NETWORK REPRESENTATION AND SHORTEST PATH
REFLECTING THE RAMP ENTRY OR EXIT DIRECTION
LIMITATION
Kazutaka TAKAO1*, Tohru HIGASHI1, Koji YASUDA2, Yasuo ASAKURA3
1. Institute of Systems Science Research, Japan
Shinmachi-I.S. Bldg., 428 Komusubidana-cho, Shinmachidori-Shijo-agaru, Nakagyo-ku, Kyoto 6048223, Japan, Tel: +81-75-221-3022, Fax: +81-75-231-4404, E-mail: {takao, higashi} issr-kyoto.or.jp
2. Regional Futures Research Center, Inc., Japan
3. Kobe University, Japan
ABSTRACT
There are some ramp entry or exit direction limitations on networks. This kind of limitation is
represented by successive passage prohibition (SPP). It indicates that the vehicles cannot
successively pass two links with SPP attributes. However, the shortest-path tree sometimes
cannot be expressed with a simple tree when a network having this limitation is represented
with single nodes. In such a case, an accurate shortest path cannot be found by simply
applying Dijkstra's algorithm. Therefore, we propose an intersection expansion to find the
most accurate shortest-path tree. It is transformed automatically during the process of finding
the shortest path.
KEYWORDS
shortest path, Dijkstra's algorithm, intersection expansion
BACKGROUND AND OBJECTIVE
The shortest path problem is one of the major topics in the field of ITS and travel engineering
because it is widely used in car navigation systems, travel behaviour analysis, traffic
assignment, etc. It has been explained for many years with simple network examples in much
of the literature, including textbooks and handbooks. In many cases, an intersection is
represented as a single node and a street is represented as a two-way single link (Figure 1).
Many of these sources explain Dijkstra's algorithm (cf. [1]) as is without any alteration. This
is adequate if the network is simple, i.e., if the ramps are connected to all directions and if
there is no turn prohibition at the intersections.
In the real world, however, networks are complex with respect to the passage of vehicles. For
example, there are incomplete ramps (easterly, westerly, etc.) that are connected to one
direction, i.e., vehicles can enter in only one direction of the expressway or vehicles from
only one direction can exit the expressway at the ramp. Moreover, there are many turn
prohibitions in urban areas, where vehicles cannot pass specific combinations of links. In such
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a case, an accurate shortest path cannot be found by simply applying Dijkstra's algorithm to
the network with single-node representation. This paper proposes a network representation
that reflects this real-world limitation on ramp entry or exit direction and reports on a method
of finding the shortest path.
Figure 1 - Typical single-node network representation
SUCCESSIVE PASSAGE PROHIBITION REPRESENTATION
Limitations on ramp connections are often expressed with indistinctive representation, making
it troublesome to maintain the data. Successive passage prohibition (SPP) representation is a
method for representing a ramp entry or exit direction limitation without yielding this
problem. SPP is realized by attaching a special attribute to specific links, namely the ‘SPP
attribute’. It indicates that the vehicles cannot successively pass two links with SPP attributes.
Ashiya ramp
G
10
109 H 110
1
Nishinomiya ramp
C
20
I
10
104
M
107 F 108
1
K
103
111
1
E
10
B
20
1
3
105 D 106
1
102
L
J
Kobe
A
10
1
101
Osaka
Mukogawa ramp
Figure 2 - ‘Successive Passage Prohibition’ network representation
Figure 2 shows an example of the Hanshin Expressway. The dotted links are with SPP
attributes, whereas the solid links are without SPP attributes. The three-digit number indicates
node ID, a one- or two-digit number indicates link travel time, and an alphabetic letter
indicates link ID. A-B-C express the urban streets and D-E-F-G-H-I express the Hanshin
Expressway. Ashiya and Mukogawa ramps are easterly ramps and they are only connected to
Osaka; vehicles can enter the expressway only toward Osaka at these ramps and vehicles only
from Osaka can exit the expressway there. This limitation is represented with SPP; vehicles
cannot pass J-D or L-H. In the same way, Nishinomiya ramp is a westerly ramp and it is only
connected to Kobe; vehicles cannot pass K-F.
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PROBLEM OF THE SHORTEST PATH
Concerning the shortest path finding problem, the direction limitation of ramps represented by
SPP is regarded as the passage prohibition on a specific combination of links that are
connected to a node. Dijkstra's algorithm seems to be usable for handling SPP by simply
ignoring the label-setting operations for the prohibited combinations. However, this does not
work in practice, since the shortest-path tree in the network with entry or exit direction
limitation cannot always be expressed as a simple tree when it is represented with single
nodes and single links. Therefore, it is insufficient to apply Dijkstra's algorithm as is.
Figure 2 illustrates an example. Consider finding the shortest path from 101 to 104. During
the process, the shortest path from 101 to 107 is searched as A-K, and thus the backpointer of
107 is labelled K. Next, since the passage of K-F is prohibited, the labelling operation of 108
as F is ignored. Consequently, path F-G-H is never used in the shortest-path tree from 101,
and wrong path A-B-L-I-M would be found. Actually, the shortest path from 101 to 104 is
J-E-F-G-H-I-M. Therefore, two branches of the shortest-path tree conflict at node 107, i.e., its
backpointer should be K when the trip terminates at 107 but E when the trip continues to 104.
This problem cannot be resolved by representing only the expressways as double links and
nodes. Vehicles often cannot U-turn at the exit of a ramp because it would be prohibited
physically or legally. Therefore, they cannot enter the expressway again immediately at the
ramp from which they have just exited. Figure 3 shows an example. Consider link C to be an
expressway link. If the U-turn at 104 is prohibited, the shortest path from 101 to 103 should
not be 101-102-104-102-103. Instead, the shortest path is E-D-C-B. On the other hand, the
shortest path from 101 to 105 is A-C-D. Therefore, the two directions of links C and D
overlap in the shortest-path tree from 101. This shows that the problem of branch overlap in
the shortest-path tree is not limited to the links that correspond to the expressways.
104
102
C
3
1
1
105
A
D
1
B
103
E
10
101
Figure 3 - Sample network with shortest-path tree branch overlap
Thus, the shortest-path tree on the network with ramp entry or exit direction limitation
sometimes cannot be found with single-link and single-node representation. Note that the turn
prohibition of intersections is also regarded as the passage prohibition on a specific
combination of links, and thus the same problem arises.
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INTERSECTION EXPANSION
Dijkstra's algorithm does not pay attention to the condition of the label that is set on a node;
all of the links connected to a node must be treated equally. As a result, the algorithm
assumes that the shortest-path tree is represented in a simple tree. The cause of the problem is
that special conditions are attached to specific combinations of links with SPP.
Figure 4 - Intersection expansion
(a) Before expansion
(b) After expansion
Figure 5 - Expansion avoiding the SPP pair
Therefore, this paper proposes a network transformation method by intersection expansion to
remove the SPP. Figure 4 shows an example of a node to which four two-way links are
connected. The four real links are transferred into eight one-way double links expressed as fat
lines; expanded nodes are connected with hypothetical links shown as thin lines. The U-turn
combinations are not connected, and thus paths that include U-turns are avoided in the
shortest path finding. Another example that involves SPP is shown in Figure 5. The SPP
combinations are simply avoided from connecting with hypothetical links. In this way, the
network is transformed to the representation without special condition. The intersection
expansion was suggested in Asakura (1987) (cf. [2]) originally for expressing the turning
delay at intersections. However, this idea can be applied to removing SPP and resolving the
non-simple-tree problem.
By expanding the intersections, the shortest-path tree can be found appropriately by simply
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applying Dijkstra's algorithm. This expansion is required only while finding the shortest path
and is transformed automatically. Figure 6 shows the expanded form of Figure 3; the
combinations of SPP links are not connected with hypothetical links; additionally,
hypothetical links are added from the origin (root). Thus, the problem of branch overlap is
resolved. This expansion is required for the entire network because the problem is not limited
to expressways as shown in Figure 3. Furthermore, note that this method also works for
handling cyclic paths, e.g. in Figure 7, 101-102-104-105-106-104-102-103.
Origin
Figure 6 - Sample network after intersection expansion
106
Loop Line
103
104
Osaka Port
102
105
101
Amagasaki
Figure 7 - Sample network that gives cyclic path
The merits of intersection expansion are that the simple form of Dijkstra's algorithm can be
applied after the transformation and that the network is transformed automatically. The
demerit is that the network size explodes, and thus the calculation time becomes long.
Therefore, it would be a trade-off decision to choose between the expanded form for
obtaining an accurate shortest-path tree or the unexpanded form for saving calculation time.
INTERSECTION EXPANSION II
With intersection expansion, one node is expanded to eight nodes when four two-way links
are connected to it. This method makes it easy to introduce an intersection delay by attaching
travel times to the hypothetical links. Moreover, as a special case, when we are not interested
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in the intersection delay, the travel time of the hypothetical links are ignored and the network
representation can be transformed into a simpler form; it is aggregated into the representation
of four nodes by joining the hypothetical links to their downstream links, namely ‘Intersection
Expansion II’. Figure 8 shows an example. Only the links that are involved in the route that
flows out and upward are shown. Consequently, each intersection is expanded into the flow-in
nodes that correspond to the connected links and the expanded links that correspond to the
connection combinations at the node.
(a) Intersection Expansion
(Eight nodes per intersection)
(b) Intersection Expansion II
(Four nodes per intersection)
Figure 8 - Intersection Expansion II
Intersection Expansion II does not cause any degeneracy because every node and link in the
expanded form has a correspondence with a real one in the unexpanded form. The effect of
Intersection Expansion II is that it can reduce the calculation time of Dijkstra's algorithm in
comparison with Intersection Expansion because the performance is strongly influenced by
the number of nodes.
COMPARISON OF SHORTEST PATH FINDING
Table 1 - Comparison of representations for shortest path finding
Representation
Conventional representation
Double-linked expressway
Intersection expansion on entire network
U-turn
avoided
arises
avoided
Shortest path
incorrect
(U-turn)
correct
Calc. time
short
short
long
A comparison of representations for shortest path finding is shown in Table 1. As described
above, the conventional representation has the possibility of yielding incorrect shortest paths.
The second representation, which represents only expressways as double links, has the
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possibility of finding U-turns. Therefore, it poses a risk for practical use, even though it may
not include a U-turn in a fortunate case. The last one is the suggested representation, and it
gives correct shortest paths. However, its calculation time would be longer than the above two
representations. Accordingly, it is examined in the next section with a real network.
EXAMPLE ON A REAL NETWORK
We carried out an all-or-nothing traffic assignment on a real network to see the calculation
time and the shortest path. The network consists of 10,850 nodes and 15,328 links before
expansion; 71,908 nodes and 124,482 links after Intersection Expansion; 41,379 nodes and
93,953 links after Intersection Expansion II. The number of shortest-path trees is 1,140. The
CPU times of the calculation are as follows (Pentium4 630 3 GHz with HT technology,
including the time to output the data for Table 2):
(A) Without intersection expansion:
22 sec.
(B) Intersection Expansion:
160 sec.
(C) Intersection Expansion II:
145 sec.
Note that a special consideration is introduced in our program for hypothetical links in (B), so
the time cannot be compared simply by the ratio of the number of nodes and links.
We compared the OD shortest travel times between (A) and (B) by indicating how much
shorter (B) is than (A). Note that the results of (C) are identical with (B). The travel times of
all OD pairs whose OD flows are not zero were calculated. The results are shown in Table 2.
Table 2 - Comparison of OD shortest travel times
Rank
Identical
Shorter up to 5%
Shorter up to 10%
Shorter up to 15%
Shorter up to 20%
Shorter over 20%
Total
Num. of OD pairs
184296 (96.8%)
5314 ( 2.8%)
659 ( 0.3%)
96 ( 0.1%)
27 ( 0.0%)
6 ( 0.0%)
190398
Thus, the suggested methods (B) and (C) give shorter OD travel times than the conventional
method (A), which confirms that (A) sometimes gives incorrect shortest paths. On the other
hand, although the calculation times of (B) and (C) are longer than (A), they are suitable
within a practical degree.
RELATED STUDIES
This problem is not limited to road networks. Fidler and Einhoff (2004) (cf. [3]) reported that
the same problem arises in computer communication networks. They explained the routing
with prohibited turns and proposed a node-link transformation. They offered no ideas on how
to handle intersection delay, and so their method corresponds to Intersection Expansion II,
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which is a special case that ignores intersection delay.
Some papers have dealt with the shortest path problem having specific constraints. For
example, Van der Zijpp and Fiorenzo Catalano (2005) (cf. [4]) proposed a set partitioning
method for finding K-shortest paths. Crainic and Florian (2005) (cf. [5]) suggested a shortest
path method with a sequential constraint. Thus, finding the shortest path under certain
constraints is a currently studied topic.
CONCLUSION
This paper described the representation and shortest path of a real-world network. The
accomplishments of this paper are briefly summarized as follows:
(a) The ramp entry or exit direction limitation is expressed distinctly with SPP.
(b) The shortest-path tree sometimes cannot be expressed as a simple tree when the network
that includes SPP is represented with single nodes and single links.
(c) This paper proposed intersection expansion to find the most accurate shortest-path tree. It
is transformed automatically during the process of finding the shortest path.
(d) When the intersection delay is not a matter of interest, Intersection Expansion II can be
used for reducing the size of the transformed network; it aggregates hypothetical links
with their downstream links.
(e) The calculation time in a real network is within the range of practical use.
The achievement of the suggested method is that the expanded representation does not have
any special conditions. Therefore, other shortest path algorithms can be used after the
transformation, such as a label-correcting algorithm. On the other hand, a remaining problem
is that the use of cyclic paths implies the production of many verbose paths in applying it to
k-th shortest path algorithms.
REFERENCES
[1] Dijkstra, E.W. (1959). A note on two problems in connexion with graphs, Numerische
Mathematik, 1, pp. 269-271.
[2] Asakura, Y. (1987). An optimal design model of network and land use considering the
traffic network equilibrium, Doctoral Dissertation, Kyoto University (in Japanese).
[3] Fidler, M., Einhoff, G. (2004). Routing in turn-prohibition based feed-forward networks,
In Proceedings IFIP (International Federation for Information Processing)-TC6
Networking, Athens, Springer, LNCS 3042, pp. 1168-1179.
[4] Van der Zijpp, N.J., Fiorenzo Catalano, S. (2005). Path enumeration by finding the
constrained K-shortest paths, Transportation Research Part B, Volume 39, Issue 6, pp.
545-563.
[5] Crainic, T.G., Florian, M. (2005). A note on computing shortest paths with logistic
constraints, In Proceedings 10th International Conference of Hong Kong Society for
Transportation Studies (10th HKSTS), Hong Kong, pp. 259-267.
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