Second Asia International Conference on Modelling & Simulation

A Novel Approach for Intelligent Route Finding through Cumulative Proximity Evaluation Yang Yaw Chang1, Stephen Yung1 and Raymond Chiong2 1 School of Engineering 2 School of Information Technology & Multimedia Swinburne University of Technology (Sarawak Campus) State Complex, 93576 Kuching, Sarawak, Malaysia [email protected], [email protected], [email protected] domain [5]. This is not only undesirable but also wasteful in terms of computation. In this paper, we present a study to examine the performance of both informed and uninformed search techniques in route finding based on some major cities and towns in Borneo Island. We also propose a novel search algorithm using better evaluation criteria and the notion of cumulative proximity score. We draw comparison on the proposed algorithm and other search techniques by evaluating their efficiency and effectiveness. The efficiency can be measured by the time and space requirement, whereas effectiveness refers to the accuracy of the first found solution. The rest of this paper is organised as follows: the next section describes some of the best known search techniques for route finding. Following which, we discuss the implementation of different search algorithms, including our novel proposal. Experimental results are then presented, and conclusion is drawn based on the analysis of the results.

Abstract Today, the increased traffic and complex modern road network have made finding a good route from one location to another a non-trivial task. Many search algorithms have been proposed to solve the problem, and the most well-known being Dijkstra’s algorithm, Johnson’s algorithm and A*. While these algorithms are effective for path finding, they are wasteful in terms of computation. In this paper, we present a study to examine both uninformed search and heuristic search based on some major cities and towns in Borneo Island. We propose a novel algorithm for intelligent route finding using better evaluation criteria and the notion of cumulative proximity score. We show that this proposed technique is able to reduce the time and space required in computation, and also produce better result in terms of accuracy and shortness of path found.

1. Introduction

2. Background

Route finding, or better known as path-finding in the field of Artificial Intelligence, is a process of finding the least-cost route given a source-destination pair. It has many uses, ranging from transportation, networking and gaming systems to helping robots and travellers navigate through an environment. To date, many solutions have been proposed for this task, and the most well-known being Dijsktra’s algorithm [1, 2], Johnson’s algorithm [3] and A* [4]. Most of these search techniques adopt flooding search approach that finds the shortest route for a given source-destination by locating all possible paths. As such, the resources requirement of these algorithms increases exponentially with the complexity of the problem

978-0-7695-3136-6/08 $25.00 © 2008 IEEE DOI 10.1109/AMS.2008.163

In 1959, Dijkstra introduced a single source shortest path algorithm which was named after himself [2, 3]. This technique is a combinatorial algorithm of bestfirst search and breadth first search [2] due to expansion based on adjacency matrix of all nodes forming a network. Dijkstra’s algorithm starts by assigning infinity as default score to all nodes except the source. Candidate nodes for subsequent computation will be stored into a priority queue according to adjacency matrix that shows directional connectivity between nodes. Priority queue integrates best first search notion for computational purposes. Each connection edge has values which imply the respective distance from one node to another.

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As the computation domain expands to the entire network, node score of adjacent nodes will be updated to hold the shortest distance from source to the current node. This process will continue until the priority queue is empty, i.e. the goal is reached. The potential of Dijkstra’s algorithm was brought to a new level by Bellman and Ford later [3]. The improved algorithm, known as Bellman-Ford algorithm, allows handling of negative edge in the network. This algorithm requires sequential labelling of all nodes in a network and each node will hold two values, the shortest path from the source and the previous node constitutes to the shortest path. BellmanFord algorithm then executes iterations to update each node in order according to respective label. The major shortcoming of Dijkstra’s algorithm is the time and space requirement. Especially for huge networks, the computational cost for it to work is far too excessive. This technique integrates greedy bestfirst paradigm in locating the local optima as a heuristic to obtain global optimum. Nevertheless, a lack of direction differentiation features greatly reduces its efficiency in route finding process. Similar characteristics are found in Johnson’s algorithm and A*. Therefore, improvement is needed.

Our experiments are carried out by comparing the first found solution of each algorithm. Backtracking is integrated into all search algorithms except hill climbing and random walk. A two dimensional array of size n x n is used to remember cities/towns that lead to dead end situation (loop in networking context), where n is the total number of cities/towns used. Backtracking feature is essential for either obtaining the global optimum or implying that a sourcedestination pair is not connected.

3.1. Uninformed Search As aforementioned, two uninformed search techniques are being considered and examined, namely random walk and depth first search. The following subsections will describe them in detail. 3.1.1. Random Walk. This algorithm will search the paths randomly, as long as the cities and towns are connected, without taking any consideration regarding the distances between the connected cities and towns. A random number is generated using rand() function to indicate the index of the next node. For example, if Town A is connected to Town B, C and D, random walk will generate a number from 1 to 3. This can be achieved with rand() % n + 1, where n is the total number of towns connected to the previous town (in this case, n = 3). If the number generated is 2, second town (Town C) will be chosen. This step is repeated if more than one town is connected to the previous town. A disadvantage of this search type is that there is a possibility of reaching dead-end since it does not utilise backtracking. High degree of inconsistency is expected from this approach.

3. Methodology In this section, we will present our implementation of each search algorithm and discuss how they work. We examine both heuristic search and uninformed search for route finding based on 100 cities/towns in Borneo Island. The road network is represented as a search tree with reference to an adjacency matrix (i.e. two dimensional array) showing the connectivity between cities and towns. Each city/town in the network holds its respective geographical coordinates. Distance between two cities/towns is obtained mathematically with respect to these coordinates based on “haversine” formula as follows:

3.1.2. Depth First Search. This algorithm traverses a tree graph by starting at the root node, and always chooses to go deeper into the graph. When it comes to the bottom of a finite graph, backtracking is performed to allow it to return to the most recent node it has not finished exploring. In our implementation, we rank successive candidate towns according to alphabetical order. This is done by comparing the ASCII code of the town’s name, as shown in Figure 1. The name of a town with lower ASCII code will be chosen as the next path. This is because when implementing depth-first search on a tree, the entire connected towns’ names are normally arranged according to alphabetical order. Therefore, by comparing the ASCII code we can actually build a virtual tree for the connected towns. Brute force search pattern of depth first search anticipates a non-optimal route.

distance = r * acos[sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1)] where longitude lon and latitude lat are in radians, and the earth’s radius r is 6378.7 km. We specifically selected some tree-based search techniques for comparison purposes, including two uninformed search algorithms i.e. random walk and depth first search, and three informed/heuristic search algorithms i.e. hill climbing, best first search and A*. Experiments will be conducted to compare these algorithms with a modified A* as well as our novel algorithm.

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based on the sum of c(Node) and t(Node) is inappropriate because it does not reflect the factors of interest – how close is the current node to the destination. Apart from that, the total distance of the route is a cumulative distance of node to node distance which is a missing element in the above evaluation function.

Figure 1. ASCII code for alphanumeric characters

3.2. Heuristic Search

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Heuristic search is implemented to provide an educated attempt for intelligent route finding. Three heuristic search algorithms will be described here: they are hill climbing, best first search, A*. In addition to that, our modified A* and the novel proposal will also be discussed.

B

X F

3.2.1. Hill Climbing. This is a simple heuristic search that continuously traverses forward based on an evaluation function. The evaluation function can be presented as f(Node) = d(Node), where d(Node) is the distance from last node to current node. Hill climbing, in theory, is possible to end up at dead end (a.k.a. local optima) since backtracking is not allowed. However, high efficiency is expected when complexity of a network is relatively low.

Y A S Legends: S – source T – destination F – current node Figure 2. Limitation of conventional A*

3.2.2. Best First Search. This is another heuristic search that has relatively low complexity. It takes into consideration the distance between the connected cities/towns to the destination. The evaluation function for best first search is f(Node) = t(Node), where t(Node) is the distance from current node to the destination. Node with a smaller score is the better one since it implies that the node is relatively closer to the destination than other nodes. Unlike hill climbing and random search, backtracking is incorporated into this technique and hence it will always reach the destination.

Figure 2 shows an example in which the limitation of A* approach produces the same score for two nodes at different locations. Considering node X and node Y which are located on the straight line between the source and destination, these nodes will have similar score due to the location differential incompatibility of the evaluation function of A*. This is because it does not determine the proximity of current node to the destination due to the summation operator. Similar situation is encountered when two nodes are located at a hemispheric curve, where diameter is the straight line between the source and destination, i.e. node A and B. Further analysis is performed upon the A* technique based on physical displacement efficiency. A node with good score is a node that is: 1. far from source – high c(Node) 2. close to destination – low t(Node) Hence, the following evaluation function is instead used for our modified A*:

3.2.3. A*. A* search is indeed an extension of best first search. Conventional A* uses a more intensive evaluation function consisting of the cost from the source to the current node and the estimated cost from the current node to the destination in choosing the next node. The evaluation function can be represented as f(Node) = c(Node) + t(Node), where f(node) is the node’s score, c(node) is the cost to travel from source to current node, and t(node) is the cost to travel from current node to destination. This evaluation function is likely to cause confusion in the context of our route finding task that requires a multiple-step solution in general. Hence, consideration

f(Node) = c(Node) - t(Node)

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Assignment of negative sign to the term t(Node) allows displacement differential compatibility. In other words, node A and B and node X and Y will have respective scores which are different from each other. As opposed to the conventional A* evaluation, higher score from the modified evaluation function is the better score. Both positive and negative scores are possible despite no negative weight existed between cities/towns. Positive score suggests that the node is closer to destination than to source and vice versa. Higher score implies that a node is travelling further away from source and closer to end node.

Legends: f1(node) = score of the 1st node. c1(node) = cost to travel from source to 1st node. t1(node) = cost to travel from 1st node to destination. d1(node) = cost to travel from previous node to 1st node. f2(node) = score of the 2nd node. c2(node) = cost to travel from source to 2nd node. t2(node) = cost to travel from 2nd node to destination. d1(node) = cost to travel from 1st node to 2nd node.

3.2.4. Custom Search. The shortest path always has the best cumulative score. In some cases A* search does not find the shortest path from source to destination. This is mainly due to the fact that A* only considers node score at a particular state and not the cumulative score for more than one state. In the case when node with the best score does not have the best cumulative score for subsequent nodes, the shortest path cannot be found. Hence, our novel proposal has placed a high emphasis in solving this weakness and allows the comparison of cumulative score for two nodes ahead. The evaluation function of node in our novel algorithm is further modified from the evaluation function of A* with the introduction of a new quantifier d(Node) – distance of current node from the previous node. The single state evaluation function is:

Figure 3 illustrates the idea in the form of a search tree. From source S, there are four possible combinations for two subsequent nodes ahead, namely A-C, A-D, B-E and B-F. Assuming that the combination of B-F has the best score, node B will be selected. Similar evaluation will be used to obtain the subsequent node. The three possible combinations from node B are E-G, E-H and F-I. Assuming E-G has the best score, node E will be assigned as the subsequent element of node B despite the fact that B-F has the best score in previous state. With the integration of backtracking and efficient evaluation function, our custom search is capable of finding the shortest path (relative to other search techniques) for most source-destination instances over the testing process.

f(Node) = c(Node) - t(Node) - d(Node)

S

The notion of introducing an extra quantifier into the equation is conceived from hill climbing search. This is because observation shows that hill climbing is capable of finding the shortest path with the least effort in many cases. The practical evaluation function for our custom search considers the cumulative score for two subsequent states. A node with high score is desired in custom search because a good node is: 1. far from source – high c(Node) 2. close to destination – low t(Node) 3. in close proximity from current node – low d(Node) The cumulative score can therefore be obtained from the following equation:

A

C

B

F

E

D G

H

I

Figure 3. A search tree of intelligent route finding

4. Experimental Results

f(Node) = f1(Node) + f2(Node)

In our experiments, 5 source-destination sets of sample size 200 and 10000 each were used for performance evaluation of all search techniques. Two quantifiers are used as evaluation criteria. The first being the average distance returned by a search

where f1(Node) = c1(Node) - t1(Node) - d1(Node) and f2(Node) = c2(Node) - t2(Node) - d2(Node).

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technique and the second being the total number of steps (iterations) required to solve all sourcedestination pairs in the sample. Experimental results are shown in Table 1. In general, the best search algorithm among all is our custom search. Hill climbing is an under-performing approach because it chooses only the node with shortest path ahead, i.e. only node with lowest travelling cost will be selected. This technique always ended up at local optima or dead end situation due to the absence of backtracking mechanism. Moreover, the effectiveness of hill climbing is highly compromised because the evaluation function does not take into consideration the relative position of a node to the destination. In situation where a node is closest to the current node but located further from target that exists, its effectiveness will drop. Best first search yields a much better result than hill climbing because of its geographically sensitive

characteristic. Its evaluation function takes into account only the straight line distance of candidate nodes to destination. As the route vector expands, the last node gets closer to the destination. Nonetheless, travelling cost from one node to another is not taken into consideration by this technique. Hence, solutions found by best first are still defective. A* search takes into consideration the cost of travelling from source and cost to find the destination with reference to the current node. This technique considers the cost efficiency of the candidate node. Node with desired score is positioned further from the source and closer to the destination. Such criteria ensure that the current node will get closer to destination as the vector expands. While it is highly effective, A* in comparison to the custom search is still slightly inferior because the evaluation function does not initiate smart routing technique by analysing the connection network at further extent.

Table 1. Performance evaluation for 10 sets of random samples each with sampling size of 200 or 10000. Efficiency (cumulative steps) and effectiveness (average distance) are being evaluated. 200 SD1 Search Type Random Walk Depth First Search Hill Climbing Best First Search Conventional A* Modified A* Custom Search

Average Cumulative Steps 1630.6 11101.4 1997.0 6977.2 8779.2 7735.2 6635.0

10000 SD2

Average Distance (km) 997.19 2308.49 656.139 1761.51 1935.52 1851.30 1585.63

From the experimental results, we see that the modified A* search has outperformed the conventional A*. The accuracy of results and timespace requirement are better off when modified A* is used. We also see that the custom search has achieved the shortest average distance and the least cumulative steps among all. This is because its evaluation function, as discussed previously, takes into account three quantifiers that affect the results: 1. Distance of current node to destination. 2. Distance of current node from source. 3. Distance of current node from previous node.

1 2

Average Cumulative Steps 83147.0 541449.4 105054.6 318264.0 420809.0 358468.8 313108.0

Average Distance (km) 1070.04 2276.34 721.32 1714.07 1953.85 1771.82 1523.25

Apart from that, we reckon that a shorter route always has a better cumulative score. Thus, by allowing the evaluation function to look two nodes ahead, we manage to reduce the probability of selecting a node that has the best score at a particular state but with poor scores at the subsequent nodes.

5. Conclusion In this paper, we studied and identified the shortcomings of some conventional heuristic search algorithms. We introduced a modified A* as well as a novel search algorithm by modifying the evaluation functions and proposing some changes for better evaluation criteria.

Random Walk and Hill Climbing did not reach the destination 164 times and 176 times in average respectively. Random Walk and Hill Climbing did not reach the destination 8456 times and 8650 times in average respectively.

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We showed that the use of cumulative proximity score can greatly improve the efficiency and effectiveness of intelligent route finding techniques. The notion of cumulative evaluation for two nodes ahead ensures optimal solutions in our study. Future work will investigate problems with higher level of complexity. The breadth of the proposed cumulative evaluation can also be further expanded.

Acknowledgement The authors are grateful to the School of Engineering, Swinburne University of Technology (Sarawak Campus), for providing computing support in conducting the experiments. The authors also wish to thank the anonymous reviewers for their comments and suggestions.

References [1] E.W. Dijkstra, “A Note on Two Problems in Connection with Graphs,” Numerische Mathematik, 1, 1959, pp. 269-271. [2] M. Puthuparampil, Report: Dijkstra’s Algorithm, tech. report, Computer Science Department, New York University, 2007. [3] J. Siek, Bellman-Ford Shortest Path, Boost C++ Libraries, Indiana University, 2000; http://www.boost.org/libs/graph/doc/bellman_ford_shortest .html [4] J. Pearl, Heuristics: Intelligent Search Strategies for Computer Problem Solving, Addison-Wesley, 1984. [5] B. Liu, “Intelligent Route Finding: Combining Knowledge, Cases and An Efficient Search Algorithm,” Proc. 12th European Conference on Artificial Intelligence (ECAI 1996), Budapest, Hungary, 1996, pp. 380-384.

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