Approximating Shortest Paths in Large-Scale Networks with
an Application to Intelligent Transportation Systems
YU-LI CHOU y T.J. Watson Research Center, IBM, Yorktown Heights, NY, Email: jychou@watson.ibm.com
H. E
DWIN ROMEIJN y Rotterdam School of Management, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam,
The Netherlands, Email: E.Romeijn@fac.fbk.eur.nl
R
OBERT L. SMITH y Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI
48109-2117, Email: rlsmith@umich.edu
(Received: January 1996; revised: January 1997; accepted: February 1998)
We propose a hierarchical algorithm for approximating shortest
paths between all pairs of nodes in a large-scale network. The
algorithm begins by extracting a high-level subnetwork of rela-
tively long links (and their associated nodes) where routing
decisions are most crucial. This high-level network partitions
the shorter links and their nodes into a set of lower-level sub-
networks. By fixing gateways within the high-level network for
entering and exiting these subnetworks, a computational sav-
ings is achieved at the expense of optimality. We explore the
magnitude of these tradeoffs between computational savings
and associated errors both analytically and empirically with a
case study of the Southeast Michigan traffic network. An order-
of-magnitude drop in computation times was achieved with an
on-line route guidance simulation, at the expense of less than
6% increase in expected trip times.
O
ur interest in this article is directed toward solving very
large-scale shortest path problems, motivated by the prob-
lem of finding minimum travel time paths within an on-line
route guidance system. Route guidance within the context of
Intelligent Transportation Systems is the task of providing
routes between origins and destinations that promise to
minimize the trip times experienced. The link travel times
that are provided as an input to this function are time-
dependent forecasts based upon current and anticipated
traffic congestion (see e.g., Kaufman and Smith,
[7]
Wunder-
lich and Smith
[9]
). Because of the rapid change of link travel
times caused by time-varying travel demands and lane
blockage resulting from incidents, the data used in comput-
ing the shortest paths information is updated periodically,
ideally every 5 to 10 minutes. During the time interval
between data updates, a shortest path must be provided for
every origin/destination (O/D) associated with trips that
begin during that time slice. Thus, the calculation of shortest
paths must be efficient enough to respond in a timely way to
trip requests on a real-time basis. Because a realistic problem
may have hundreds of thousands of nodes, all of which may
be potential origins and destinations, a fast heuristic that can
provide good approximations in a limited amount of time
may be preferred to exact methods.
We present here a hierarchical approach for finding ap-
proximate solutions for shortest path problems. A key idea
behind our development is the imposition of a hierarchical
network structure. Our approach is motivated by the obser-
vation that traffic networks have a hierarchical structure that
divides links (and nodes corresponding to intersections of
these links) into two or more classes (Yagyu et al.
[10]
). The
higher level corresponds to longer links (i.e., highways)
where the frequency of decision opportunities is low, with
routing decisions being correspondingly more important.
The lower level corresponds to links of limited duration (i.e.,
surface streets) and, correspondingly, more opportunities to
correct routing decision errors. The links and nodes com-
prising the higher level subnetwork correspond to a macro-
scopic representation of the whole network. This high-level
subnetwork partitions the original network into a set of
subnetworks at the lower level in a way to be described
later. By solving for all pairs of shortest paths in the higher
level subnetwork and interfacing with gateways into the
lower level subnetworks, we achieve significant economics
of computation, albeit at the expense of a loss of optimality.
We explore in this article the magnitudes of the computa-
tional gains and associated errors from optimality.
A similar approach is taken by Shapiro, Waxman, and
Nir,
[8]
who classify each of the arcs in the network, and
approximate shortest paths based on the assumption that it
is desirable to spend as little time as possible on lower level
arcs. Their approach can be very efficient, especially when
only a few shortest paths need to be computed. Moreover, it
has the advantage that it is not necessary to explicitly con-
struct a hierarchy of subnetworks. Alternatively, our ap-
proach has the advantage that much of the preprocessing
computations can be done in parallel, so that the computa-
tion of an additional shortest path is very cheap. Our ap-
proach thus seems more suitable for cases where at least a
moderate number of shortest paths have to be computed.
Habbal, Koutsopoulos and Lerman
[6]
propose a parallel
decomposition method for solving the all-pairs shortest path
problem. Their algorithm, however, is an exact one, and
Subject classifications: Distance algorithms
Other key words: Networks, graphs, traffic models.
163
INFORMS Journal on Computing 0899-1499y 98 y0902-0163 $05.00
Vol. 10, No. 2, Spring 1998 © 1998 INFORMS
seems to be less suitable for use in an on-line route guidance
system, which is the motivation for our study.
We begin by introducing the hierarchical algorithm (HA)
in Section 1. We show how the nature of traffic networks
induces a natural choice for how the hierarchical network
should be modeled. In Section 2, we discuss the issue of the
efficiency of HA for solving the all-pairs shortest path prob-
lem, as well as the case where only a limited number of
nodes can occur as origin or destination. Furthermore, con-
ditions for efficiently implementing HA are presented. In
Section 3, the efficiency of HA when applied to an on-line
route guidance system is investigated. A numerical experi-
ment is reported in Section 4, which is based upon a real
traffic network, the Southeast Michigan road network.
1. A Hierarchical Approach
1.1 Overview
In this section, we introduce a hierarchical approach to
solving for shortest paths in a large-scale network. The main
idea is to decompose the network into several smaller (low-
level) subnetworks. When a shortest path needs to be found
between two nodes in the same subnetwork, we approxi-
mate this shortest path by the shortest path having the
property that all nodes on the path are contained in that
same subnetwork. The resulting path is an approximate
shortest path in that the true shortest path could leave the
subnetwork at some point, and re-enter it before arriving at
the destination.
To provide approximations of shortest paths between
nodes not contained in the same subnetwork, we define a
(high-level) network, which is actually a subnetwork of the
original network, whose nodes intersect all (low-level) sub-
networks. The nodes that are present in both the high-level
network and in one or more low-level subnetworks are
called macronodes. The high-level network is called the ma-
cronetwork, reflecting the fact that it gives a macroscopic
view of the original network. Correspondingly, the low-
level subnetworks are called microsubnetworks, reflecting
the fact that they give a microscopic view of a part of the
original network. Given this decomposition, we can approx-
imate a shortest path between two arbitrary nodes of the
original network by constraining the path to pass out of the
microsubnetwork containing the origin, through the ma-
cronetwork, and into the microsubnetwork containing the
destination. Variants of this general procedure are consid-
ered, based upon the departing macronode out of the origin
subnetwork and the entering macronode into the destina-
tion subnetwork chosen.
1.2 The Hierarchical Algorithm
1.2.1 Definition
Let G 5 (V, A, C) be a strongly connected directed graph
with a set of nodes V, a set of arcs A # V 3 V, and a
non-negative arc length function C: A 3 R. (A directed
graph is strongly connected if there exists a directed path
from each node to each node of the network.) Our objective
is to approximate shortest paths within the network G.
We will first describe the HA in its most general form,
consisting of the following three phases. In Sections 1.2.2
and 1.2.3, we will briefly discuss some of the choices that can
be made within this general framework. These choices will
then be illustrated in Section 4.
Phase 0: Decomposition
Extract a strongly connected, but not necessarily fully
dense, macronetwork G
˜
5 (V
˜
, A
˜
, C
˜
), where V
˜
# V, A
˜
# V
˜
3 V
˜
,
and C
˜
: A
˜
3 R. The nodes in V
˜
will be called macronodes, and
correspondingly, the arcs in A
˜
will be called macroarcs. Each
macroarc (I, J) [ A
˜
corresponds to some directed micropath
from I to J in the original network G where the correspond-
ing arc length C
˜
(I, J) is defined as the length of this path.
Note that the micropath corresponding to an arc (I, J) [ A
˜
is
not necessarily the shortest path between I and J in the graph
G, but rather depends on the structural properties of the
network. Finally, note that all macronodes are micronodes
as well and that there exists such a subnetwork associated
with every choice of macronode set V
˜
# V because G is, by
assumption, strongly connected.
Divide the original network G into H strongly connected,
but not necessarily disjoint, microsubnetworks G
h
5
(V
h
, A
h
, C
h
), where V
h
# V, V 5 ø
h
V
h
, A
h
5 A ù (V
h
3 V
h
),
and V
˜
ù V
h
Þ A for h 5 1,...,H. Note that every cover (V
h
,
h 5 1, 2, ..., H)ofV (i.e., a set of subsets satisfying V 5
ø
h
V
h
) is allowed as long as each V
h
in the cover contains at
least one macronode.
Phase I: Constrained Shortest Paths
Find all required shortest paths (which can either be all
shortest paths or a suitable subset thereof, depending on the
type of shortest path problem that needs to be solved) in the
macronetwork G
˜
and in the microsubnetworks G
h
, h 5
1,...,H. Let the function
f
˜
: V 3 V
˜
3 R
return the shortest path lengths in the macronetwork G
˜
.
Similarly, let
f
h
: V
h
3 V
h
3 R
return the shortest path lengths in microsubnetwork G
h
, h 5
1,...,H.
Phase II: Combination
Consider a pair of micronodes i [ V
h
, j [ V
,
.Ifh 5 , (i.e.,
the micronodes are contained in the same microsubnet-
work), then the corresponding shortest path between i and j
is approximated by the shortest path from i to j computed in
Phase I with all intermediate nodes contained in this sub-
network V
h
. Otherwise, the shortest path is approximated by
combining three parts computed in Phase I:
(i) the shortest path (in G
h
) from i to some macronode
I [ V
h
ù V
˜
;
(ii) the shortest path (in the macronetwork G) from I to
some macronode J [ V
,
ù V
˜
; and
(iii) the shortest path (in G
,
) from macronode J to j.
Note that there always exist macronodes I and J in steps (i)
and (iii) that are also micronodes in G
h
and G
,
, respectively,
by construction in Phase 0. Variants of HA depend upon
164
Chou, Romeijn, and Smith
how these nodes I and J are selected when there are multiple
candidates. Note also that the combination phase can be
simplified somewhat when the origin node i and/or the
destination node j are macronodes. In particular, in those
cases we can skip step (i) and/or (iii) in Phase II. However,
depending on the particular choice of macronode made in
these steps, it may be fruitful to treat an origin or destination
macronode as any other micronode. We will come back to
this later when discussing strategies for choosing the exiting
and entering macronetwork macronodes for a given O/D
pair.
1.2.2 Comments on the Decomposition Phase
In the decomposition phase of the Hierarchical Algorithm,
there are obviously many possible ways of choosing the
macronodes, macroarcs, and microsubnetworks. However,
in a specific application, there usually is a natural hierarchy
within the arcs. For example, in a traffic network, the ma-
cronetwork can be chosen to be the network consisting of
highways and freeways, with the macronodes being a suit-
ably chosen subset of the entrances to and exits from these.
The macroarcs will then be micropaths consisting only of
highways and freeways, and the microsubnetworks could
be chosen in a natural way as the subnetworks enclosed by
the macroarcs. In Section 4, we will illustrate this procedure
of forming the macro- and micronetworks using the actual
Southeast Michigan road network.* We will also see there
that usually some adjustments are necessary to obtain a
macronetwork that is connected and sufficiently dense in the
original network for the HA algorithm to be successful.
1.2.3 Comments on the Combination Phase
If, in the combination phase, different microsubnetworks
containing the origin and destination contain only one
macronode each, it is unambiguous as to how to construct
the approximating path from the description of HA. How-
ever, often there will be more than one macronode that can
be used to connect the microsubnetworks to the macronet-
work. In that case, we need to make a choice among these
macronodes. An intuitively appealing choice would be to
choose the macronodes that allow us to enter the macronet-
work as soon as possible. In other words, we choose the
closest macronode that can be reached from the origin, and
the macronode from which the destination can be reached as
quickly as possible. More precisely, if i [ V
h
is the origin,
and j [ V
,
(, Þ h) is the destination, then the connecting
macronodes are chosen to be
I* 5 arg min
I[V
h
ùV
˜
f
k
~i, I!
and
J* 5 arg min
J[V
,
9V
˜
f
,
~ J, j!.
The version of HA corresponding to this rule for the selec-
tion of macronodes will be called Nearest HA.
Obviously the best possible selection rule among all HA
variants in terms of quality of the solution obtained (but also
the most expensive one in terms of computation time) is to
select the pair of macronodes that yields the shortest approx-
imate path. In other words, choose the pair of connecting
macronodes as follows:
~I*, J*! 5 arg min
~I, J![~V
h
ùV
˜
!3~V
,
ùV
˜
!
$ f
h
~i, I! 1 f
˜
~I, J! 1 f
,
~ J, j!%.
This variant of HA will be called Best HA.
At times there will be more than one microsubnetwork
containing the origin (and/or destination). In that case we
would adjust the choice of macronodes as follows. For Near-
est HA, we set
I* 5 arg min
I[ø
h:i[V
h
V
k
ùV
˜
f
k
~i, I!
and similarly for J*. Similar adjustments would be made for
finding (I*, J*) for Best HA.
2. Computational Complexity Analysis
2.1 Notation and definitions
For the heuristic HA to be useful, there obviously needs to
be time savings to balance the loss in precision in computing
the approximate shortest paths produced by it when com-
pared to an exact algorithm. In this section we investigate
the relative efficiency of both Nearest HA and Best HA
under various conditions. Before analyzing the complexity
of HA, we first review the concepts of 2, V, and Q functions
(see e.g., Aho, Hopcroft and Ullman
[1]
).
Definition 2.1. Let f and g be functions from the non-negative
integers to the non-negative reals.
(a) f(n) is said to be 2(g(n)) (or f(n) 5 2( g(n))) if there exist
positive constants c and n
0
such that
f~n! < cg~n! for all n > n
0
.
Note that the 2-notation is used to specify an upper bound on the
rate of growth of some function. Analogously, the V-notation is
used to provide a lower bound on the rate of growth, whereas the
Q-notation is used if the upper and lower bounds coincide.
(b) The function f(n) is said to be V(g(n)) (or f(n) 5
V(g(n))) if there exist positive constants c9 and n
0
such
that
f~n! > c9g~n! for all n > n
0
.
(c) The function f(n) is said to be Q(g(n)) (or f(n) 5
Q(g(n))) if there exist positive constants c, c9 and n
0
such
that
c9g~n! < f~n! < cg~n! for all n > n
0
.
Now, assume that the number of nodes uVu in the graph G
is Q(N), where N is an input parameter characterizing the
size of the graph. Because we are focusing on road networks,
which are sparse (i.e., the degree of each of the nodes can be
* SEMCOG, 1985. Survey of Regional Traffic Volume Patterns in
Southeast Michigan. Technical Report, Southeast Michigan Council
of Governments, Private Communication.
165
Approximating Shortest Paths in Large-Scale Networks
bounded from above by a constant, independent of the size
of the network), the most efficient algorithm for computing
all shortest path lengths exactly is Dijkstra’s algorithm (see
e.g., Denardo,
[4]
Dreyfus and Law
[5]
). The number of oper-
ations needed by this algorithm is 2(N log N) and V(N) for
solving for the shortest paths from a single origin to all
destinations. We will repeatedly make use of this result in
the remainder of this section.
To gain insight into the potential gain in computation
time achievable by HA and how we should decompose the
original network to realize this gain, we assume within this
section that we decompose the network in such a way that
the number of macronodes is Q(N
m
) (for some 0 , m , 1),
and that the number of microsubnetworks is Q(N
k
) (for some
0 , k , 1). The actual performance gain under real-world
conditions will be explored in Section 4. Each microsubnet-
work is further assumed to be of roughly the same size, so
that the number of nodes in each subnetwork is Q(N
12k
).
Moreover, we assume that each macronode is shared, on
average, by a constant number of microsubnetworks, so that
the number of macronodes per microsubnetwork is Q(N
m2k
)
(so that the number of macronodes is roughly equal for all
microsubnetworks). Because we need at least one macro-
node per microsubnetwork, we obviously need that k ¶ m.
The relative magnitudes of m and k to achieve the greatest
computational savings will be explored under two scenarios
in the next two subsections.
2.2 The All-Pairs Shortest Path Problem
We begin by investigating the efficiency of HA for solving
the all-pairs shortest path problem. Let C
N
(N) denote the
number of operations needed for approximating all shortest
path lengths in a network structure defined above using
Nearest HA, C
B
(N) the corresponding number using Best
HA, and C
D
(N) the number of operations needed for com-
puting all shortest paths using Dijkstra’s algorithm in the
same network. The following theorem gives the relative
efficiency of Nearest HA with respect to Dijkstra’s algo-
rithm.
Theorem 2.2 The relative efficiency of Nearest HA with respect
to Dijkstra’s algorithm for the all-pairs shortest path problem
satisfies
C
D
~N!
C
N
~N!
5 2~log N! and
C
D
~N!
C
N
~N!
5 V~1!.
Proof. See the appendix. n
The proof of the above result shows that the combination
phase is the most time-consuming one, and, in fact, that the
complexity for that phase is independent of the particular
decomposition chosen. Notwithstanding this fact, we could
try to minimize the effort necessary for Phase I, even though
this phase is already much less time-consuming than Phase
II. Recall from the proof of Theorem 2.2 that Phase I takes on
the order of N
max(22k,2m)
operations (disregarding the “log
N” term). Recall also that m Ä k. Therefore, it seems optimal
(with respect to computation time) to choose m 5 k. The
computation time for Phase I then becomes of the order
N
max(22k,2k)
, which is minimized if 2 2 k 5 2k,orifk 5
2
⁄
3
.
This means that to most efficiently implement Nearest HA,
the network should be decomposed in such a way that
(i) the number of subnetworks should be of the same
order as the number of macronodes, i.e., the number
of macronodes per subnetwork should be
independent of the size of the network; and
(ii) the size of each of the subnetworks should be smaller
than the size of the macronetwork.
However, note that a pure minimization of the computa-
tional effort does not take into account any natural structure
present in the network, or the magnitude of the error in-
curred by the corresponding decomposition.
The following theorem is the analogue of Theorem 2.2 for
Best HA.
Theorem 2.3 The relative efficiency of Best HA with respect to
Dijkstra’s algorithm for the all-pairs shortest path problem satis-
fies
C
D
~N!
C
B
~N!
5 2~N
2~k2m!
log N!
and
C
D
~N!
C
B
~N!
5 V~N
2~k2m!
!,
where m Ä k.
Proof. See the appendix. n
Unlike in the case of Nearest HA, the only instances where
savings might be obtained occur when k 5 m, i.e., when the
number of macronodes per subnetwork does not grow with
N. Whenever m . k, the complexity “savings” actually be-
come a complexity “dissavings” in the sense that Dijkstra’s
algorithm is more efficient than Best HA! However, if k 5 m,
then the relative efficiency of Best HA is the same as the
relative efficiency of Nearest HA.
2.3 The Limited Shortest Path Problem
We have seen that HA does not always perform more effi-
ciently than Dijkstra’s algorithm for solving all shortest
paths in a network defined as above. Fortunately, however,
in traffic networks, most nodes cannot actually occur as
origin or destination. Consider, for instance, surface street
intersections or highway entrances or exits. To model this
observation, we assume that there are Q(N
g
) nodes that can
occur as origins or destinations, for a total of Q(N
2
g
) shortest
paths to be computed (where 0 ,
g
¶ 1). Moreover, we
assume that these origins (destinations) are spread out uni-
formly over the network, and that every microsubnetwork
contains at least one origin, so that
g
Ä k.
In this subsection, we will be concerned with computing
all shortest paths between the nodes that can occur as origin
or destination. Note that, in traffic networks, we rarely need
to compute all such shortest paths because network data
(e.g., link travel times) is being updated periodically and not
all possible trips will take place between any two updates.
This situation will be analyzed in Section 3.
166
Chou, Romeijn, and Smith
Similar to the notation in the previous section, let
C
D
(N;
g
), C
N
(N;
g
), and C
B
(N;
g
) denote the number of op-
erations needed for computing or approximating the re-
quired shortest paths in the network structure defined above
using Dijkstra’s algorithm, Nearest HA, and Best HA, re-
spectively. In particular, it is easy to see that
C
D
~N;
g
! 5 2 ~N
11
g
log N!
and
C
D
~N;
g
! 5 V~N
11
g
!.
Theorem 2.4. The relative efficiency of Nearest HA with respect
to Dijkstra’s algorithm for the limited-pairs shortest path problem
satisfies
C
D
~N;
g
!
C
N
~N;
g
!
5 2~N
,
1
~k, m,
g
!
log N!
and
C
D
~N;
g
!
C
N
~N;
g
!
5 V
S
min
S
N
,
2
~k, m,
g
!
log N
, N
12max~
g
,m2k!
DD
,
where
,
1
~k, m,
g
! 5 min~k 2 max~0, m 2
g
!,
1 2
g
2 2max~0, m 2
g
!!
,
2
~k, m,
g
! 5 min~k 2 max~0, m 2
g
!,
1 2
g
2 2~m 2
g
!!.
Proof. See the appendix. n
As before, we can investigate the optimal number of
macronodes and microsubnetworks in terms of computa-
tional efficiency. If, for simplicity, we only consider the
2-expression for the relative efficiency of Nearest HA, we
see that
C
D
~N;
g
!
C
N
~N;
g
!
5 2~N
min~k, k2m1
g
,12
g
,11
g
22m!
log N!.
Clearly, it is optimal to choose k as large as possible. Because
k ¶ m by assumption, we will let k 5 m. The relative
efficiency then becomes
C
D
~N;
g
!
C
N
~N;
g
!
5 2~N
min~m,
g
,12
g
,11
g
22m!
log N!.
The optimal relative efficiency can now be obtained if m 5
1 1
g
2 2m,orm 5 (1 1
g
)/3. It is easy to see that (1 1
g
)/3 Ä min(
g
,12
g
), so that the optimal relative efficiency
is 2(N
min(
g
,12
g
)
log N), for k 5 m 5 (1 1
g
)/3.
Now consider the case of Best HA. Then the following
theorem is the analog of Theorem 2.4.
Theorem 2.5. The relative efficiency of Best HA with respect to
Dijkstra’s algorithm for the limited-pairs shortest path problem
satisfies
C
D
~N;
g
!
C
B
~N;
g
!
5 2~N
,9
1
~k, m,
g
!
log N!
and
C
D
~N;
g
!
C
B
~N;
g
!
5 V
S
min
S
N
,9
2
~k, m,
g
!
log N
, N
12
g
22~m2k!
DD
.
where
,9
1
~k, m,
g
! 5 min~k 2 max~0, m 2
g
!,
1 2
g
2 2~m 2 k!!
,9
2
~k, m,
g
! 5 min~k 2 max~0, m 2
g
!,
1 2
g
2 2~m 2
g
!!.
Proof. See the appendix. n
From Theorem 2.5, we have the following relative efficiency
for the case of Best HA
C
D
~N;
g
!
C
B
~N;
g
!
5 2~N
min~k, k2m1
g
,12
g
22m12k!
log N!.
Again, it is optimal to choose k 5 m, yielding
C
D
~N;
g
!
C
B
~N;
g
!
5 2~N
min~m,
g
,12
g
!
log N!.
So the optimal relative efficiency is again
2(N
min(
g
,12
g
)
log N), attained for k 5 m Ä min(
g
,12
g
).
Table I summarizes the complexity results derived in this
section.
3. On-Line Shortest Path Calculations
In this section, we explore the complexity of Best HA and
Dijkstra’s algorithm when implemented in an on-line short-
est path route guidance system. In an on-line route guidance
system, the arc length function C is updated regularly, on
the basis of recent information concerning the link traversal
times (see Kaufman and Smith
[7]
). In such a system, a path
will have to be provided for a certain O/D pair if a request
is made for that pair during the time slice during which the
arc length function is valid. Moreover, such a path will only
have to be computed if and when the O/D pair is requested
for the first time during a time slice. Unlike the problems
discussed in the preceding section, where the number of
O/D pairs requiring solutions was assumed known, the
Table I. Summary of Complexity Results
Operation Complexity
Number of shortest paths
solved
2(N
g
)
All-pairs shortest paths in
micronetworks
2(N
max(
g
,m)112k
log N)
All-pairs shortest paths in
macronetworks
2(N
2m
log N)
Combination phase
Nearest HA 2(N
m2k
)
Best HA 2(N
2(m2k)
)
167
Approximating Shortest Paths in Large-Scale Networks
