AbstractCompared with a Dijkstra-based or partially Astar-based one, a totally Astar-based algorithm is proposed in the paper for vehicle navigation systems. It has a better performance such as computing speed and veracity in a large-scale road network than a Dijkstra-based one because the computational complexity of Astar algorithm has little connection with the overall scale of a road network. To recognize all the reasonable routes between a specific OD pair, this algorithm takes all the geometrically reasonable routes into account and considers several constraints that meet the drivers’ preferences like circuitous route, the number of turns and traffic control strategy (for example, no left turn). Two numerical examples demonstrate the operation and efficiency of the algorithm

Yang, Xiaoguang

Yin, Wei

Elsevier - Publisher Connector 

 Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
1877-0428 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and peer-review under responsibility of Chinese Overseas Transportation Association (COTA).
doi: 10.1016/j.sbspro.2013.08.123 
ScienceDirect
13th COTA International Conference of Transportation Professionals (CICTP 2013) 
A Totally Astar-based Multi-path Algorithm for the Recognition 
of Reasonable Route Sets in Vehicle Navigation Systems 
Wei YINa*, Xiaoguang YANGa 
a .School of Transportation Engineerin, Tongji University, No. , Jiading District, Shanghai,601804, China 
Abstract 
Compared with a Dijkstra-based or partially Astar-based one, a totally Astar-based algorithm is proposed in the paper for 
vehicle navigation systems. It has a better performance such as computing speed and veracity in a large-scale road network 
than a Dijkstra-based one because the computational complexity of Astar algorithm has little connection with the overall scale 
of a road network. To recognize all the reasonable routes between a specific OD pair, this algorithm takes all the 
geometrically reasonable routes into account and considers several constraints  like 
circuitous route, the number of turns and traffic control strategy (for example, no left turn). Two numerical examples 
demonstrate the operation and efficiency of the algorithm. 
Keywords: Multi-path Algorithm, A-star, Route navigation systems 
© 2013 The Authors. Published by Elsevier B.V.  
Selection and/or peer-review under responsibility of Chinese Overseas Transportation Association (COTA). 
Keywords: Multi-path Algorithm, A-star, Route navigation systems 
1. Introduction  
The reasonable route set is defined as a set of all paths that drivers may choose between a specific OD pair and 
it is more meaningful than a single shortest path in dynamic vehicle navigation systems. Given uncertainty of link 
travel time and interference of on-trip events, a fixed shortest path does not exist in a road network. And 
delivering multiple paths instead of single ones is a feasible way to handle  preferences and 
to avoid the overload of specific paths. 
Traditionally, multi-path algorithms can be divided into two types: the k-shortest path algorithms proposed by 
Eppstein (1998) and Martins et al (1999), and disjoint path algorithms proposed by Dinic (1970) and Torrieri 
(1992). Chen et al (2007) indicate the drawbacks of k-shortest path algorithm that it becomes inefficient for 
finding alternative paths when the number of paths between an OD pair is large. And they design a partially 
Available online at www.sciencedirect.com
 13 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
l ction and pee -review under responsibility of Chinese Overseas Transport tion Association (COTA).
1070   Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
Astar-based algorithm to solve this problem. Bell (2009) designs a multi-path Astar algorithm called Hyperstar 
for vehicle navigation with risk aversion.
The pathfinding algorithm used in a navigation system has a high request in computing speed because the
algorithm is carried out frequently to find changeable real-time optimal paths due to the random on-trip events.
LuFeng et al (1999) set an oval restricted area on the network prior to the multi-path search by the k-Dijkstra 
algorithm. The area is efficient because the number of nodes traversed by algorithm is decreased. Wagner and 
Willhalm (2006) review the methods for speeding up pathfinding algorithms and summarize two classical speed-
up techniques, which are bidirectional search and goal-directed search (Astar algorithm). 
In summary, the main factor influencing the efficiency of algorithm is the number of nodes that have to be
traversed. LuFeng et al (1999) use a prior area to reduce it but the drawback of the method is that some feasible
paths may be lost and the border is hard to define because some circuitous paths are allowed. And directional 
search mentioned by Wagner and Willhalm (2006) use directional potentials to decrease it automatically. For 
example, Astar algorithm introduces node potentials to modify the priority of the nodes in Dijkstra s open list, 
thereby raising the efficiency of search Bell (2009).
Deletion Algorithm (J.A.Azevedo et al, 1990) finds a shortest path, deletes one link on it and search for 
another reasonable path, thereby organizing a multiple paths set. The drawback of it is that some similar paths and 
loops may exist in the results. And Barra (1993) finds a new reasonable path by increasing the impedances of 
links on the current reasonable path.
D preference, which is another factor, must be considered. Szczerba et al (2000) discuss the setting of 
constraints for various mission scenarios in a pathfinding algorithm. In the paper, the constraints on drivers
preference are listed as follows, maximum path duration, minimum path reliability, maximum number of turns
and maximum ratio of minor road to the entire path length.
In this paper, a new multi-path algorithm is designed which integrates deletion algorithm and Astar algorithm
to weaken the negative affection of large-scale network, considers and traffic control
strategies and allows some reasonable detours.
The remainder of this paper is organized as follows. In Section 2, we give the key procedure of the algorithm
and describe the constraints. In Section 3, two numerical examples are designed. The example 1 expresses the
operation of the algorithm in a small network. And the example2 shows the efficiency of it in a relatively large 
grid network. Finally some conclusions are presented in Section 4.
2. Algorithm framework
2.1. Algorithm layout
To express the algorithm, a road network consisting of a set of links A and a set of nodes I is considered. And 
the following sets and variables are defined.
A Set of links
The link from node a to node b
I Set of nodes
Set of adjacent nodes of node i
1071 Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
The flow comes into the node i from node a and go out to node b
Set of forbidden
Set of forbidden links,
The number of times that link from node i to node j is chosen.
Linear distance between node i and end node
Linear distance between node i and start node
The number of circuitous links in one path of which the end point is node i
The maximum number of circuitous links in one path
Travel time of 
Least travel time from start node to node i
Travel time from start node to node i
con parameter of path type 
H Set of nodes attached to possible circuitous links
G Set of nodes attached to normal links
Variables have been initialized prior to the algorithm:
;
As is shown in Fig.1, the algorithm begins from the start node and then chooses some adjacent node as next 
node iteratively (section 2.2). If the node is the end node, a new path formed by all chosen nodes adds into paths
set. And then the algorithm comes back to start node to search a next path. If a path doesn t satisfy corresponding
constraints (section 2.3), it breaks and starts a new path search from the start node. The algorithm terminates
when the iterative limit is satisfied.
1072   Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
Locate start node
Find next node by the
choice algorithm (2.2)
End node
or not?<
Start
Over Iterative limit
or not?
1
1
Initialize variables (2.1)
End <
Satisfies constraints
Or not?(2.3)
Form a new path
and add it into paths
set(2.3)
<
1
Fig.1 Layout of the algorithm
2.2. Iterative choice of next node
The iterative choice of next node is a key component of the algorithm because a new path which contains those 
nodes must be reasonable and different from the others in the paths set.
As is shown in Table 1, the choice algorithm is divided into four layers whose priority rank from high to low.
If the upper layer keeps neutral, the lower one makes a decision.
Table 1 four layers of decision-making
Priority Objective
LevelA Generate a restricted area
Level B Eliminate forbidden flows
Level C Find a different path
Level D Search the next link with high contribution to reach the target
e as current node, as next chosen node and as previous node. And the following algorithm shows the 
procedure of next node choice.
0. Level A:
#1 
#2 then ;
#3 then
1073 Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
#4 then ;
#5 then
1. Level B:
#6
#7
#8
#9 go to Step 1
2. Level C:
#10 find ;
#11 if ; ;
#12
#13 if {z}>1 then go to Step 3
3. Level D:
#14 Find 
#15 then ; //a normal link; 
#16 then ; ;
4. Get next node
#17 ;
#18 ;
#19 ;
#20 minimum 
#21 If and ; then  ;con=0;
Conventional Astar Algorithm is the combination of directional potentials and Dijkstra Algorithm. In this 
paper, directional potentials are integrated with Deletion Algorithm and two potentials are introduced into 
the algorithm to recognize the type of links. Drives are supposed to be logical and believe that the next chosen
road should make them more close to the target and some detours are allowed if they can reach the target faster.
The potentials of which the inequality relation divides three kinds of conditions describe such actions as is shown 
in Fig.2. A normal link is preferred because such link has a direct contribution for drivers to reach the destination.
A possible circuitous link is chosen if no normal link exists. And an invalid link is ignored in any condition.
L is the set of possible next nodes of . And the element z (each possible next node) in L is eliminated by the
choice of four levels until is found. The Level A is used to form a restricted area, which is different from an
artificial geometrical area mentioned in LuFeng et al (1999). The formation of the area is dynamic and non-
geometrical. As is shown in Fig.3, the border of a reasonable area (dark gray) and circuitous area (light gray) is 
formed gradually with the iteration of nodes because the invalid links and possible circuitous link over the limit
of are eliminated in each node. And the border of the area is calculated by directional potentials and so
the area is not just a simple shape. The vital importance of area formation raises the computing speed by
controlling the number of nodes especially in a large-scale network.
1074   Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
Fig.2 direction-oriented definitions of links
Fig.3 restricted area generated by directional potentials
Level B eliminates all the forbidden flows and links caused by the accident, road maintenance or traffic
control policies like turn left .
In Level C, is a key parameter which records the search frequency of and only links with minimum 
are allowed in L (#10). Such method is similar to the Delete Algorithm (J.A.Azevedo et al, 1990). The 
difference is that all the links of the current path are labeled (#18) instead of labeling one of them to find a next 
one. As one kind of breadth first search, it can traverse nodes rapidly and decrease the similarity of paths. The 
label could also be replaced by the impedance of roads, which is discussed in section 4, If more than one node is 
existed in L after Level C, is decided by Level D.
Drivers prefer high grade roads, which is describe by the design speed. This variable is adapted and a new
index is proposed in Level D. The meaning of the length of a road is replaced by the decrease
of linear distance contributed by the road. The algorithm assumes that drivers choose the road with a higher index
(#14).
1075 Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
A normal path (con=0) is defined as a path consisting of normal links. And a detour (con=1) must contain any 
possible circuitous links. When a circuitous link with negative index is chosen, the condition is added that the 
circuitous path has a shorter travel time than that of any normal one (#15, 16, 19~21). 
2.3. Path constraints 
The drawback of this algorithm is that the iteration of a next node may create some invalid, inefficient or 
overlapping paths. So a series of path constraints is proposed in the paper which includes maximum path 
duration, minimum path reliability, maximum number of turns and maximum ratio of minor road to the entire 
path length. 
The maximum number of paths in the reasonable paths set is determined first. And then a constraints system is 
established, which depends on specific environment (Szczerba et al ,2000). If a path fails to satisfy any 
constraints, it is eliminated and a new path search starts. If it is more excellent after comparing with other paths 
in the set, it will replace the worst one. 
3. Numerical examples 
3.1. Example1 
 
Fig.4 the numerical example of the algorithm 
 
1076   Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
To simplify the presentation, we only show a key network consisting of 12 nodes and 17 links , omit the others
by the radial lines connected with Node 1, 2,3 and set to 1.  The of all links are initialized to 0. And the
travel time of each link is labeled (see Fig.4A). The start and end node are s t and one forbidden flow exists
in Node 5. Maximum path duration is considered and is set to the value of the fourth shortest path.
According to the algorithm, the choice of first path depends on Level D. And the second one is totally decided 
by Level C due to the labels of first one (see Fig.4B,C).  And then a possible circuitous link is chosen and 
node 1 is set to the current node so are three candidates of next link. , which connects with
the other part of the network, is eliminated in Level A because > and becomes a border of the restricted 
area. is chosen due to a higher index in Level D but the entire path breaks because the travel time of the
circuitous path ( ) is longer than the path ( ) (see Fig.4D).Such situation also occurs in Fig.4F.
In Level B is a forbidden flow so Node 3 is chosen as the next node when the algorithm calculates the Node
5. And the path ( ) becomes a reasonable circuitous path (see Fig.4E). An odd situation
that one path may be found repeatedly occurs in Fig.4G , which is discussed in next section. All the reasonable
paths are presented in Fig.4H. And the path ( ) is eliminated because it fails to satisfy path
constrains.
3.2. Example2
To test the efficiency of the algorithm, a randomly generated grid network of 1024 nodes is discussed (see
Fig.5). One cell symbolizes one intersection. And the real coordinates of it and the travel time of corresponding 
links are random in a finite interval to simulate a real road network. The start and end node are respectively A
B and set to 3. Four optimal paths are required in this test. 
Fig.5 the calculated results of a large grid network
Table 2 Results Analysis of two algorithms
1077 Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
Items Mean value of travel time(min) 
Number of 
Iterations Variance 
Multi-path A-star 56 10000 8.94 
K-Dijkstra 49 10.4 million 3.25 
 
As is shown in Fig.5 and Table 2, the results of the multi-path Astar based algorithm is prominent in the 
independence among chosen paths and the number of iterations. However, the optimality of the chosen paths is 
lower than that of K-dijkstra one.  
4. Discussion 
In future, more people will install navigation devices to handle complicated and changeable traffic 
environment. It leads to the risk that people may encounter a new kind of congestion due to the same path choice. 
The essence of this problem is that the pathfinding algorithm is simple and the calculation is mutually 
independent. To solve the above-mention problem, future vehicle navigation system may centralize navigation 
demands from drivers, calculate together and deliver different navigation information. The algorithm is proposed 
against this background because it satisfies two basic requirements of future navigation systems which are high 
computing speed and multiple reasonable paths. 
The computing speed of the proposed algorithm rises significantly because the strategy of the algorithm is 
different from that of conventional multi-path algorithm. Original K-Dijkstra Algorithm whose complexity is 
O(n3) searches all paths and then compares them to find a reasonable path set. It traverses all nodes each time 
and cannot get results until termination so it is inefficient in both spatial and time dimension. And the 
improvement of the algorithm cannot essentially raise efficiency. However, the proposed algorithm use goal-
direction search which decrease the number of calculated nodes. And most suboptimum or optimum result is 
found in the initial stage of iteration. The algorithm increases the efficiency in both sides.  
Updating search frequency of links and updating impedance of links are two methods to get different paths. 
The first method as a breadth-first search can traverse nodes rapidly and increase the reliability of paths but many 
inefficient and repetitive paths are found. And the second one as a depth-first search can get more reasonable 
paths but some key links may lose and the reliability of paths decreases. The second one will be discussed in 
further works. 
The drawbacks of the algorithm is the same as most heuristic method that it lacks a steady process of 
optimization and a specific convergence condition and the results are not strictly optimal. However, finding a set 
of suboptimum paths with relatively little iteration are more practical than searching optimum paths set with huge 
calculation in modern vehicle navigation system. 
Acknowledgements 
This research 
China(Grant No. 2011AA110404), the National Natural Science Foundation of China (Grant No. 60974093), 
Science and Technology Commission of Shanghai Municipality (Grant No. 10530705100). The writers wish to 
thank Dr Yue Liu From University of Wisconsin  Milwaukee and Dr Wanjing Ma from Tongji University for 
their helpful suggestions and useful comments on an earlier version of the paper. 
1078   Wei Yin and Xiaoguang Yang /  Procedia - Social and Behavioral Sciences  96 ( 2013 )  1069 – 1078 
References 
Chen, Y., M. G. H. Bell, et al. (2007). "Reliable pretrip multipath planning and dynamic adaptation for a centralized road navigation system." 
Intelligent Transportation Systems, IEEE Transactions on 8(1): 14-20. 
Eppstein, D. (1998). "Finding the k shortest paths." SIAM J. Comput. 28(2): 652-673. 
Martins, E. Q. V., M. M. B. Pascoal, et al. (1999). "Deviation algorithms for ranking shortest paths." International Journal of Foundations of 
Computer Science 10(3): 247-262. 
E. A. Dinic (1970). of maximum flow in a  11. 248
264. 
D. Torrieri, (1992).  
40(11): 1698 1702.  
Bell, M. G. H. (2009). "Hyperstar: A multi-path Astar algorithm for risk averse vehicle navigation." Transportation Research Part B: 
Methodological 43(1): 97-107. 
LuFeng ,LuDongmei ,CuiWeihong(1999). Time Shortest Path Algorithm for Restricted Searching Area in 
Transportation Networks  Journal of Image and Graphics,4(10): 850-853. 
Szczerba, R. J., P. Galkowski, et al. (2000). "Robust algorithm for real-time route planning." Aerospace and 
Electronic Systems, IEEE Transactions on 36(3): 869-878. 
J.A.Azevedo, J.J.E.R.S.Madeira, E.Q.V.Martins and F.M.A.Pires (1990). "A shortest paths ranking algorithm" 
Proceedings of the Annual , Models and Methods for Decision Support, Operational 
Research Society of Italy,1001-1011  
De La Barra, T., B. Perez, et al. (1993). "Multi-dimensional path search and assignment." PTRC-
PUBLICATIONS-P: 307-307. 
 


A Totally Astar-based Multi-path Algorithm for the Recognition of Reasonable Route Sets in Vehicle Navigation Systems 

file:///data/remote/core/dit/data/elsevier/pdf/967/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMTg3NzA0MjgxMzAyMjQ5MA==.pdf

A Totally Astar-based Multi-path Algorithm for the Recognition of Reasonable Route Sets in Vehicle Navigation Systems

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector