1 Spatial Search in the Context of Urban Studies

Urban studies is a transdisciplinary field that encompasses different academic fields, including urban geography, urban sociology, urban economics, urban housing and neighborhood development, urban environmental studies, urban governance, politics and administration, urban planning, design, and architecture (Bowen et al. 2010; Harris and Smith 2011). Search is ubiquitous in these focused research areas (Ballatore et al. 2016). In its most general form, spatial search is the search for information in a spatial and temporal context (Miller 1992). The introduction of the spatial dimension in the search problem can be viewed from two perspectives: one is as part of the information sought (i.e. the search for a place) and the other is as the context in which the search is carried out (e.g. the network of roads to be routed through with an optimal route; Miller 1992).

Spatial search in urban studies carries different connotations depending on the root subject and the application. In the context of technology and geoinformatics, spatial search includes spaceless point search, range search, k-nearest neighbor search, and aggregated spatial search (e.g. total area or total count). In economics and sociology, spatial search can be seen as a decision problem and behavior. The spatial search problem is formatted as a connected graph with physical dimensions (e.g. two-dimensional space). The spatial search problem can vary with options (e.g. perfect knowledge with fixed sample set, online without recall, online with recall, with imperfect information). In the environment of linked open data (LOD), spatial search can be described as a process of identifying the place (converting into geographic information), modeling the spatial dimensions, indexing spatially for improved performance or heuristic results, formulating the search problem, and searching for results in constrained cases.

Spatial search in urban studies involves the following components to manage and maintain a spatial information system:

  • Geocoding: a process to parse and extract spatial references from a query request.

  • Spatial indexing: a process to improve the performance of spatial information retrieval.

  • Spatial search algorithms: a set of algorithms to achieve the efficient and effective discovery of spatial information for different applications.

  • Catalog and federated catalog: a system to manage spatial metadata.

The chapter is organized as follows. The next section reviews the geocoding process. Information about popular geocoding approaches and tools is introduced in this section. This is followed by a review of the approaches and data structures used in indexing the spatial information. The third section describes the spatial search problem as expressed in computer algorithms, while the fourth section reviews the cataloging strategies of spatial data and their approaches in distributed environments. The final section briefly touches on some of the recent advances and research directions in spatial search.

2 Geocoding

In urban studies, place names and street addresses are commonly used in referencing data geospatially (Dueker 1974). Geocoding is the step to relate location to descriptive text or place names. In early literature, it was termed place naming (Dueker 1974; Tobler 1972). In urban areas, geocoding can be efficiently referenced using different approaches for different datasets. Street geocoding, parcel geocoding, and address-point geocoding are three of the commonly used approaches in geocoding to associate an address with spatial coordinates (Zandbergen 2008; Owusu et al. 2017). As more and more types of geocode have emerged, the levels of detail can be associated with geocodes at different granularities. Table 37.1 shows the major generations of geocoding technologies along with major software or services for the corresponding generation. Geocoding has evolved along with the development of geographic information systems (GIS). At the beginning of GIS development, in the 1960s, the simplest geocoding schemes and systems became available. Geocoded area units could be matched to a representative point. Because these geocodes (e.g. demographic information, economic metrics) can associate with many attributes, they can be used effectively as base areal units for analyzing spatial differentiation in urban areas.

Table 37.1 Brief history of geocoding development
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In the Web environment or connected applications, the approach is to use the API provided by geocoding services. All these services support both geocoding and reverse geocoding. The responses of these APIs are mostly in JSON, which can be easily incorporated and used by JavaScript in the Web environment (Table 37.2).

Table 37.2 List of selected geocoding web services
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A place name may evolve over time, and sometimes, a place may carry multiple alternative names. In such cases, a gazetteer (a searchable database of toponyms) is useful and may be adapted to provide specific geocoding assistance. A gazetteer also contains basic information about the place in addition to geographic coordinates. This basic information may include demographic statistics, physical features, literacy, and economic conditions. The NGA GEOnet Names Server (GNS) is one of the sources used in these services. These services from gazetteers have been found very useful in urban studies (Janowicz et al. 2019; Dimou and Schaffar 2009). Table 37.3 lists a few of the most widely used gazetteers for retrieving geographic dimensions or coordinates of a place name and basic information about the place. The capabilities of gazetteers in disambiguating place names and putting place in context have led to many applications in the semantic analytics of urban studies (Janowicz et al. 2019).

Table 37.3 List of selected gazetteers
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3 Spatial Indexing

Spatial indexing is the process of creating an effective and efficient data structure to help in speeding up spatial queries. Spatial indexing differs from common database indexing in having spatial properties: the object is not just one value but has two or more dimensions, and the size of an object may be non-zero (that is, a line, area, or volume; Kriegel and Seeger 1988). These properties lead to spatial relationships that are more complex than simple linear relationships. Many spatial indexing schemes have been developed along with the development of computer technologies (Kriegel and Seeger 1988; Lu and Ooi 1993). The basic goal of such spatial indexing is to reduce the computation required to retrieve matched spatial objects, given a set of geometrical criteria.

To create a spatial index, it is first necessary to identify the features to be indexed. For example, in a 2D spatial world, geographic features are commonly expressed as points, lines, or areas. Points can be represented as a pair of coordinates, which can be treated as fields to be indexed in a spatial database. Most spatial indexing approaches are specially designed for points (Lu and Ooi 1993). Lines and areas cannot be represented accurately as fields fit for indexing in a spatial database without losing information. Representative features need to be either selected or extracted for complex geographic objects. The processes are analogous to feature selection and feature extraction in machine learning, statistics, and information theory. In other words, the selection of features does not change the values which can be interpreted as dimensions. For example, the minimum bounding rectangle (MBR), the two-dimensional case of the minimum bounding box, can be treated as a selected feature, since its value can be found in the array of coordinates representing the geographic object. Any selected coordinate from the represented arrays (e.g. start point, end point, or middle point) can also be selected as the basis of indexing. The process can be generalized as one of transforming a k-dimensional space to a 2 k-dimensional space as described by Kriegel and Seeger (1988). For example, a rectangle aligned with the axes in 2D space can be defined by four coordinates. One encoding can be the corner coordinates (either upper left coordinate plus lower right coordinate or lower left coordinate plus upper right coordinate) or the center coordinates plus extent distances to each side (Kriegel and Seeger 1988). The grid file could be a four-dimensional grid, with the rectangle snapped to the closest cell in the grid file. On the other hand, the extraction of features goes through a computerized process to compute a set of values from the objects. For example, a hashing value is computed from the object using a hashing function. A centroid can also be computed from the object. The object can be represented as the first n principal components using principal-component extraction algorithms. These derived features can be used as indexed fields in a spatial database.

The next question for spatial indexing is how to handle the overlapping of spatial objects defined by the indexing spatial feature. Two schemes are available to deal with the partition: a clipping scheme (C-scheme) and a bounding scheme (OR-scheme) (Kriegel and Seeger 1988). For example, when an MBR is used as the spatial feature, the coverage defined by one MBR may overlap with that of another MBR. One example is shown in Fig. 37.1. With the clipping scheme, the object is duplicated with both partitions when the partition line crosses the region. For example, Object R3 is duplicated in both partitions (Fig. 37.1a). With the OR-scheme, Object R3 is only included in one partition S1 (Fig. 37.1b). The advantages and disadvantages of the two schemes are described in Table 37.4.

Fig. 37.1
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Partition scheme for overlapping regions

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Table 37.4 Schemes for overlapping regions in a partition
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The computerized data structures for spatial indexing are as follows:

  • Fixed grid index: The simplest example is uniform grid scheme where the space is partitioned uniformly into regular grids by value ranges along each axis. The grid system can be predefined with specified intervals or units. Retrieval time for the closest spatial rectangle would be O(1), and on average for any spatial rectangle would be O(nCells + n), where nCells is the number of grid cells and n is the number of spatial objects, that is, the rectangles in the example. The memory requirement is O(nCells + n).

  • Spatial hashing: Because the distribution of spatial objects is often sparse, a uniform grid would result in many empty cells. A hash table can be used to store the index, and multi-level multi-key grid files can be used to index the multi-dimensional spatial data (Bentley and Friedman 1979).

  • Spatial data partitioning trees

    • Binary space partitioning (BSP) tree: This is a general partition approach to partition space recursively into two convex sets using a hyperplane. It was developed as a general method in 3D video image processing (Schumacher et al. 1969). The k-dimensional binary search tree (k-d tree) is constructed by using one axis to split data at the median of the points along the axis (Bentley 1975). The Local Split Decision tree (LSD tree) is designed to handle both points and intervals (Henrich et al. 1989). The K-D-B tree is a derived tree structure that combines properties from the k-d tree and the B-tree (balanced tree) (Robinson 1981).

    • Quad tree: A quad tree builds a hierarchical representation of spatial data by dividing recursively into four quadrants (Finkel and Bentley 1974).

    • Octree: An octree is a hierarchical data structure that extends the quadtree to 3D, with all internal nodes having eight children (Meagher 1980).

    • Balltree: A balltree is “a complete binary tree in which a ball is associated with each node in such a way that an interior node’s ball is the smallest which contains the balls of its children” (Omohundro 1989).

    • R-tree: An R-tree uses a minimum bounding rectangle (MBR) to determine its children (Guttman 1984). It is a balanced tree. Its variant trees include the Hilbert R (Kamel and Faloutsos 1984), R + (Sellis et al. 1984), Priority R (Arge et al. 2008), R* (Beckmann et al. 1990), GiST (Hellerstein et al. 1995), and G-tree (Zhong et al. 2015).

    • Metric tree: The vantage-point tree (vp-tree) is a space-partitioning algorithm to construct a tree with a sphere-like bounding area to partition the metric space (Yianilos 1993). Each part is defined within a threshold to each vantage point. A multi-vantage-point tree (MVP tree) is a variant of vp-tree which uses more than one point to partition at each level (Bozkaya and Ozsoyoglou 1999). The cover tree algorithms construct a leveled tree where each parent covers the extent of all children (Begelzimer et al. 2006). The Bukhard-and-Keller tree (BK-tree) is adapted to discrete space by arranging points that are close to each other (Burkhard and Keller 1973).

4 Search Algorithms

A spatial search in urban studies can be viewed from different perspectives and formulated differently for different subject domains. In this section, two perspectives are examined. First, from the perspective of geography, spatial search is treated as a technology and method, and typical spatial queries and corresponding search algorithms are reviewed. Second, from the perspective of urban economics and urban sociology, spatial search is treated as a form of decision-making, generalized spatial search is formulated with graph theory, and related search algorithms are reviewed.

4.1 Spatial Queries

The following are the common types of spatial search used in urban studies:

  • Nearest neighbor search: This is termed the k-nearest neighbor (k-NN) search. Typical questions can be “Find the k stores that are closest to a given point or current location” or “Find the closest restaurant.”

  • Range search: Range search is also common in urban studies. Example queries: “Find all the restaurants with 5 miles range” and “Find all the zones that can be reached between a half hour and a hour.”

  • Aggregate search: Questions can be often asked in urban studies that involve spatial aggregation. Examples are: “Get the number of hospitals for travel distance zones of under 10, 10–50, 50–100, and above 100 miles” or “Find the total area of green space in an urban district.”

The k-NN search is well studied in computer science and geographic information systems (Knuth 1997). There are a suite of algorithms designed to solve the problem. There are two major categories of algorithms: exact search and approximate search. The simplest approach to find the k-nearest neighbors is sequential search that does not require any preprocessing of the spatial data (Bentley and Friedman 1979). The search time is O(kn), where k is the dimension and n is the total number of features. The storage requirement is also O(kn).

Spatial indexing can be used in preprocessing the data, creating a data structure that can be easily retrieved. BSP-trees, metric trees, and R-trees are three types of commonly used tree data structures in indexing spatial data. The kd-tree, one of the BSP-trees, uses axial rays to partition (ending up as rectangles), while the vp-tree, one of the metric trees, uses equidistance circles to partition data. The R-tree structure uses rectangles but has a focus on keeping the geographic object in a hierarchical structure. Most of these data structures lead to improvements by reducing the time to search to approximately O(log n) on average.

Different geographic information systems may support different spatial indexing algorithms. The R-tree and its variants are the most popularly implemented spatial indexing algorithms in geographic information systems, including PostGIS, MySQL, and Oracle. A grid-based spatial indexing scheme is popularly implemented in many geospatial databases, including Esri geodatabase, Oracle, and Microsoft SQL, due to its data-driven spatial indexing scheme.

Spatial search (k-NN, range search, or aggregate search) has been applied in many urban studies. Alternative site selections, such as the “spatial search” of Massam (1980), analyze spatial interactions and require range searches to assess the effect of selecting one alternative over another. For example, a firm searching for a location may consider the labor force that is available within a certain distance of each alternative location. In choosing a location for a retail store location, the analyst may need to conduct spatial queries on household purchasing power within a certain distance of each of the location alternatives. The results of such spatial queries would help in evaluating alternatives and making better plans.

4.2 Spatial Search with Graph Theory

Spatial search can be seen as a decision problem in urban studies, especially those studies with roots in economics. Economic Search Theory is well studied and has been used in studies of urban migration, urban markets, and urban agglomeration effects (Meier 2009,2010). Adding the spatial context, a generalized spatial search model can be formulated (Meier, 1995,2010). The spatial search problem is effectively defined within a connected graph. The vertices of the connected graph are alternatives at discrete locations in two-dimensional space. The edge connecting two vertices represents the cost, which may be a function of distance. The goal is to maximize the expected utility when the decision is to move from one vertex to another. Each alternative may be visited once.

The model of spatial search results from the tight bounding and integration of spatial context with a domain-specific model. In economics, this spatial model is tightly integrated with a model of economic search. This approach of integrating the spatial context with models in urban studies effectively converts the spatial search problem into an optimization problem on a graph.

The traveling salesperson problem is NP-hard. However, most problems in urban studies have a limited size, making them soluble. There are also heuristics to help in solving the optimization problem efficiently.

With the conversion of the spatial search problem to an optimization problem in a graph, the commonly used graph search algorithms become applicable to the spatial search model. These algorithms include breadth-first search, depth-first search, greedy best-first search, heuristic A*, and Dijkstra’s shortest path algorithm. The spatial search model has found applications in market area analytics, firm location, urban effect analysis, and urban modeling (Meier 1995). The simple distance or fuel cost-based spatial search model may be used in urban transportation planning and commercial truck routing (Zarezadeh et al. 2018; Moreno-Monroy and Posada 2018; Monte et al. 2018).

5 Distributed Search and Interoperability in the Web Environment

The abundance of geospatial information has grown beyond anyone’s ability to manage it be properly. The introduction of live sensors and fast updating of information also suggests that the monolithic geographic information system cannot satisfy the requirements of spatial search in urban studies. Yet the data resources available for urban studies continue to grow.

There are several approaches to enable spatial search and geoprocessing to leverage the growing volume of information for urban studies. First, the information can be harvested and ingested into a local spatial catalog system through the harvesting of spatial metadata and data from different sources. The local spatial catalog system has to manage all the information. Each harvester may be updated or re-started (if incremental harvest is not supported by remote services). After each harvest, spatial indexing needs to be updated or re-built. The advantage for such a system is that the existing spatial indexing techniques are already supported. The major drawbacks are that the data can grow out of control and are not always current.

Second, the information is harvested, integrated, and indexed in a distributed manner. In this case, the local catalog system is replaced with a distributed catalog that clusters multiple cloud-computing instances. Each cloud-computing instance may handle a strip of information. A distributed spatial indexing scheme needs to be adopted to support the spatial search in such a distributed system (Priya and Kalpana 2018). The advantage for such a system lies in its capability to handle large datasets in a scalable cloud-computing environment. The major limitations are: (1) the freshness of the metadata and data cannot be warranted, (2) the remote services may not allow the duplication of their metadata and data for various reasons, and (3) the maintenance of a large distributed spatial catalog system can still be a challenge, and the distributed spatial search capability is still in development.

Third, a federated spatial catalog system can be adopted to support the on-the-fly integration of distributed search (Shao et al. 2013; Bai et al. 2007). The development of a federated spatial catalog depends on the adoption of open geospatial standards. The standard interface and response from catalogs make it possible to do translation on the fly. The idea of federated catalog is to set up a series of plug-in translators that handle the translation of request to and response from the remote catalog services. When a user sends in a spatial query, the query request is first translated into a format that matches the remote server and the translated request is sent out. The response from the remote service is then translated and integrated in the mediator to be sent back to the user. The advantages of such a federated catalog are: (1) it does not need extensive resources in manage the metadata and data since most of the resources are still maintained by the original provider; (2) the contents are in complete synchronization with remote services; and (3) spatial search is completed in a distributed environment. The drawbacks are: (1) the spatial search function and responses are tied to what the remote services offer, and (2) duplicates may not be removed properly if two remote services offer the same content.

6 Trends

The spatial search problem is a hard problem to solve. The performance of current solutions is acceptable only because either one of the following assumptions stands: (1) the size of data is limited, (2) optimal heuristics exist for the dataset, or (3) the best option executes in an acceptable time. This section reviews two frontiers in solving the spatial search problem: a quantum spatial search algorithm and semantic spatial search.

Quantum algorithms have emerged in solving the spatial search problem with improvements. Quantum computing is seen as the future of computing, to improve non-deterministic algorithms that consider multiple superpositions of states (Venegas-Andraca 2008; Chakraborty et al. 2016; Ambainis 2008). The spatial search problem is seen as one of the hard problems to be solved with classic computers (Meier 1995,2010), or as a decision problem to find the target vertex in a connected graph (Meier 1995). In a fully connected lattice graph of n vertices, the worst time to find the marked target is O(n log n) using a random walk in a classic computer. New algorithms in quantum computing have shown that the search can be improved many fold with quantum random walks (Portugal 2018). A discrete-time quantum walk (DTQW) algorithm improved the time to O(√n log n) (Ambainis et al. 2005). A controlled quantum walk (CQW) algorithm on a lattice using an ancilla qubit improved the time complexity to O((n log n)1/2) (Tulsi 2008). An improved version of DTQW also achieved the same time complexity (Ambainis et al. 2015). Portugal described an approach to the design of quantum algorithms for the spatial search problem that explains how Grover’s algorithm (Grover 1996), the quantum algorithm for searching a database, “can be seen as a spatial search problem on the complete graph with loops using the coined model and on the complete graph without loops using the staggered model” (Portugal 2018).

The application of semantic technology improves the accuracy of spatial search with more explicit spatial semantics. Most current spatial search solutions treat spatial objects as a spaceless point. Spatial extents and spatial relationships are not taken into full consideration with current solutions. The augmentation of linked geodata (Stadler et al. 2012) with spatiotemporal semantics enables a semantic spatial search (Neumaier and Polleres 2019). A Transportation ontology domain can be added to a semantic-based public transportation geoportal to support semantic spatial search on concepts, relationships, and individuals (Gunay et al. 2014). Ontology provides additional semantic constraints in semantic spatial search (Jones et al. 2004,2001). A spatial entity can be described by its sub-components, and the search for a spatial entity can be modeled as a multi-component spatial search problem (MCSSP) (Menon and Smith 1989, Menon 1990). This effectively formulates the spatial search problem as a constraint satisfaction problem (CSP) in computer science. The suite of heuristic CSP algorithms can be applied to help in finding the best match, including backtracking, graph-based backjumping, arc consistency, and forward checking (Frost 1997).

7 Conclusion

Spatial search has been one of the most intensively researched topics in urban studies, and can be traced back to a pre-computer era. The classic spatial search in dealing with connectivity between spatial objects or entities has been thoroughly researched and supported by most geographic information systems. The spatial search problem can be integrated with models in urban studies to put the research in spatial context. Extending studies with spatial dimensions increases the complexity of problem solving. In a fully connected graph depicting the relationships among entities in a spatial context, the problem is NP-complete and is therefore difficult to solve. However, in actual applications in urban studies, the data size is often manageable and heuristics can be applied to solve the spatial search problem within a reasonable time interval.

New developments in alternative computing environments shed light on solving the spatial problem more efficiently. One of the most researched alternatives is to leverage random walk with quantum computing. Several algorithms have been proposed to solve the spatial search problem efficiently with quantum walks. Another frontier is the use of semantic Web technology in dealing with big data and heterogeneous data in the spatial context.