3.1. Hierarchical Clustering Tree

Unlike spatial indexing techniques, where objects are organized in a certain order, clustering techniques group objects by their characteristics. One important characteristic of road networks is the connectivity of road segments which can be used to cluster roads.

In the family of clustering algorithms for different types of applications, the two most common branches are the partitioning branch and the hierarchical branch (Huang et al., 1998; Kotsiantis and Pintelas, Reference Kotsiantis and Pintelas2004). Partitioning algorithms create a ‘flat’ decomposition of a data set into a set of clusters. Examples of partitioning algorithms include k-means, k-medoids and density-based algorithms. They generally need some input parameters which specify, either the number of clusters that a user intends to find, or a threshold for point density in clusters. However, it is difficult to determine what parameters are needed and what values they have, as the parameters may not even exist.

In contrast, hierarchical clustering algorithms do not actually partition a data set into clusters. Instead, an hierarchical representation of the data set is computed to reflect the possible hierarchical clustering structure of the data set. Hierarchical clustering algorithms are more robust and less influenced by cluster shapes; they are less sensitive to largely differing point densities of clusters, and they can represent nested clusters (Sander et al., Reference Sander, Qin, Lu, Niu and Kovarsky2003). Since hierarchical clustering algorithms can partition a data set with no prior knowledge of the number of clusters (e.g., distribution of road segments in a road network), they are seen as an alternative to spatial indexing techniques in this research. Clustering road segments in a hierarchical clustering tree by using an average-linkage method is described in the next section.

3.2. Clustering Road Segments

In graph theory, a graph is formally represented by the term <V, E>, where V represents vertices in the graph and E represents edges in the graph. A general adjacency matrix in graph theory is A={a_ij}, where a_ij represents the weight (e.g., distance) between i^th vertex and j^th vertex. Consisting of intersections and segments, a spatial network can be represented as a graph where intersections are the vertices and segments are the edges. To better perform map matching for navigation applications, segments are clustered instead of intersections, because intersections only represent geometric information, whereas segments represent both geometrical and topological information.

To build the hierarchical clustering tree of a spatial network, a new matrix A′ is introduced in this algorithm. Generalizing a spatial network as a non-directional graph clustered by distance, A′ becomes a symmetric matrix. In order to define A′ two matrices are defined first. One matrix represents the topology of a spatial network, denoted by T={t_ij}, where t_ij is 0 if the i^th and the j^th vertices are on the same segment (this occurs when i and j represent the same vertex or i and j are the two vertices of a segment) and t_ij is 1 when the i^th vertex is not directly connected to the j^th vertex. The other matrix is an adjacency matrix A={a_ij}, which computes the Euclidian distance between any two vertices to represent the closeness of the intersections in geometric space. As a result, a new matrix A′=A*T={a_ij*t_ij} is defined to combine the two factors, geometrical distance and topological relationship from matrices A and T. The weight of each element in the matrix A′ not only considers the distance between the vertices (i.e, intersections), but also considers the topology of the spatial network.

A road network can be very large but sparse, since a road link is only connected to few links in topology. Therefore, topological relationship matrix T is sparse and could be built in a more compact data format by traversing only physical connected links. In addition, although real world road networks may contain one way segments, the road network is treated as a symmetric graph, which can simplify the clustering calculation and make the matrix A′ compact. A ‘one-way’ attribute can be added in the process of map matching.

Based on matrix A′, we build a hierarchical clustering tree from the bottom-up. Segments on the lowest level of this tree, as the smallest units in the structure, are grouped together based on the distances, one from another, and the clustering process is continued until the root of the tree is reached.

To illuminate the clustering procedure, Figure 1 shows 10 segments, labelled by their identities from 0 to 9. Every segment has two vertices, so 10 segments have twenty vertices, from 1 to 20. Matrix A′, therefore, becomes a 20-by-20 matrix. Since it is symmetrical, only it's upper or lower triangular (off diagonal) needs to be stored. The weights in the upper triangular and the lower triangular are all set as 0 s, as shown in Figure 2.

Figure 1. An example of road network.

Figure 2. Corresponding matrix (20-by-20).

Given the matrix A′, a hierarchical clustering tree is built using the average-linkage clustering method. The average linkage clustering is based on measuring the proximity between two groups of vertices. Here we use the average distance between all pairs of vertices in cluster r and cluster s as the measurement. Its definition is as follows:

(1)$${\rm d} ({\rm r,s}) = \displaystyle{1 \over {\rm n_r n_s}} \sum\limits_{{\rm i} = 1}^{\rm n_r} {\sum\limits_{\,{\rm j} = 1}^{\rm n_s} {{\rm dist}({\rm x_{ri}}, {\rm x_{sj}} )}} $$

where n_r is the number of vertices in cluster r and n_s is the number of vertices in cluster s, and x_ri is the i^th vertex in cluster r, and x_sj is the j^th vertex in cluster s.

After applying the average-linkage clustering method, Figure 3 shows how the hierarchical clustering tree is built from the matrix A′ in Figure 2. The clustering process starts from the bottom where average distances of two vertices on the same segment as a cluster are set to zero in A′. Segments are grouped together in an order that clusters those with closer distances until all the clusters are grouped together and the root is reached. The root of the clustering tree represents the entire spatial network as one group.

Figure 3. Corresponding clustering tree.

4.1. A Binary Tree Structure from the Clustering Tree

Since each group only joins one of the two members (one of which may be compound) in the clustering tree shown in Figure 3, the clustering tree can be structured as a binary tree. Therefore, the map matching process will search a binary tree to identify candidate segments. Considering that road segments each have a different position, length and orientation, Minimum Bounding Rectangles (MBR) are used to delineate cluster boundaries. Each MBR in upper levels of the tree represents a group of segments where sibling nodes may cover overlapping areas. Construction of the binary tree is from bottom to top, so in the end, the root node represents a MBR covering the entire road network. Figure 4 shows a portion of the binary tree. Figure 5 shows the MBR of a group of road segments, segments 1, 2 and 9.

Figure 4. Data structure of a binary tree for road segment clustering.

Figure 5. Distance from a GPS point to a MBR.

4.2. Searching Algorithm

As shown in Figure 4, the range of MBRs is narrowed down from top to bottom. Therefore, in order to determine the search window for map matching, the binary tree is traversed from top to bottom and will stop at a group node where a halt criterion is met. The halt criterion is important for determining the size of the search window. The searching algorithm is as follows:

1. 1. Input the binary tree and a GPS point.

2. 2. Set a criterion to halt search.

3. 3. If the GPS point is within the boundary of the current group node and the halt criterion is not met, then search its sub-trees. Otherwise, search the sibling node.

4. 4. If the halt criterion is met, then traverse all the leaf nodes of this sub-tree for the candidate road segments.

The halt criterion is critical in this algorithm. Since our purpose in the algorithm is to define the set of candidate road segments by a given GPS position, the halt criterion must consider the relationship between the GPS position and the segment groups. We describe a GPS position as p(x, y), and a bounding box as B(minx, miny, maxx, maxy). Then, c is noted as the centre of the bounding box. Distance (p, c) calculates the distance between the GPS position (p) and the centre point (c) and within (p, B) evaluates whether p is within the bounding box or not. With this, the criterion could be:

$${\rm IF\; Distance} \ ({\rm p, c}) \lt {\rm threshold\;and\;Within \ (\,p, B)\;THEN\;stop\;search}.$$

The smaller the threshold is, the closer the GPS position is to the centre of a selected group. In order to avoid misidentifying road candidates if the GPS position is located on the boundary of two clustered sub-groups, the threshold is set relatively small in order to guarantee that the GPS position is close enough to the centre of a selected group. With the same threshold, road segments in one cluster would be selected as candidates more often in dense urban areas than in less dense, rural areas.

4.3. Adaptive Search Window Set

In spatial indexing techniques, a search window for candidate segments is normally a fixed-size rectangle centred at a given GPS position. In order to retrieve those segments within the search window, searching algorithms need to pass through the indexed tree several times. In contrast to such indexing techniques, a clustering tree provides an adaptive search window by clustering spatially close segments into groups and only passing through the tree once. The output of the searching algorithm is a list of candidate segments.

The number of segments selected in the list of candidate segments depends on three factors. The first factor is the density of the road network. For example, in a rural area, road networks are sparse and candidate segments may only include one or two segments of highways, whereas in urban (downtown) areas, more segments are included in the candidates list. Second is the relevant dynamic position of GPS positions in a clustered group. The final factor is the threshold in the halt criterion, which is set based on GPS error range.

In common with spatial indexing techniques, nodes in the clustering tree have overlapping MBRs, which is the reason why we set a threshold to justify whether the map matching of a GPS position is related to this clustered segment group or not. If a GPS position is located between groups, then the search window will be adaptively adjusted for a higher-level clustered group that includes the clustered lower-level groups. For instance, in Figure 5, if a GPS position is located on the corner of Segment 2 and Segment 6, the search window should cover the higher group that includes segments 9, 1, 2 and 6. Instead of only searching the group corresponding to the MBR shown in Figure 5, the search should be stopped on the group that contains segments 9, 1, 2 and 6. This search procedure starts from the root of the tree, as shown in Figure 3, and once the halt criterion is met it will stop at the level that includes segments 9, 1, 2 and 6, rather than searching the lower level.

4.4. Adaptive Search Window Update

To track the movement of a vehicle, a map matching search window must follow its motion. Whether to update a search window or keep searching in the previous window depends on the direction and speed of the vehicle's movement. With the previous candidate road segments in memory, and by knowing the successive positions to estimate the vehicle's movement, searching in the same segment group is continued. However, when the vehicle is moving out of a known segment group, a new sub-tree has to be initiated. Under this circumstance, the search window must be updated by repeating the initial search procedure with new parameters.

5.1. Datasets

We employed road maps in the Pittsburgh area from the US TIGER data files and collected some GPS positions on the University of Pittsburgh's main campus and then tested our algorithm on different GPS locations, simulating a vehicle's movement. Our algorithm was implemented in Matlab and tested in a PC environment with Intel Core 2, 2·13 GHz CPU and 2 GB memory.

5.2. Construction Cost

Memory usage and the time complexity involved in constructing a clustering tree is dependant upon the extent of a road network. In order to save both memory capacity and time, our strategy was to divide a large-scale road network into a set of sub-networks and then build a corresponding clustering tree for each sub-network. Thus the original road network corresponds to a forest structure. Indexing each tree in the forest is straightforward. As shown in Figure 4, intermediate memory is needed to build matrices to calculate the average distance of clusters. Each matrix, before compression, is n by n, so the memory usage is n ², where n is the number of intersections.

In our algorithm, average-link clustering merges, in each iteration, the pair of clusters with the highest cohesion. Based on this recursive computation of cohesion, the time complexity of average-link clustering is O(n²logn). However, Murtagh (Reference Murtagh1992) compared various hierarchical clustering approaches in computational time complexity and concluded that O(n²) time implementations exist for most of the widely known hierarchical clustering methods, and some methods can even perform close to O(n) expected time for hierarchical clustering. Therefore, the construction cost of our hierarchical clustering tree can be further reduced by more sophisticated techniques. In this paper, we use a recursive approach to construct the hierarchical clustering tree. Table 1 provides features of the constructed clustering trees corresponding to the two different network scales. Constrained by memory limitation in MATLAB®, we could not conduct experiments with large-size road networks. In spite of this, our algorithm can be expanded to any large-scale network by splitting it into sub-networks and organizing it as a forest structure rather than a large tree structure. For example, Allegheny County geographically includes Pittsburgh City, so the city road network can be structured as a sub-network of the county road network.

Table 1. Tree features of two road networks.

A balanced binary tree has a depth, log₂n, but the hierarchical clustering trees in our algorithm are not balanced, which is why maximum and minimum depths are considered. Figure 6 shows the result of the tree construction by using, as an example, the University of Pittsburgh's main campus which has the maximum depth 14 and minimum depth 10 as shown in the first row in Table 1.

Figure 6. The clustered tree for the University of Pittsburgh's main campus.

5.3. Searching Cost

In theory, if n is the number of segments, then the time complexity of searching binary trees is O(log₂n). However, two factors influence query efficiency. One is the density of a road network and the other is the changing of a moving object from an intersection to its linked road segments. Although geometry and topology of road networks decide the tree clustering, once a road network is built, its corresponding graph morphology hardly changes. Thus, the structure of a clustering tree corresponding to the road network is essentially fixed, based on the clustering methods. Therefore, for a given road network, we mainly consider the query cost as influenced by the location of GPS positions.

As vehicles move on the road, they transfer from one road segment to another. As vehicles approach an intersection, or move on a relative short segment, a search window will cover more road candidates than those covered when moving on a long segment. Query performance is most inefficient when a vehicle moves on a boundary between two large-scale clusters. Figure 7 shows how candidate road segments change with different positions when a vehicle is moving. The red square in each figure shows the received GPS position and the red points on the map show the intersections of candidate segments as the result of searching the clustering tree. Furthermore, by testing the same points in a larger area, e.g. in Oakland, which covers the entire University of Pittsburgh's main campus, it can be seen that the nature of the road network itself determines similar candidate roads, as shown in Figure 8, although different clustering trees for different road networks were searched. Since our average-linkage clustering approach is based on the average difference of groups, and the same area has constant road network structure, the searching algorithm produces results with certainty. Given a GPS position, the experiment shown in Figure 8 indicates that selected candidates in the large-scale network are consistent with the searching results in small-scale network, as shown in Figure 7.

Figure 7. Query results changing with a vehicle's movement.

Figure 8. Example scenario on a large-scale network.

As discussed previously, in order to evaluate various map matching situations, experiments were conducted in three scenarios: approaching an intersection, moving on a relatively long segment, and moving around a boundary of two clusters. Table 2 shows the average searching cost for the two situations.

Table 2. Results of the searching cost.

In summary, since the hierarchical clustering tree built for the road network is not a balanced binary tree, we tested on real road networks to observe the actual construction structures and computed average search costs in different situations. Compared with multi-pass searching an object in spatial indexing techniques, searching on the clustering tree of road segments requires only one pass.