Figure 61 shows the Sydney schematic as laid out using our method. This can be compared to Figure 62, produced by the previous multi-criteria hill climb-ing method by Stott et al. Our new result illustrates how our additional mod-ifications have increased the line straightness and octilinearity in many areas, for example the central green line has now been kept horizontal and periphery sections contain fewer bends thanks to the new clustering technique. The city centre area to the right side of the schematic, typically very dense, has also been expanded slightly due to the new balance criterion. In addition to these points, the effect of bend points can be seen as lines now change direction more often along an edge rather than at junctions.
Along with significant layout improvements, an improvement can be seen in the positioning of labels. Our implementation that allows rotation of labels has allowed for fewer occlusions and an improved level of label position con-sistency overall.
An interesting aspect of our layout is that it appears to be more compacted vertically. This is possibly attributable to the fixed screen size with which we
Figure 62: Sydney metro map as optimised by the previous multi-criteria hill climber by Stott at al.
worked not letting the schematic expand to its desired size, and has caused the horizontal section with many parallel lines (to the right) to be positioned in very close proximity to nearby separate lines. Along with this drawback, there are also a couple of strangely sharp bends (centre – yellow, purple), caused by bend points attempting to meet the octilinear criterion in a confined area.
4.8 Summary
This chapter has detailed our implementation of a multi-criteria hill climbing technique for the automated layout of metro map style schematics.
We have explained the modifications made to the original optimiser in (Stott et al. 2010), including 1) New schematic objects – bend points, in order to move line bends away from junctions/stations. 2) A change from operating on sta-tions to bend points to reduce the number of objects for which to evaluate po-sitions. 3) Reduced the number of potential positions for objects to move to without hindering layout in order to improve optimisation speed. 4) A new clustering method in order to ensure periphery line sections are straightened.
5) A number of new criteria: line straightness through junctions, line straight-ness at peripheries, enforced parallel lines and schematic balance. 6) Recalcu-lated criteria weightings. 7) An improved labelling technique allowing much greater flexibility by rotation of labels. We believe the modifications we have made allow the method to produce layouts of a higher quality as explained in the previous section.
Although we employ techniques such as node clustering in order to allow our method to escape common occurrences of suboptimal layout; as pointed out in our examples, our layout technique still suffers from a number of occur-rences of this. Sections of suboptimal layout are a common feature of search based optimisations, caused by the necessity to restrict the search space due to performance limitations.
We noticed that modifying the optimiser parameters had a large effect on the resulting output, and often resulted in different sections of the schematic not achieving an optimal configuration. We were therefore interested in explor-ing if a methodological approach to modifyexplor-ing parameters would produce any pattern of suboptimal layout sections in the output for specific schematics. If this was the case, we hoped that we could then identify optimal parameter set-tings based upon characteristics of the input schematic. The following chapter covers our work exploring how optimiser parameter manipulation affects situ-ations of suboptimal layout in the final output.
Chapter 5
Exploring the Effects of Parameter Manipulation
In search-based graph drawing methods there exist a number of parameters which control the operation of the search algorithm. These parameters do not affect the fitness function, but nevertheless have an impact on the final layout.
This chapter covers our work exploring how varying three such parameters (grid spacing, search distance and cooling schedule) affects the fitness value of the resultant diagram in our multi-criteria hill climbing optimiser.
By doing this, we hoped to identify patterns in the resulting layouts which were attributable to specific changes in parameter values, and to correlate these with characteristics of the input schematic. If such a relationship exists it would then be possible to automatically set optimiser parameters based on the input schematic, helping to produce a more optimal result than would otherwise be achieved using default values, without broadening the search space.
5.1 Testing Procedure
In order to perform the required testing, we implemented a testing rig into SchemaSketch which is capable of batch-optimising a large number of schemat-ics whilst varying the parameter settings for each. Table 2 lists the schematschemat-ics
Table 2: Schematics used
Schematic Junctions Stations Edges
Washington 9 77 53
Vienna 10 80 63
Mexico City 24 123 120
Sydney 24 151 103
Table 3: Parameters and values used
Parameters Values
Grid Spacing 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 Search Distance 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 Cooling Schedule None, Linear, Exponential
on which the testing was performed, along with the number of junctions, sta-tions and edges in each. These schematics were chosen as being representative of reasonably sized schematics demonstrating different characteristics and for which we could easily access the data. The table is listed by ascending order of the total number of junctions and stations.
We examined the following three key parameters of the method. Table 3 shows the values we chose to test for each.
1. Grid Spacing: A grid is placed over the canvas, and each node must be positioned onto a grid intersection. This parameter defines the grid reso-lution in pixels. When altered, this parameter affects the starting layout as the nodes are moved slightly when snapped to the grid. It will also alter the number of potential sites that nodes can be positioned in when they move, and the absolute distance by which they can move.
2. Search Distance: This parameter defines the initial (and maximum) dis-tance that nodes can be moved, in terms of grid positions, each iteration.
Increasing this parameter allows greater movement flexibility for nodes during layout. The distance by which nodes can be moved decreases over time as defined by the cooling schedule.
3. Cooling Schedule: The cooling schedule affects the speed and pattern of
the decrease in the node movement distance and can be set to one of the following three styles (listed in ascending order of speed of reduction):
None: The distance is not reduced and stays at the initial start distance value.
Linear: The distance is reduced by one grid space each iteration.
Exponential: The distance is reduced exponentially each iteration and is calculated using Equation 21.
Distance= StartDistance
Iteration (21)
The cooling schedule parameter that we are testing is a modification to our layout method which was necessary to obtain more valuable test data. Previ-ously, we used a predefined number of iterations for the layout to complete in, and the distance by which nodes could move was linearly decreased over this number of durations down to one, after which the optimisation was stopped.
However, when testing the effect of parameters upon the fitness of the final lay-out, we did not want to use a fixed number of iterations. Instead, we wanted to let the optimisation run for as many iterations as required to achieve the best layout (down to a fitness function accuracy of 3dp), as we felt this was necessary to accurately see the effect of parameter modifications upon the final fitness. To replace the reduction in search distance based upon a fixed number of itera-tions, we implemented a cooling schedule to reduce the search distance over time. Our cooling schedule supports 3 different schemes, as explained previ-ously.
Using the schematics and parameters listed in Tables 2 and 3 respectively, our testing rig optimises each schematic with every possible combination of parameter values (363 variations of each schematic). It then outputs images of each resulting layout and a file containing the fitness value and number of iterations required.