional algorithmic complexity.
-9-
Section II
The NXN Algorithm
The NXN algorithm for solving the shortest route problem is actually
a special case of the following new 2N 3 operation algorithm:
1)
For every i£{1,2,...,N-2} in order, do step a:
a)
For every j,kE{i+l,i+2,...,N}, do step b:
b)
2)
djk
+
min (djk,dji+dik)
For every i£{N-2,N-3,...,1} in order, do step a:
a)
For every j,kc{i+l,i+2,...,N}, do steps b and c:
b)
c)
dji
.
J1
dij
min (dik+dkj
dij)
min (djk+dkif d.i)
jk~dkii
An intuitive proof of this algorithm will be helpful in understanding the
NXN algorithm.
By inductive reasoning similar to that for Floyd's algo-
rithm, when step 1 has been completed for i=io, then djk (for j,k > io)
represents the conditional shortest j to k distance subject to all intermediate nodes being elements of the set {1,2,...,i }.
Consequently, when
o
step 1 has been completed, then the djk
(for j,k > N-2) represent uncon-
ditional shortest distances d*.
Note than an arbitrary i to j path (for j > i) must be of the form
i,...,r,...,j where r is the first element in the path such that r > i;
and, if this path is the shortest path, then its length is d* +d*..
lr rj
performing step 2 for i = N-2, d*. is known and d*
ir
rJ
When
must be the same as
-10-
the minimal dir conditional on all intermediate nodes being elements of
the set {1,2,...,N-3}; it follows that at the end of step 2 for i = N-2,
d.. = dt., and similarly dji = d*. for every jE{N-l,N}.
1]
ij
]iIj
reasoning shows that at the end of the algorithm D
The NXN algorithm will now be presented.
=
Clearly, inductive
D*.
However, in order to simplify
the discussion, it is assumed that all of the arcs are duplex, i.e. if d.i.<1J
Define Ci, called the ith connection set, as follows:
then d.ji<.
Ji
j > i and there exists a path P from i to j such that L(P)
intermediate node k satisfies k < i.
j6C. if
1
< - and every
Notice that the C. are functions of
1
topology only (implicitly assuming the length assigned to an arc is - if
and only if the arc does not exist in some sense).
In step 1 of the above algorithm, dji = C if jCi and dik = X if
kiCi.
Furthermore, in step 2 of the above algorithm, dik =
if kgCi.
X
and d
=
The corresponding operations are clearly unnecessary; the algo-
rithm obtained by deleting them is called the NXN algorithm:
1)
For every iE{1,2,...,N-2} in order, do step a:
a)
For every j,kEC i, do step b:
b)
2)
djk
+
min (djk, dji+dik)
For every i£{N-2, N-3,...,1} in order, do step a:
a)
For every js{i+l,i+2,...,N} and keCi, do steps b and c:
b)
dij
c)
dji
+
min (dik+dkj, dij)
min (dj+d
dji)
-11A decomposition is defined as an ordering of the nodes.
Since the
connection sets are a function of the decomposition, the number of
operations which the algorithm requires is also a function of the decomposition, as will be demostrated in the following section.
In the case where some of the arcs are not duplex, two alternatives
are available.
The first is to change the definition of Ci as follows:
jC.
Ciif j > i and there exists a path P from i to j or from j to i such
that L(P)
< - and every intermediate node k satisfies k < i.
causes unnecessary operations for the algorithm.
This approach
The alternative is to
define two connection sets for each node--one for the incoming connections
and one for the outgoing connections.
In the latter case, one must alter
the NXN algorithm to incorporate the efficiencies of the additional connection
sets.
The increased algorithmic complexity of the second approach and the
resultant additional computer steps must be weighed against the number of
unnecessary operations of the first approach for the problem at hand.
-12-
Section III
Decomposing the Network for the NXN Algorithm
This section is introduced via an example.
Consider figures 1 and 2
in which the same network has been decomposed two ways.
C. = {i+l,N-l,N} when i{l1,2,...,N-3} and CN
2
For the first,
{N-1,N}; the number of
operations for the NXN algorithm is calculated in a straightforward fashion
as:
N-3
Step 1,
(
(2) (3) (3))
+ (2) (2) (2)
i=l
N-3
Step 2,
(C
(2) (2) (3) (N-i))
+ (2) (2) (2) (2)
i=l
which totals 6N +12N-66.
By contrast, for the decomposition of figure 2,
Ci = {i+l,i+2,...,N} which is exactly the same as if the network was fully
connected, and it follows immediately that the NXN algorithm requires 2N3
operations.
This example makes it clear that the choice of decomposition
can have a profound effect of the efficiency of the algorithm.
For an arbitrary network, finding the optimal decomposition in the
sense of minimizing the required number of operations for the NXN algorithm
is not a trivial problem and probably can only be solved by exhaustive
comparision.
The method of choosing the decomposition for the examples
which are presented later in Section IV deviated only slightly from the
following heuristic procedure:
-13-
J-
Figure 1.
44
An N node network with an NXN decomposition
implied by the numbering of the nodes.
N- 4
Figure 2.
The same N node network as in Figure 1 with
a distinct NXN decomposition.
-14-
1)
Label a node "1" such that the cardinality of C
is minimized.
1
2)
For every i£{2,3,...,N} in order, do step a:
a)
Given the nodes which have been labeled "1","2",...,
"i-l", label an unlabeled node "i" such that the
cardinality of Ci is minimized.
The effort in finding the decomposition via the above procedure is on
the same order as doing a shortest route computation via Floyd's algorithm, and as a consequence computer time savings are realized only when
the NXN algorithm is iterated several times for the same topology.
There are a large number of networks such that the computation time
does not vary widely with the decomposition.
Such networks could be termed
"locally connected" and have the property that the nodes to which there are
direct arcs from any given node are very likely to have direct arcs to one
another.
In this case, the nodes could be numbered very rapidly by eye
with little degradation in efficiency (nodes must at some point in time
be assigned a number anyhow in order to communicate the topology to the
computer), and in the first shortest route computation the connection sets
could be established with very little effort.
In fact, the only modification
to the NXN algorithm is an additional step which is included just before
step la:
aa)
Initially C i = 0; for jC{i+l,i+2,...,N}, do step bb:
bb)
Include j in Ci if dij < A.
1The
additional
operations
required
this
by
step
number2
The additional operations required by this step number -t which is quite
modest for the potential gains.
-15-
Section IV
The IHU Algorithm
The presentation of the IHU algorithm requires some additional definitions.
For this algorithm, a network decomposition is defined as a division
of the network's nodes into ordered subsets S1, S2,...PSk such that for
every ieSm and jESS,
dij =
X
if
jm-Yj > 1.
Every
node of the network
belongs to exactly one subset.
The submatrix D S
contains all the distances
i m
of arcs from elements of Si to elements of Sm and has dimension Isil x ISmi
(where ISj
means the cardinality of set S).
entries in the case li-mi
Evidently, D S
1m
has no finite
> 1 (see figure 3).
Various matrix operations will be performed on the submatrices to generate the desired shortest distance matrix.
Let D S.S
DS
mean DS
is replaced by the shortest distance matrix computed from the submatrix
DS S.
Define A°B =
S1U S2 U...US
then 2m =
D
=
k A 2.
[min(aim+bmj)
] and min(A,B) = [min(aij,bij) ].
Also let
, and if m = k (where k is the number of ordered sets)
Define the conditional shortest distance submatrix,
S.(m2), as the shortest distance submatrix under the restriction that
all the intermediate nodes on the respective conditional shortest routes
are members of 2 .
i
In the case m = k, D*
(S2)
(S)
S.S. m =D*
S.S.
D*S
S.S
Under the assumption that an allowable decomposition has been given,
the following algorithm generates all the shortest distances in the network
(the parenthetical equality to the right of each step is the claim of what
each step accomplishes):
-16-
7
|D/BS
/7/7/
/7
u3 /
Figure 3.
T
S3
S3/
/
/7/'
~/7
Ds4
I/
s/S4S
f
/t
o
533
_
'
3
The form of the D matrix for the IHU algorithm in
the case k = 5. If the decomposition is to be
acceptable, the shaded submatrices have no finite
entries prior to the algorithmic operations on
the D matrix.
-17-
1
(
1)
D1
2)
1S1
S1S 1
1
For every i{1l,2,...,k-l}in order, do steps a, b, c, and d:
a)
D
D
S
Si+l i
b)
D
c)
D
d)
D
(1) ))
D
SiSi+l
S
Si+1 +l
(i ))
(= D*
i
i+lS i
i+lS
D
(= D*
i
S i si+
min (DS.
S1+1i+l
Ds
i
(i))
SiSi+l
DS
+l
ii+l i
ii+l
i
Si+lSi+l
C
s~~ c
Si+i+
i+Sl
Si+lSi+l
3)
i+1lS i+l
For every iE{k,k-l,...,3,2} in order, do steps a, b, and c:
a)
D
DD*S
i
b)
C)
D
S
i
D
i
i-1 i
4
D
(= D*
1
iSi
S
S.S.
iSi-1
DS
SiSi-1
(=D*
SS
ii
-min (D
SiS
4)
i
(= D*
S
i-1S i
,D
Si-1i
D
SiSi
(=D*
SiSii-1
S
iSi_
l
For every rC{2,3,...,k-1} in order, do step a:
a)
For every i,je{l,2,...k} if ji-j
b)
D
S.S.
i
D
s.SS
i p
D
S S.
pj
-= r, do step b:
(=D
)
S.S.
3
where p is an element of the set Q = {s+l,s+2,...,t-2,t-2} for s = min(i,j)
and t = max (i,j) such that Ispl < ISm | for every mCQ.
A
rigorous proof of the algorithm would be very lengthy and repetitious,
and the interested reader is referred to Hu's work [6] for exposition of a
similar proof.
Steps 1 and 2 are bootstrapping successive diagonal and
first off-diagonal submatrices, so that at the end of step 2, DSkSk
Skk
~~·113~ll~so---·rrrsl~rrrassk
D*
SkS k
kk
~__
Step 3 is essentially a backwards form of step 2 and replaces the diagonal
and first off-diagonal submatrices with the respective unconditional shortest
distance submatrices.
Step 4 is one method for finding the unconditional
shortest distance submatrices corresponding to decomposition sets which are
separated by at least one intermediate set.
The ordering in step 4 allows
p to be any element of the set Q, and the particular choice of p minimizes
the number of operations.
If one assumes that the shortest distance calculations for submatrices
are done via Floyd's method (requiring 2p3 operations for a p x p submatrix)
and that the pseudo-multiplications are done in a straightforward manner
(requiring 2pqr operations to calculate A-B where A is dimension p x q and
B is q x r), then the number of operations required by the IHU algorithm is:
Step 1,
21Sl13
k=l
Step 2,
2 i~l
(2ISi.xi Isii 2 + Isi
Step 3,
2
(21Si1 2 ISi_11 + ISi_-ll Sil)
Step 4,
2
.
Isi+l12 + 1Si+xl3)
i=2
Z
ISi
ISj
I
i,j
such that li-jl > 1
The total number of operations is then
k-l
2(
i=l
k-l
i i usi l 3
3 +
Isii
-
i=2
Isil
i,j
such that
Isjl Isp I)
li-jl > 1
-19-
One may compare the IHU algorithm to other versions of Hu's algorithm.
For any given decomposition, the IHU algorithm requires fewer operations
than the fastest version of Hu's algorithm known to the author, which is
that due to Yen
[11].
For purposes of comparision, an example which
commonly appears in the literature [5,6,11]
for i even and ISij = t for i odd.
sets, be odd.
2(mt3
Define m =
k+l
2 .
is presented.
Let JSij
= 6
Assume 6< t, and let k, the number of
In this case, the new algorithm requires
2
3
2
2
2
2
3
+ (m +5m-6)t2 6 + (2m2+2m-6)t62 + (m2-4m+5)63) operations.
Yen's
modification requires 2(mt 3 + (m2+6m-7)t26 + (2m2+10m-20)t62 + (m2+6m-14)63).
The new algorithm is faster for the entire range of interest, i.e.
t > 6 > 1 and m > 2.
As a particular case, let 6= t and m = 3; the IHU
algorithm requires 82t3 operations, Yen's modification requires 128t3
operations, and Floyd's algorithm requires 250t3 operations.
-20-
Section V
Decomposing the Network for the IHU Algorithm
Perhaps even more important than the numerical
gains of the new algo-
rithm are the insights it provides into optimal decomposition of a network.
Assume that Floyd's method is used for shortest route computations on submatrices, and that pseudo-multiplications are done by the straightforward
technique.
It follows that for a given decomposition, if a further decom-
position exists by partitioning of existing sets, then the computation
time of the further decomposition is less than that of the given decomposition.
This "more the better" fact suggests a heuristically good decomposition
technique which can be performed by the computer or quickly guessed at by
eye.
If the decomposition is to be done automatically by the computer,
however, it should probably be limited to those cases where many shortest
route computations for the same topology will be performed, as in column
generating linear programs.
An algorithm for finding a good network decom-
position for the IHU algorithm is:
a)
find two nodes, j and k,
such that the minimal number
of arcs, d, connecting them is maximal over all pairs
of nodes; i.e. find the diameter of the network and an
associated pair of nodes;
b)
construct d+l sets by letting S 1 = {
and d
rm
< - or d
mr
} and Si+l = {mlm{Q-0i}
< o for some reS }.
1
This procedure was used to generate the IHU decomposition sets for the
examples of the next section, and the reader may want to look at the figures
associated with that section at this point.
-21-
Section VI
Some Examples Using the IHU and NXN Algorithms
In this section several examples are given which provide insight into
the classes of networks for which the NXN and IHU algorithms can substantially reduce shortest route computation time.
Although no examples
are presented for which the IHU algorithm is faster than the NXN algorithm, they do exist.
Such networks form a rather small and special class
of networks, and typically may be decomposed in such a manner as to be a
variation on the following theme:
ISil for i odd is large compared to ISil
for i even, and if jeSi and ksSi then j and k are very likely to have direct
arcs to one another.
The first example is an old version of the ARPA net which is shown in
figure 4.
In that figure, the NXN decomposition is defined by the numbering
of the nodes, and the IHU decomposition is defined by the partitioning of
the nodes with broken lines.
This network lends itself to NXN decomposition
due to the high number of nodes which have arcs directly to only two other
nodes--a fact which keeps the cardinality of connection sets very low.
The second example is the 47 node symmetric network shown in figure 5.
This network is not "locally connected" to a very high degree, but still
the NXN algorithm is (perhaps surprisingly)
efficient.
-22-
The final example is the 64 node network displayed in figure 6.
The
density of arcs is perhaps greater here than in the other examples, but a
high degree of local connectivity promotes the efficiency of the NXN
algorithm.
Efficiency is measured with Yen's implementation of Dijkstra's algorithm as the standard.
Theoretical efficiency refers to the relative
savings in the number of operations required to perform a shortest route
computation.
The computation times for the IBM 370-168 to execute the
Fortran programs of various algorithms were noted, and relative savings
are referred to as the measured efficiency.
The comparisions of the
various algorithms in performing shortest route calculations on the three
sample networks are summarized in table 1.
The Fortran programs were
complied by the IBM G1 compiler; and each algorithm not only computed the
shortest distance matrix, but also computed a routing matrix which specified
the next node from each node on the shortest route to any other node.
-23-
"
2 r3
I0
,.
c
4
2
X~/I
Figure 4.
a
Z4O2
I-2
-
The topology of ARPA network (at one stage of its
evolution) with decomposition information.
-24-
~8
332
and
=
13,27,l6,l9,45,35,1,24}
35
36.
44
77
19
38
.31
18
Figure 5.
A 47 node symmetric network (nodes are connected
to first and seventh nearest neighbors by arcs). NXN
decomposition is indicated by node labeling. IHU decomposition sets: S =
{1},S = {21,47,28,39}, · S = {34,
40,30,26,2,4,s,8
36,43}' S
and S
=
s =1417,18,29,22,23,42,4
32,25,
{46,33,~5,37,31,38,44,3,6,7,9,29,11,12},
=13,27,16,19,45,35,10,2
4}.
-25-
t2
08S
-
25
50
39
49
5
47
27
'23
$17
.Z4
45
\
344
Fe
>\>\/
26
64 node
\
55,/
vo31
510,
\/2 (
,l/--!89
Figure 6.
v
37t
138 1/
14+8
2)
\/
z2
54
4
29
t65
\
n
44
\
e,9
n
-63--
infrai
42-
41
/\,115
/
52
53
/3
5
Z?.
3_
1
34\.
2
A 64 node network with decomposition information.
\
-26-
ARPA
network
of figure
4
47 node
network
of figure
5
27040
154630
393216
1.00
1.00
1.00
.090
.450
1.115
Measured
Mieasured
efficiency
1.00
1.00
1.00
Number of
Numbera of
operations
6948
83880
124722
3.89
1.84
3.15
Computation
time (seconds)
time (seconds)
.020
.195
.290
Measured
Measured
efficiency
4.50
2.31
3.84
Number of
operations
operations
2828
28608
42416
efficiency
9.56
5.41
9.27
Computation
time (seconds)
.015
.115
.165
Measured
efficiency
efficiency
6.00
3.91
6.76
Number of
operations
operations
Dijkstra's
Theoretical
shortest
route
efficiency
algroute luComputation
time (seconds)
Ioritm
algorithm
HTheoretical
efficiency
64 node
network
of figure
6
NXN
algohNXNmTheoretical
algorithm
Table 1.
Comparative performance of three different shortest
route algorithms on the three sample networks.
-27-
Appendix
Section IA
Introduction
This appendix describes and lists the program which provided the
computational experience cited in this paper.
The program of section VIIA
reads the topology of the network, finds a decomposition for the IHU and
NXN algorithms, solves a sample shortest route problem via each algorithm
and the Dijkstra algorithm in order to compare computation times, and
calculates the number of operations required by each.
Typically, an
application of these programs requires at most two of the listed subroutines-one to decompose the network and one to calculate all the shortest routes.
The decomposition subroutine needs to be called only one time for any given
topology since a new set of data cards are punched by the decomposition subroutines which record the appropriate decomposition information.
In this
appendix, a hybrid notation will be employed which is a combination of
that used in the body of this report and that used in the Fortran programs.
The definitions of all Fortran terms are given in the comment cards at the
beginning of the program listing that is found in section VIIA.
-28-
Section IIA
Bookkeeping for Shortest Routes
The algorithms which are listed not only find the shortest distances
between every pair of nodes in the network, but they also record the
shortest routes.
The method which is used for this purpose is establishing
a "next node" matrix where NX(I,J) is the next node on the shortest path
from node I to node J.
Initially, NX(I,J) = J for every existing arc
(I,J), and every time the operation, dij
1
min (dij, dik+dkj
is performed
such that dik+dkj is the distinct minimum, then the algorithm makes the
replacement NX(I,J) + NX(I,K).
For the remainder of this appendix, the
algorithms are discussed only in terms of the shortest distance problem.
-29-
Section IIIA
The Main Program
The main program reads in the topology, assigns arc numbers and
provides the control for its specific purpose, i.e. to compare the
various algorithms.
In figure 7, an example network is presented.
Table
2 lists the data cards which communicate the topology of the network to
the program.
The first card is a header which provides the name of the
network and the values for NN, MIHU, MNXN, MAXPRI and NFORBD.
The second
card says that node "1" has "2" outgoing arcs which terminate on nodes
"12"and "3".
There is one such card for each node in succession.
-30-
Figure 7.
An Example Network With Seven Nodes And
Thirteen Arcs.
-31-
7 NODE, 13 ARC EXAMPLE NT
1
2
2
3
2
2
1
4
3
2
4 5
4
1
5
5
2
6
7
1
7
2
6
6
7
2
2
Table 2.
7
2
2
7
0
Topology Cards For The Network
of Figure 7
-32-
Section IVA
Subroutine DIJKST
Subroutine DIJKST is an implementation of Dijkstra's algorithm suggested by Yen [12].
The algorithm can be floated to perform the operation,
in the case that if j is a node number such that
DSiSi
iDsis
i
min k < j < max k, then jeS.. When the call to the subroutine DIJKST is
kcS.
kcS.
1
1
made, for this case, then NB = min k and NF = max k. If the operation,
kES.
kES.
D + D*, is to be performed via Dijstra's algorithm, then NB = 1 and
NF = NN.
-33Section VA
Subroutines DECIHU and IHU
Subroutine DECIHU decomposes the network for the IHU algorithm which
is implementated in subroutine IHU.
of Section V.
The method of decomposition is that
Figure 8 shows the network as decomposed by DECIHU with
the new node numbers as printed out.
Table 3 shows the cards punched by
DECIHU which record the decomposition information and describe the topology
in terms of the new node numbers.
Again, the first card is a header with
the title of the network, a "1" which says the cards were punched by DECIHU
and a "3" which is the number of IHU sets.
The second card says that node
"1" has "2" outgoing arcs, is a member of set number "1" (the next two
zeros have no significance), and the outgoing nodes are to nodes "2" and
"3"; and so forth.
on the next card.
The ninth card is a header for NTWIXT which starts
From them, NTWIXT(l,l) = "O", NTWIXT(1,2) = "0",
NTWIXT(1,3) = "2", NTWIXT(2,1) = "0", etc.
The information on these cards
define the variables found in the common block IHUSTF, and these values
are given in Table 4.
Subroutine IHU is a straightforward implementation of the IHU algorithm
as presented in Section IV.
via subroutine DIJKST.
The operations, DS S. +
DS.S .
are performed
-34-
J
Figure 8.
4
Node Renumbering and Partitioning by
Subroutine DECIHU for the Network of
Figure 7.
-35-
7
NODE, 13
1
ARC
EXAMPLE
7 - NODE,13
ARC
EXAMPLE NT
0 0
0
0
0
2
3
22
2
0
0
1
6
3 2
2
0
0
6
5
4
2
2
0
0
1
7
5 2
3
0
0
4
7
6
1
3
0
0 5
7 2
3
0
0
FOR
NTWIXT
00
Table 3.
2
0
3
7
1
2
1
NT
2
4
3
0
Cards punched by subroutine DECIHU which relate
the IHU decomposition information and the topology
in terms of the new node numbers for the network
of Figure 7.
-36-
Nl(l) = 1
N1(2) = 2
Nl(3) = 5
N2(1) = 1
N2(2) = 4
N2(3) = 7
NTWIXT(1,l) = 0
NTWIXT(2,2) = 0
NTWIXT(1,3) = 2
NTWIXT(2,1) = 0
NTWIXT(2,2) = 0
NTWIXT(2,3) = 0
NTWIXT(3,1)
NTWIXT(3,2) = 0
NTWIXT(3,3) = 0
= 3'
NS = 3
Table 4.
Values of the variables in labeled common
block IHUSTF which may be deduced from cards
in Table 3 for the network of Figure 8.
-37-
Section VIA
Subroutines DECNXN and NXN
Subroutine NXN is a general implementation of the NXN algorithm for
the case in which all the arcs in the network are not necesarily duplex.
Two connection sets are established for each node--one for outgoing connections and one for incoming connections.
Define CI as the outgoing
connection set, i.e. jeCC if there exists a path P from i to j such that
1
C(P) < - and every intermediate node k satisfies k < i.
define C. as the ith incoming connection set.
Similarly,
The NXN algorithm takes
this form:
1)
For every i£{1,2,...,NN-2} in order, do step a:
a)
For every j£ CI and k£ C?, do step b:
1
b)
2)
1
djk + min (djk, dji+dik)
For every iC{NN-2,NN-3,...,1} in order, do stepa:
a)
For every jc{i+l, i+2,...,NN}, do steps b and c:
b)
For every mCC., dij + min (dim.
+dmj
1
c)
1J
For every kOCI, dji
1 ji
in
dij)
1j
min (dji, ddjk+dki )
i jk ki
The method DECNXU uses for decomposing the network is given in Section
III with the alteration that nodes are chosen in order to sucessively
minimize ICoi + ICI|.
i
'
For the network of Figure 7, the new node number-
ing which implies the decomposition is shown in Figure 9.
The cards
punched by DECNXN which contain topology information in terms of new node
numbers and the decomposition information are shown in Table 5.
-38-
The interpretation of the cards is now more difficult but should be
clear by the program in Table 6 which reads in the cards of Table 5, sets
up arc numbers, and prepares the decomposition information for DECNXN.
One feature of the program not yet discussed is that of NFORBD which
is an input variable.
If a network is "locally connected" except for a
few nodes, they should be numbered last and suppressed from being assigned
new node numbers which are low by establishing NFORBD as the cardinality
of the set of such nodes.
-39-
4
Figure 9;.-
The topology of Figure 7 with new node numbers
as assigned by DECNXN
-40-
7
NODE, 13 ARC
EXAMPLE
1
2
3
2
32
2
1
2
1
3
3
2
2
2
4
2
5
NT
7
2
3
2
5
3
5
5
4
3
3
4
1
5
4
6
2 1
3
3
2
5
6
7
2
1
267
6
7
6
2
10
0
3
7
7
2
10
0
4
6
2
Table 5.
Cards punched by subroutine DECNXN for
the network of Figure 7 which contain decomposition information for the NXN algorithm
and topology information in terms of the new
node numbers.
V-4
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-42-
Section VIIA
Program Listing
The program of this section provided the computational results of
this paper.
The program is generously commented and should be transparent
when studied along with this appendix.
In general, clarity was sacrificed
for speed only in the subroutines DIJKST, NXN, and IHU.
-43-
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-71-
BIBLIOGRAPHY
1.
G. Dantzig, All Shortest Routes in-a Graph, Operations Research
House, Stanford University, Technical Report 66-3, Nov. 1966.
2.
E. Dijkstra, "A note on two problems in connexion with graphs",
Numerische Mathematik, Vol. 1, pp. 269-271, 1959.
3.
S. Dreyfus, "An appraisal of some shortest-path algorithms",
Opns. Res., Vol. 17, pp. 395-411, 1969.
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R. Floyd, "Algorithm 97, shortest path", Comm. ACM, Vol. 5, pp. 345,
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5.
T. C. Hu, "A decomposition algorithm for shortest paths in a network", Opns. Res., Vol. 19, No. 3, pp. 983-983, 1971.
6.
T. C. Hu, Integer Progranming and Network Flows, Addison-Wesley,
Reading, Mass., 1969.
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A. Hoffman and S. Winograd, "Finding all shortest distances in a
directed network", IBM J. Res. Develop., Vol. 16, pp. 412-414,
July 1972.
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NAC 4th Semiannual Tech. Report Project "Analysis and optimization
of store-and forward computer networks", Def. Doc. Cen., Alexandria,
Va., Dec. 1971.
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A. R. Pierce, "Biblography on algorithms for shortest path, shortest
spanning tree, and related circuit routing problems", Networks, Vol. 5,
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P. Spira, "A new algorithm for finding all shortest paths in a graph
of positive arcs in average time o(n 2 1og2 n)", SIAM J. Comput., Vol. 2,
No. 1, March 1973.
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J. Yen, "On Hu's decomposition algorithm for shortest paths in a network", Opns. Res., Vol. 19, No. 3, pp. 983-985, 1971.
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J. Yen, "Finding the lengths of all shortest ~aths in n-node nonnegative-distance complete networks using 1 n additions and n3
comparisons", J. Assoc. Comput. Mach., Vol. 19, No. 3, pp. 423424, July 1972.
-72-
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