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In this chapter, we present two approaches for limiting the search space using a window-based heuristic algorithm to compute shortest routes and analyze their solutions and performances

Window-Based Heuristic Algorithm for Real-Time Routing in Map-Based Mobile Applications

Hassan A KARIMI1, Peter SUTOVSKY1, Matej DURCIK2

1School of Information Sciences, University of Pittsburgh

2 SAHRA-HWR, University of Arizona

Abstract. The demand for routing algorithms that produce optimal solutions in real time is continually growing Real-time routing algorithms are needed in many existing and emerging applications and services An example is map-based mobile applications where real-time routing is required Conventional optimal routing algorithms often do not provide acceptable real-time responses when applied to large real road network data As a result, in certain real-time applications, especially those with limited computing resources (e.g., mobile devices), heuristic algorithms that can provide good solutions, though not necessarily optimal, in real time are employed In this chapter, we present two approaches for limiting the search space using a window-based heuristic algorithm to compute shortest routes and analyze their solutions and performances using real road network data The results of a set of experiments on the two approaches show that the window-based heuristic algorithm produces aceptable response times using real road network data and that window sizes and orientations impact accuracy and performance of the algorithm

12.1 Introduction

Routing is a fundamental function in numerous map-based mobile applications Example applications are navigation, location-based services, and automatic vehi-cle location In such real-time map-based mobile applications, the overall accuracy and performance of the underlying routing algorithm is of particular interest This

is because routing accuracy influences the user’s confidence on and routing time performance impacts the real-time response time of the maps produced Providing

a reasonable level of confidence on the routes (which in some applications are the only maps produced) to the users and having a fast routing solution (which im-proves response time to real-time activities), will enhance the usability of real-time map-based mobile applications Therefore, it is imperative to realize the

Routing belongs to a class of problems widely known as optimization and is computed by using either conventional algorithms or heuristic algorithms Con-ventional routing algorithms are suitable in applications where the network size is not large and there is no real-time processing constraint However, in applications where the network size is very large and routes must be computed in real time, heuristic routing algorithms are considered, which may not guarantee optimal so-lutions This is because conventional routing algorithms’ performance lowers as the size of the network increases resulting in unacceptable response times; most conventional algorithms have O(N2) time complexity in the worst case scenario, where N is the number of nodes (intersections) in the road network Routing algo-rithms with such a time complexity are impractical in map-based mobile applica-tions where real-time optimal routes, some on large networks, are needed and where they typically feature mobile devices with limited CPU, memory, and power Therefore, heuristic routing algorithms play a major role in map-based mobile applications and their proper design requires a thorough understating of their accuracy and time performance

Much research in the past few decades has been focused on developing fast gorithms resulting mostly in heuristic routing algorithms that produce local-optimal solutions For example, see Fu et al (2006) for a survey of heuristic short-est path algorithms and Cherkassky et al (1996) for an overview of theoretical and experimental studies of various shortest route algorithms Any real-time rout-ing algorithm can be of practical use in map-based mobile applications only when

al-it produces reasonable solutions in an acceptable response time wal-ith real road network data To date, there have been very few new routing algorithms that are tested for real-time processing using real road network data (Jacob et al 1999; Ja-gadeesh et al 2002; Kim et al 2005a) A review of the differences between using real road network data and randomly generated network data along with the com-putational study of routing algorithms using realistic networks can be found in Ja-kob et al (1999) and Liu (1997)

Chabini and Shan (2002) adapted A* algorithm to shortest path problems in dynamic deterministic networks They evaluated the algorithm on the randomly generated network containing 3000 nodes and 10,000 links, and 100 time inter-vals They compared the dynamically adapted A* algorithm and the dynamically adapted Dijkstra’s algorithm and reported that the dynamically adapted A* algo-rithm resulted in a saving ratio of 11 in terms of nodes selected and a saving ratio

of five in terms of computational times Huang et al (2006) extended Lifelong Planning A* algorithm to solve dynamic deterministic shortest path problems They suggest the use of an ellipse to prune the unnecessary nodes to be searched and experimentally showed that the number of examined nodes could be 70-80% less than that of the A* algorithm This corresponds to 17-31% savings in compu-tational time needed to calculate shortest paths using test road networks in the ex-periments Kim et al (2005b) proposed to model the problem of computing dy-namic stochastic shortest paths with traffic congestion information as discrete-time finite horizon Markov decision process They developed decision-making accuracy and time performance of routing algorithms and their impact on such ap-plications

procedures for determining optimal driver attendance times, optimal departure times, and optimal routing policies under time-varying traffic flows for just-in–time delivery services They tested their method using real-time traffic congestion information for the road network in southeast Michigan They achieved reduction

in travel time of approximately 9.8216.19% (Kim et al 2005a) also improved ficiency of their previous algorithm

ef-Map-based mobile applications’ effectiveness is often measured by the racy and time performance of the routing algorithms on which they are based Ja-gadeesh et al (2002) have suggested a routing approach that combines hierarchi-cal and heuristic techniques based on road classification They tested their approach using the road network of Singapore On average, the routes computed

accu-by their approach were 3.31% longer than the shortest routes Zhao and mouth (1991) proposed an adaptive route guidance technique that alternates be-tween two different heuristic search algorithms based on the time available for route computation Jung and Pramanik (2002), Chabini and Shan (2002), and Karimi (1996) developed a heuristic routing algorithm that limits the number of nodes used in computation by devising a window with two of its sides parallel with the straight line connecting the origin and destination nodes However, while this window-based heuristic routing algorithm provides reasonable time perform-ance, further research was required to realize accuracy and time performance of the algorithm based on a variety of possible windows (or subnetworks); windows can geometrically have different size and structure

Wey-The work presented in this chapter is based on the window-based heuristic ing algorithm developed by Karimi (1996), and is focused on different approaches (window size and orientation) in limiting the search space that contains real road network data The objective of this work was to test the hypothesis that the size and orientation of a subnetwork impact the accuracy and time performance of the window-based heuristic routing algorithm To test this hypothesis, two subnet-work approaches (window size and orientation) were taken to reduce the search space in the window-based heuristic routing algorithm The impacts of different sizes and orientations of windows on accuracy of the solutions and the time per-formance by the window-based heuristic routing algorithm were analyzed Both approaches were tested using real road network data providing meaningful results applicable to real-world map-based mobile applications

rout-The main contributions of this chapter are: (a) analysis of different approaches for reducing the search space in the window-based heuristic routing algorithm and (b) testing of the window-based heuristic routing algorithm using real road net-work data Realization of the impact of window size and orientation on accuracy and time performance by the window-based heuristic routing algorithm will help developing optimal solutions that meet the requirements of map-based mobile ap-plications The structure of the chapter is as follows The window-based heuristic algorithm is described in section 12.1 In section 12.2 the experiments conducted are discussed The results are discussed in section 12.3 Conclusions and ideas for future research are given in section 12.4

12.2 Window-based heuristic algorithm

Since the approaches presented are based on Dijkstra’s algorithm, the following brief description of the algorithm is given A detailed description of Dijkstra’s al-gorithm can be found in standard books on algorithms (e.g., Bertskas and Gallager 1992) Variations of Dijkstra’s algorithm can be found in several publications, in-cluding Fredman and Tarjan (1987) and Gallo and Pallottino (1988) Dijkstra’s algorithm computes the shortest route from a single, source, vertex to all other vertices in a weighted, directed network G N A , , where N is a finite set of nodes and A is a finite set of arcs Associated with each arc(i,j)A is its length (the cost is a function of length) l i,jt0 The time necessary to compute a shortest route is approximately a quadratic function of the network size, O(N 2 ), where N is the number of nodes in the network However, a time complexity of O(A+NlogN), with A arcs and N nodes in the network, is possible by implementing a priority

queue with a Fibonacci heap (Fredman & Tarjan, 1987)

The main idea behind the window-based heuristic algorithm is to limit the number of nodes used for computation by using a subnetwork of the original net-work (a window overlaid over the original network) The window-based heuristic algorithm first creates a subnetwork by constructing a rectangle that includes the origin and destination nodes and then applies Dijkstra’s algorithm to the subnet-work Since there are conceivably different approaches to devise such subnet-works, in this work we tested the following hypothesis Subnetworks with differ-ent sizes and orientations will impact accuracy of the solutions and the time performance by the window-based heuristic algorithm To test this hypothesis, we took two approaches to limit the search space and compared the solutions and the performance by the window-based heuristic algorithm In the remainder of this section the two approaches are described

12.2.1 Orientation-based window (OBW)

OBW was described by Karimi (1996) as a window oriented in the direction of the line connecting the origin and destination nodes The window is constructed so that the origin and destination nodes lie on the long axis of symmetry of the rec-tangle The size of an OBW is determined by the Euclidean distance, L, between the origin and destination nodes and by the parameters fsx and fsywhich spec-ify the percentage of increase in the size of the edge with respect to L, so that the former is for the longer side and the latter is for the shorter side Therefore, the shorter side of the rectangle is 2 Lfsy and the longer side is 2 Lfsx  L as shown in Fig 12.1

Fig.12.1 An example of OBW

12.2.2 Parallel-based window (PBW)

PBW has sides parallel with the horizontal and vertical axes of the geographic tent (i.e., parallel to the x and y axes of the coordinate system in which the road network is presented) of the road network PBW is constructed as depicted in Fig 12.2 First, a rectangle whose sides are parallel with the x and y axes and the ori-gin and destination nodes are on its diagonal is constructed Each side of the rec-

ex-tangle is then increased by 2b to construct a larger recex-tangle PBW constructed in this way has a buffer b with the length of L+2b and the width of 2b A special case

of PBW is when the abscissa connecting the origin and destination nodes lie on the vertical or horizontal axis

Fig 12.2. An example of PBW

Several cases may occur when selecting links (road segments) for OBW and PBW Example cases are illustrated in Fig 12.3 Note that there are usually two types of roads in a road network database, the end nodes (intersection nodes) and the shape nodes (the nodes making the geometry of the road) While the intersection

Fig 12.3 Possible cases for OBW and PBW

12.3 Experiments

An empirical study was carried out to realize the solutions and performances of the two approaches (OBW and PBW) using a real road network data set All tests were implemented on a 2GHz Pentium 4 with 500MB of memory All program codes for simulations were written in Java The road network data set used in the experiments was obtained from the Topologically Integrated Geographic Encod-ing and Referencing (TIGER) database (TIGER/Line Files 2002a; TIGER/Line Files 2002b); data from Record Type 1 (RT1) and Record Type 2 (RT2) of Alle-gheny County RT1 files contain intersection nodes (end nodes) and RT2 files contain shape nodes The road network used in the experiments consists of 59,861 nodes with a ratio of 1.33 between the links and the nodes A total of 190 origin-destination (O-D) pairs were randomly selected The Euclidean Distance (ED) be-tween an O-D pair is one factor that determines the window (PBW or OBW) size and the number of nodes in the window

Shortest routes generated by Dijkstra’s algorithm on the road network were used as the baseline for performance and accuracy assessment Twelve different window sizes were used (see Table 12.2 and Table 12.2) Note that for OBW (Ta-ble 12.1) the scale factor in both x and y directions are indicated while for PBW (Table 12.2) the buffer width is indicated Dijkstra’s algorithm yielded, for each computed route, the number of roads in the route, the route length, and the compu-tation time The average time and error (the difference between the shortest route using the entire road network and the shortest route using the subnetwork for a given origin-destination pair) rate for each window size using all computed routes were calculated The computation time for each of the following was measured:

nodes are needed to provide the extent of a road segment, the shape nodes are needed to represent the geometry of the road segment In Fig 12.3, road segments

a, b, c, and d are included and road segments e and f are not included in the network

sub-1 Creating the window (OBW or PBW)

2 Selecting the nodes and links for the window (OBW or PBW)

3 Creating the adjacency matrix

4 Running Dijkstra’s algorithm

The accuracy of the results by PBW and OBW, which include incomplete routes and local-optimal routes, was assessed An incomplete route is defined as one which misses one or more roads necessary to make up the route between the origin and destination points A local-optimal route, using the subnetwork, is defined as a route that is longer than the shortest (optimal) route, using the entire network, be-tween the same origin and destination points

Table 12.2. Window sizes for OBW

Table 12.3. Window sizes for PBW

The number of nodes in the window also depends on the locations of the origin and destination points and the network density (number of nodes and links) within the window Tables 12.3a and 12.3b show average number of nodes for different window sizes, ratios between number of links and number of nodes in each win-dow, and basic statistics on the data These ratios are from 0.93 to 1.47 for PBW and from 0.88 to 1.48 for OBW The ratio between the numbers of links and nodes

is low compared to the ratio of 2.56 presented by Jagadeesh et al (2002) for the Singapore road network Anderson et al (1998) also analyzed road networks using TIGER/files and found the ratio between links and nodes as 1.38 The reason for the low ratio between the links and nodes in road networks from TIGER is that in addition to the intersections (end nodes), the shape nodes between the intersec-tions (these are for representing the geometry of the roads which are stored in RT2) as well as the nodes that lie on the window border are considered

Table 12.3a Average number of nodes in a PBW, ratios between number of links

and number of nodes, and basic statistics where NoR is the number of routes

Table 12.3b Average number of nodes in an OBW, ratios between number of

links and number of nodes, and basic statistics where NoR is the number of routes

12.4 Analysis of results

Shortest routes calculated by Dijkstra’s algorithm using the entire network were used as a baseline to compare the two approaches Of the 190 O-D pairs randomly selected, Dijkstra’s algorithm was able to find routes between 188 O-D pairs The reason why no routes were computed between two of the O-D pairs is that one of the origin or destination nodes in each pair was located on the county border, re-quiring road data from the adjacent county which is not part of the Allegheny County road network The average time necessary to find the optimal routes be-tween all 188 O-D pairs applying Dijkstra’s algorithm to the entire network was 334.1 seconds with a standard deviation of 6.7 seconds

Two types of errors were analyzed using OBW and PBW: (1) an incomplete route and (2) a local-optimal route A route is incomplete when the algorithm could not connect the origin and destination nodes A local-optimal route, using a subnetwork, occurs when the length of a route between the O-D pairs is longer than the optimal route using the entire network The total error is defined as the sum of incomplete and local-optimal routes

Fig.12.4. Computed routes between an O-D pair for different window sizes in PBW

Fig.12.5. Computed routes between an O-D pair for different window sizes in OBW

Fig 12.4 and 12.5 show examples of incomplete routes (in small windows), cal-optimal routes, and optimal routes for PBW and OBW, respectively Fig 12.4 shows the results of PBWs: incomplete routes for windows 1 and 2, local-optimal routes for window 3, optimal routes for the other windows (4-12) Fig 12.5 shows the results of OBWs: incomplete routes for windows 2, 3 and 5, local-optimal routes for windows 6 and 7, and optimal routes for the other windows

lo-Table 12.4 Basic statistics for route lengths and ED ratios

Dijkstra 1.22 0.12 1.05 1.87 PBW 1.25 0.11 1.05 1.87 OBW 1.24 0.11 1.05 2.28

Table 12.4 shows the basic statistics of route length and ED ratios The average ratio for PBW and OBW, 1.25 and 1.24, respectively, is slightly higher than for the baseline route calculated for the entire network (1.22) This is because the

1

1

2

2 3 3

4

4

5

5 6

6

7

7 8

8 – 12 (Optimal Routes)

O

D

Fig 12.7. Distributions of percentage of routes (NR), percentage of total error (Error), and percentage of error divided by number of routes (Error/NR) based on EDs: (a) PBW and b)

OBW.

As expected, smaller windows produced larger errors Fig 12.6 shows that the total error distribution depends on the window size and the ED between the origin and destination nodes The errors are more concentrated in the middle of ED scale for PBW (see Fig 12.6a) In Fig 12.6b, it is shown that the errors are more prob-able for shortest EDs The selected routes were grouped into equidistant intervals with a length of 2 km based on the ED Fig 12.7 shows the distribution of the total error and the distribution of error rates (the distribution of errors divided by the

Fig.12.6. The number of errors depends on window size and ED: (a) PBW and (b) OBW

1

9 17 25 33 41

1 9 17 25 33 41

1 3 5 7 9 11

0 1 2 3 4 5 6 7 8 9 10

distribution of EDs) EDs were not uniformly generated More O-D pairs with an

ED in the range between 8 km and 24 km were generated which can partially explain the distribution of the total error The distribution of errors follows the

Fig 12.8 and Table 12.5 show percentages of incomplete and local-optimal

routes for each window size used in PBW and OBW The percentage of errors was

calculated for each window size separately using the number of calculated routes

(188) in the window As expected, more errors occurred for smaller windows,

es-pecially using OBW where both errors are larger than those for PBW, but they

drop faster The experiment showed that the number of incomplete and

local-optimal errors in OBW is 20% higher than PBW for all window sizes

Fig.12.8. Percentage of incomplete and local-optimal routes for each window size

Table 12.5 Percentage of errors and performance

PBW OBW Percentage of errors [%]

mance Percentage of errors [%]

mance Window

Perfor-number

Incom-plete

rior total

infe-Times [s]

plete

Incom- rior total

infe-Times [s]

0 10 20 30 40 50 60

PB W

OB W

distribution of EDs for PBW (see Fig 12.7a) This means that errors are generated

homogeneously independent from EDs The distribution of total errors for OBW is

skewed to the smallest ED and the error rate is descending as a function of ED

(see Fig 12.7b) The result of the experiments showed that errors are more

prob-able for smallest windows than for larger windows using OBW

Fig.12.9. (a) Performance versus error and (b) dependence of relative total errors on the

number of nodes in windov (dashed line – OBW; solid line – PBW).

Fig 12.9a and 12.9b present the dependence of the total error for each window size on the average time (performance) and the average number of nodes, respec-tively From these figures and Table 12.5, which summarizes the values of errors and average performances for all window sizes used for PBW and OBW, it can be concluded that for smaller windows, the performance of OBW is better than the performance of PBW, but the error in this region is higher than 20% When the er-ror is small there is no statistically significant difference to prefer OBW or PBW

to reduce the search space in the window-based heuristic algorithm to compute timal routes in real time

op-Fig 12.10 shows the distribution of local-optimal and optimal routes ratios It can be seen that over 50% of the local-optimal routes are less than 3% longer than the optimal routes and about 90% of them are not longer than 10% of the optimal routes The average ratio of the lengths of local-optimal and optimal routes is 1.046 and 1.042 for OBW and PBW, respectively

Since the performances of both OBW and PBW partially depend on the lying road network size and density, an analysis of road networks is given A typi-cal road network is irregularly distributed and unequally developed in the various parts of any geographic extent This can be caused by the terrain or settlement in some regions Although the highest road network density is expected in urban ar-eas, there can be heterogeneity in road distribution, especially in the areas between neighborhoods, which are usually divided by terrain obstacles such as hills (parks and forests) and rivers (where the density of bridges is important) The road net-work of Allegheny County has similar characteristics and its density is high in the center of the network (downtown area) and gradually decreases from the center to

out-Fig.12.10. Local-optimal and optimal route lengths ratio distribution for OBW and PBW

Knowledge about a road density can be employed in the window-based tic algorithm as one of the factors determining optimal window sizes Generally, the optimal window size can be specified as s x y , f U , , , t ] H where U

heuris-is the density of road network, t is the computing time, H is an error function and

] is the degree of connectivity defined as the number of links to the number of nodes ratio This means that an optimal window size is the smallest area (subnet-work) of the original road network where acceptable solutions and performances can be obtained

This idea was used in an example to show how knowledge about networks could improve performance of the window-based heuristic algorithm (see Fig 12.11) The information about the network density is presented byU x y , , where x, y coordinates are used to specify conditionally the width of the window

Let abscissa AB that passes through O-D pairs be described by equation ax+b

where its length is l 1.25 OD First, we divide abscissa l into K intervals

performance, a route may not be found and if a route is found the accuracy may not be high

each with l

L

K

' length Then we divide the road network into regions by lines

that are perpendicular to the abscissa For each interval > i i ,  1 @, where point i has coordinates x yi, i on the boundary of the interval, we can specify the dis-

y x  h h  y x h  h from abscissa l that will specify the area from which the nodes used for a given interval using the follow-ing inequality:

of the window (bold in Fig 12.11) The windows for each interval are determined

in the same way This example shows how the relevant nodes can be selected more accurately when knowledge about the network is used Future work could include testing these methods on different road networks

Fig.12.11 Example of selecting intervals and width of the window

12.5 Conclusions and future research

Two approaches (OBW and PBW) based on a window-based heuristic algorithm

to find shortest routes between given O-D pairs were presented It was empirically shown that OBW and PBW reduce the average time of computation by a factor between 5 and 87 This means that they are potential for real-time computation of shortest routes in very dense networks The analysis of the results by OBW and PBW showed that there is not a statistically significant difference between them However, with a subnetwork of less than 10,000 nodes (with an average comput-ing time of less than 15 seconds), OBW has better performance and accuracy than PBW and for a subnetwork with more than 10,000 nodes the difference between the two approaches is not significant This indicates that there are more nodes in small OBWs than in small PBWs that are part of the optimal solution The same does not hold for large windows, as the size of the window increases the influence

of its orientation diminishes This is because in both approaches the geographic extent is a factor determining the number of nodes in windows Another issue is the number of nodes that are part of optimal solutions The proportion of the nodes that are part of optimal solutions to the nodes that are not part of optimal solutions

is higher for smaller windows and lower for larger windows For small windows a higher accuracy change at the expense of a lower performance change is achieved, and the opposite is true for large windows as can be seen in Fig 12.9(a)

The performance and accuracy of OBW and PBW depend on the location of the origin and destination points, the ED between them and their orientation, and the road network density If the origin or the destination point is located close to the border of an open road network (selected from a larger network), a route between this point and any other location in the network can not be computed as the con-nection to this point is through a neighboring county This problem can be over-come by taking into account all possible routes including those in the neighboring counties between all the points near or on the boundary of the road network One area for future research is determination of optimal window sizes to elimi-nate incomplete routes The results of the experiments showed that the elimination

of incomplete routes and the acceptance of local-optimal routes lead to routes which may be 4.6% longer on average than optimal routes (those computed by us-ing the entire network) Another area for future research is the utilization of other criteria than shortest distance, for example travel time, road type, road difficulties, and traffic conditions (Bovy and Stern 1990; Lotan 1997; Pang et al 1999) The proposed method can be naturally extended to constrained shortest path problems when the shortest distance criteria are combined with other conditions such as road type, a-autonomy shortest distance (Terrovitis et al 2005), and curvature of the road segments This method of limiting search space by OBW and PBW can also be applied to the problems of stochastic and dynamic shortest routes

The work presented here also provided an insight into finding a balance tween performance and accuracy This insight can be used in future to design sys-tems that will be able to choose parameters of the window based on user’s prefer-ences and knowledge of the specific road network

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lo-Table 12.4 Basic statistics for route lengths and ED ratios

Dijkstra 1.22 0.12 1.05 1 .87 PBW 1.25 0.11 1.05 1 .87 OBW