In this application, adaptive crystal growth Voronoi diagrams were generated using both raster-based and hexagon-based weighted planes of the same study area with the same socioeconomic context. The results of the HACG will be compared with those of the RACG in delimiting service areas of middle schools using adaptive crystal growth Voronoi diagrams based on accessibility-weighted and population-weighted planes. The delineation results of HACG and RACG are compared in four different spatial resolutions of the weighted plane.

#### 3.2. Weighted Planes

Two types of weighted planes, namely an accessibility-based plane and a population-based plane, were generated for each cell structure (raster grid and hexagon grid). In order to explore the scale effects (how the weighted planes in different scales will affect the delineation results), four different spatial resolutions were used for both raster-based and hexagon-based methods; there are thus one accessibility-based plane and one population plane for each cell structure in each spatial resolution. The accessibility-weighted planes take into account the effect of the transport network and water bodies (as natural barriers) on the accessibility of the schools in the study area. The population-weighted planes represent the distribution of population based on distinct types of residential buildings in the study area. The raster and hexagon cells in these weighted planes have the same size in one spatial resolution to ensure consistency and comparability of the results. Four different spatial resolutions of the raster-based and hexagon-based weighted planes are generated to compare the delineating performance. Specifically, these four spatial resolutions include cell sizes of 50, 100, 150, and 200 m^{2}.

For the accessibility-weighted planes, illustrated in

Figure 4a, seed point cells are assigned a cell value of 10 with a growth speed of 1 cell/cycle. The natural barrier cells (e.g., rivers and lakes) are assigned a value of 0 with a growth speed of 0 since they are difficult to pass without a road or bridge. The transport cells are assigned a value of 1 to 2, and the growth speed varies with their average travel speed. The growth speed of the primary transport network (e.g., highways, freeways, and primary trunk roads) cells is 6 pixels/cycle for an average driving speed of about 30 km/h, while the growth speed of the secondary transport network (e.g., secondary trunk roads, collector roads, and local roads) cells is 4 pixels/cycle for an average driving speed of about 20 km/h. For all other cells, including walkable area cells, the growth speed is 1 pixel/cycle for an average walking speed of about 5 km/h. The cell attributes of the accessibility-weighted plane are listed in

Table 1.

The geographic distribution of the population is one of the major considerations for school service area delimitation in addition to accessibility. Thus, another type of weighted plane for the adaptive crystal growth Voronoi diagrams in this study is based on population distribution. As discussed in Wang et al. (2014), due to the modifiable areal unit problem when using census tracts as the zonal scheme, creating better representations of the continuous distribution of the population is crucial for better results in service area delineations. Therefore, the population-weighted planes in this study were created based on census and land use data, which were used to estimate the population distribution of the research area using the same method in Wang et al. (2014). In the method, the continuous population distribution was simulated by distributing the population from census tracts to each cell in the weighted plane based on the distribution and attributes of residential areas. Residential areas are classified according to an urban land use map and urban planning data into three categories: low-rise residential areas, medium-rise residential areas, and high-rise residential areas. With maximum likelihood estimation, population density is 0.0680 persons/m

^{2} for low-rise residential areas, 0.1135 persons/m

^{2} for medium-rise residential areas, and 0.0937 persons/m

^{2} for high-rise residential areas. Because four different spatial resolutions are used in this study, the cell weights of the population were calculated separately by multiplying the population density with the unit cell size. For instance, with the resolution of 100 m

^{2}/cell, the weights of the cells for the three types of residential areas are 6.8 persons/cell (low-rise residential areas), 11.35 persons/cell (medium-rise residential areas), and 9.37 persons/cell (high-rise residential areas) on the weighted plane. Residential cells are assigned a value of 4 (low-rise residence cells), 5 (medium-rise residence cells), or 6 (high-rise residence cells) for the three types of residential areas with a growth speed of 1 cell/cycle.

Table 1 lists all the possible cell values and growth speeds on the weighted planes, and

Figure 4b illustrates the population-weighted plane.

#### 3.4. Hexagon Data Structure

The raster-based weighted planes for RACG are generated based on the widely utilized raster data structure. Nevertheless, there is no mature data structure for hexagon grid computation to the best of our knowledge. The hexagon tessellation implemented by ArcGIS is in the vector format (collection of polygon features), which leads to very low computational efficiency. To address the limitation, we designed a hexagon data structure (we named it “Hexter”) in a similar format to a raster grid but specifically for the crystal growth algorithm to increase the computational efficiency in this study. As illustrated in

Figure 5a, the original hexagon grid is indexed as “ColID-RowID”, in which “ColID” represents the index of the column (ranging from “A” to “ZZZ”) in the grid, while “RowID” indicates the index of the row (ranging from “1” to “

N”) in the grid. With the indexing, the hexagon grid in the vector format is converted to a raster-like data structure (see the Hexter table in

Figure 5b) with their “ColID” as the columns and “RowID” as the rows in the Hexter table. With the change of the data format, the spatial relationship in the table is also changed. Since the only spatial relationship used in the crystal growth algorithm is the neighbor cells, we defined the neighboring relationship of the Hexter based on the original relationship of the hexagon grid. Illustrated in

Figure 5b, for the cells in the odd columns (e.g., C, E, and G), the neighbors are the cells in their top, bottom, left, right, bottom-left, and bottom-right (e.g., cells with indices “C-2” and “C-7” in

Figure 5), while for the cells in the even columns (e.g., B, D, and F), the neighbors are the cells in their top, bottom, left, right, top-left, and top-right (e.g., cells with indices “H-2” and “H-7” in

Figure 5). The rows do not influence the neighboring relationship of cells. Note that the Hexter table does not represent the spatial relationship of hexagon cells. It is only used to read the vector-formatted hexagon grid into the memory in the format of a table for crystal growth computation. After the computation, the crystal growth results (cell values) will be projected to the original hexagon grid by matching the “ColID-RowID”.

#### 3.5. Results

In the study, middle school service areas were delimited by RACG and HACG in four different spatial resolutions (R1: 50 m^{2}; R2: 100 m^{2}; R3: 150 m^{2}; R4: 200 m^{2}) with different $\omega $ settings ranging from 0.001 to 0.1. In order to compare the results, two types of measurements are utilized: (1) how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school; and (2) how accessible each middle school is within its service area.

The root-mean-square error (RMSE) and maximum error (Max. Error) are employed to compare the commensurateness between the enrollment capacity of a school and the population of its service area.

Figure 6 shows the RMSE of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas delimited by RACG and HACG. It can be seen from the figure that, with different spatial resolutions, the HACG (solid lines in the figure) generally has smaller RMSE compared with the RACG (dashed lines in the figure). In addition, the finer the spatial resolution, the better the results considering the RMSE. In the figure, all lines present a U shape. The best results for different methods and different resolutions are generally achieved when

$\omega $ has a value between 0.015 and 0.065. If

$\omega $ is larger or smaller than these values, the RMSE increases. For the HACG, the best observed result is achieved when the spatial resolution is 50 m

^{2} and

$\omega $ is 0.055 or ranging from 0.015 to 0.035. Similarly, the best observed result is achieved when the spatial resolution is 50 m

^{2} and

$\omega $ is between 0.015 and 0.035 for the RACG. However, comparing the two curves of HACG and RACG with the spatial resolution of 50 m

^{2}, the delimitation results of the hexagon-based method generally perform better with lower values of RMSE.

Figure 7 illustrates the Max. Error of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas delimited by the RACG and HACG with different resolution and

$\omega $ settings. Similar with the results of RMSE, the HACG (solid lines in the figure) generally has smaller Max. Error compared with the RACG (dashed lines in the figure) with different spatial resolutions. For both RACG and HACG, the finer the spatial resolution, the smaller the Max. Error. According to the figure, the best observed results for RACG are achieved when

$\omega $ is ranging from 0.001 to 0.015, while the best observed results for HACG are achieved when

$\omega $ is ranging from 0.02 to 0.08. Considering both the RMSE and Max. Error, the best observed result is obtained when

$\omega $ is ranging from 0.02 to 0.035 for HACG, while it is achieved when

$\omega $ is 0.015 for RACG. With the best observed

$\omega $ values, the HACG has much lower RMSE and Max. Errors compared with the RACG, as shown in the figure. According to the results, both the Max. Errors and RMSE of the differences between the proportion of enrollment capacity and the proportion of population in the service areas are lowest when using the hexagon-based method after calibration of the parameter

$\omega $. Therefore, the results indicate that the hexagon-based method performs better considering the enrollment capacity of middle schools and the population in their service areas.

The longest travel time (

$LTT$) to a middle school from any location in its service area is used as a measure of the accessibility to the middle school within its service area in this study. The maximum

$LTT$ indicates the lowest accessibility of the middle school in its service area. As discussed above, the best observed results are achieved for both RACG and HACG when the spatial resolution is 50 m

^{2} considering the enrollment capacity of middle schools and the population in their service areas; therefore, the accessibility to the middle school within its service area of the delineation results is only compared using a spatial resolution of 50 m

^{2}.

Figure 8 shows the maximum

$LTT$ of the service areas of all middle schools in the study area based on the delimitation results obtained with different

ω settings. For the HACG, the maximum

$LTT$ varies with different

$\omega $ settings: it stays at high values of around 24 min when

$0.035\le \omega \le 0.05$ , while it is as low as 21 min with other

$\omega $ values. Regarding the RACG, the lower values of maximum

$LTT$ (around 25 min) are achieved when

$\omega $ is smaller than 0.02 and larger than 0.045. Comparing the two curve lines of RACG and HACG, it is clear that HACG always performs better than RACG with smaller maximum

$LTT$.

The mean

$LTT$ is the average value of the longest travel time to the middle schools within their service areas, which is used as a general indicator of the accessibility of all middle schools in their service areas based on the delimitation results.

Figure 9 illustrates the mean

$LTT$ of the service areas of all middle schools based on the delimitation results obtained with different

ω settings. The figure shows that the mean

$LTT$ for the HACG stays around 8.4 min while it is around 9.4 min for the RACG, but the mean

$LTT$ slightly increased with the increase of the value of

ω for both methods. Like the maximum

$LTT$, the mean

$LTT$ obtained with the RACG is always higher than that obtained with the HACG.

The standard deviation of

LTT measures the amount of variation of the longest travel time to the middle schools within their delineated service areas.

Figure 10 illustrates the standard deviation of the

$LTT$ for all middle schools based on the delineated service areas with different

ω settings. Similar to the maximum

LTT, for the HACG, the standard deviation of

LTT stays at high values of around 4.9 min when

$0.035\le \omega \le 0.05$. For all the other

ω settings, it stays at low values of around 4.2 min. On the contrary, the standard deviation of the

LTT obtained with the RACG is always larger than 5 min. Again, the standard deviation of

LTT obtained using the RACG is always higher than that obtained with the HACG.

According to the calibration results of the

ω settings, the best delimitation results were selected based on how commensurate the population in each service area is with the enrollment capacity of the middle school in the service area and how accessible the middle schools are within their service areas. Thus, the observed best value of

ω for the HACG is 0.02, while it is 0.015 for the RACG.

Figure 11 illustrates the service area delineation results of the adaptive crystal growth Voronoi diagrams using the raster-based and hexagon-based weighted planes with their respective best observed

$\omega $ values. It can be seen from the figure that the results are generally different.

Further, the RMSE and Max. Error of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas, as well as the maximum, mean, and standard deviation of the

LTT to a middle school from any location in its service, are compared in

Table 2 with the observed best results of both methods. As the table shows, the RMSE, Max. Error, Max.

LTT, mean

LTT and S.D.

LTT are all smaller when using the hexagon-based method compared with their values obtained when using the raster-based method, which also indicates that the HACG performs better than the RACG considering both the commensurate the population in each service area is with the enrollment capacity of the corresponding middle school and the accessibility of each middle school within its service area.

Considering calculation efficiency,

Figure 12 illustrates the number of cells for the zoning with different spatial resolutions and their computation time for both raster-based and hexagon-based adaptive crystal growth Voronoi diagrams. It can be seen from the figure that the number of cells increases (from 0.7 to 2.8 million) with finer resolution (from 200 to 50 m

^{2}), and the computation time for both methods rises. The computation time for RACG increases from 0.09 to 0.3 min while that for HACG increases from 0.35 to 2.34 min. With the longest computation time of about 2 min with a personal computer, the calculation efficiency for HACG is acceptable and feasible to be utilized in service area delineation problems.