Map-based Mobile Services Design,Interacton and Usability Phần 8 pot

12 Accuracy and Performance Assessment of a Window-Based Heuristic Algorithm for Real-Time Routing in Map-Based Mobile Applications Hassan A. KARIMI 1 , Peter SUTOVSKY 1 , Matej DURCIK 2 1 School 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 12 Accuracy and Performance of a Window-Based Heuristic Algorithm 249 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 solutions. 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(N 2 ) time complexity in the worst case scenario, where N is the number of nodes (intersections) in the road network. Routing algorithms with such a time complexity are impractical in map-based mobile applications 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 algorithms resulting mostly in heuristic routing algorithms that produce local- optimal solutions. For example, see Fu et al. (2006) for a survey of heuristic shortest path algorithms and Cherkassky et al. (1996) for an overview of theoretical and experimental studies of various shortest route algorithms. Any real-time routing algorithm can be of practical use in map-based mobile applications only when it produces reasonable solutions in an acceptable response time with 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 computational 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* algorithm 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 computational time needed to calculate shortest paths using test road networks in the experiments. Kim et al. (2005b) proposed to model the problem of computing dynamic 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 applications. 250 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 ef- ficiency of their previous algorithm. Map-based mobile applications’ effectiveness is often measured by the accuracy 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 by their approach were 3.31% longer than the shortest routes. Zhao and Wey- mouth (1991) proposed an adaptive route guidance technique that alternates between 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 performance, 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. The work presented in this chapter is based on the window-based heuristic routing 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 subnetwork 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 performance 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. 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 network 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 applications. 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. Hassan A. KARIMI, Peter SUTOVSKY, Matej DURCIK 251 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 algorithm 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, including 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  ,GNA , 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,j t0. 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 network (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 subnetwork. Since there are conceivably different approaches to devise such subnetworks, in this work we tested the following hypothesis. Subnetworks with different 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 rectangle. The size of an OBW is determined by the Euclidean distance, L, between the origin and destination nodes and by the parameters and s xsy f f which 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 sy Lf2 and the longer side is  LLf sx 2 as shown in Fig. 12.1 12 Accuracy and Performance of a Window-Based Heuristic Algorithm 252 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 extent (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 origin and destination nodes are on its diagonal is constructed. Each side of the rectangle is then increased by 2b to construct a larger rectangle. 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 Hassan A. KARIMI, Peter SUTOVSKY, Matej DURCIK 253 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 f iles 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) between 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 computation 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: 12 Accuracy and Performance of a Window-Based Heuristic Algorithm 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 subnetwork. 254 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, between 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 window, 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 intersections (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. Hassan A. KARIMI, Peter SUTOVSKY, Matej DURCIK 255 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. 12 Accuracy and Performance of a Window-Based Heuristic Algorithm 256 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. Hassan A. KARIMI, Peter SUTOVSKY, Matej DURCIK 257 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 between 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. 12 Accuracy and Performance of a Window-Based Heuristic Algorithm 1 1 2 2 3 3 4 – 12 (Optimal Routes) 4 O D [...]... 25.1 29.71 34.61 40.05 45 .89 52.43 57.02 Percentage of errors [%] Incomplete 55.3 21.3 11.7 6.9 4 .8 2.7 2.1 1.1 0.5 0.5 inferior 32.5 34.0 17.0 11.7 6.4 4.3 1.6 0.5 1.1 total 87 .8 55.3 28. 7 18. 6 11.2 6.9 3.7 1.6 1.6 0.5 Performance Times [s] 2.10 4.71 8. 51 13.43 19.59 26.63 34.67 43.01 51.52 60.76 69.16 78. 30 12 Accuracy and Performance of a Window-Based Heuristic Algorithm 90 80 80 70 70 Total Error [%... 5 6 7 8 9 10 11 12 1 2 3 Windows 4 5 6 7 8 9 10 11 12 Windows Fig.12 .8 Percentage of incomplete and local-optimal routes for each window size Table 12.5 Percentage of errors and performance PBW Window number 1 2 3 4 5 6 7 8 9 10 11 12 OBW Percentage of errors [%] Incomplete 37 .8 29.3 16.5 3.7 1.1 0.5 inferior 19.2 16.5 18. 6 16.0 10.6 5.3 1.6 1.6 0.5 0.5 0.5 0.5 total 57.0 45 .8 35.1 19.7 11.7 5 .8 1.6... routes for windows 2, 3 and 5, local-optimal routes for windows 6 and 7, and optimal routes for the other windows Table 12.4 Basic statistics for route lengths and ED ratios Method Average STD Min Max 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,... the kiosk map’s presentation and the presentation on the mobile device We observe the impact of interactivity and presentation on satisfaction, task completion, and acquired survey and route knowledge This is accomplished by varying the level of interactivity between the mobile device and the kiosk maps, and varying the presentation style of the route information on the mobile device We also observe... window size and ED: (a) PBW and (b) OBW 9 Total Error [%] 8 7 6 5 4 3 9-10 8- 9 7 -8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 10 9 8 Total Error [%] 9-10 8- 9 7 -8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 10 7 6 5 4 3 2 2 1 41 33 0 1 25 3 5 WN 17 7 9 9 ED [km] 1 41 33 0 1 25 3 5 WN 11 1 (a) 17 7 9 ED [km] 9 11 1 (b) Fig 12.7 Distributions of percentage of routes (NR), percentage of total error (Error), and percentage of error divided... modalities, limited bandwidth, communication costs, and energy consumption issues (Nivala and Sarjakoski, 2003; Reichenbacher, 2003; Zipf, 2002) Maps for mobile devices have to consider these limitations and restrictions The design parameters for mobile maps are quite different than those for large kiosk maps While kiosk maps must reflect their broad setting and a range of purposes, mobile maps often... Laakso, Elting and Coors, 2003; Meng, 2005) There are many contextual factors that must be taken into account in mobile map design, including the users themselves, their task, their location, and the available technical resources (Bieber and Giersich 2001; Nivala and Sarjakoski, 2003; Kray and Baus, 2001; Zipf, 2002) This includes factors such as time of day (Bieber and Giersich 2001) and whether the... of interactivity between the mobile device and the kiosk maps, and varying the presentation provided on the mobile device itself (the various interface configurations are detailed in section 13.5.1) We hypothesized that: a) direct interaction with public kiosk maps would promote survey and route knowledge, and b) reflecting the presentation used in public kiosk maps in the mobile phone interface would... recognize commands and background noise interference In cases where information is provided verbally, users can feel more like passive observers (Nivala and Sarjakoski, 2003), which may not suit the purpose of the kiosk Other systems have incorporated multimodal interfaces, such as MATCHKiosk (Johnston and Bangalore, 2004), GAUDI (Kray, Kortuem and Krüger, 2005) and Kimono (Huang, Pulli and Rudolph,... 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 . Times [s] 1 37 .8 19.2 57.0 11.71 55.3 32.5 87 .8 2.10 2 29.3 16.5 45 .8 12.13 21.3 34.0 55.3 4.71 3 16.5 18. 6 35.1 14.34 11.7 17.0 28. 7 8. 51 4 3.7 16.0 19.7 17.35 6.9 11.7 18. 6 13.43 5 1.1. nodes in each window, 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. road network. Anderson et al. (19 98) 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

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