Other Heuristic Approaches

Heuristic solutions for the SPED problem are an interesting topic. Thus – after the analysis of the approximation quality of Simple–Podevsnef – one may be tempted to look for other – perhaps better – algorithms.

The following is an incomplete description of concepts and experiments regarding other networks: it shows some easy approaches and eventually points out why these simple ideas do not work. After all, the classic Simple–Podevsnef seems to be quite a good heuristics for non-planar graphs.

The problem with SPED is the property of forcing a right bend “somewhere” on the hyperedge. Simple–Podevsnef circumvents this complexity by putting the right bend – if one is needed due to a 0° opening angle – on the first subedge of the hyperedge. But this also removes the possibilities of dummy merging.

6.3.1 Bend–on–End

A quite obvious idea would be to force this right bend on the last such subedge.

Such a network – named Bend–on–End – can easily be implemented by setting the capacities of all arcs that generate bends on the hyperedge and its bundle partner to zero, except for the arcs on the end face (Fig. 6.5).

Figure 6.5: The network of a hyperface for the Bend–on–End heuristics

Thus no downflow can exist anymore; upflow can only be emitted from the source node of the hyperface, and has to be transmitted completely through the entire hyperface.

Hence it is possible to use a construction similar to the one described for the alternative Simple–Podevsnef network (Sec. 4.3). This would lead to drawings where a hyperedge has to be either totally collapsed until its end face, or has an opening angle of 90°.

This is not just a bad heuristics, but even an invalid one: for graphs with a high density of hyperfaces – e.g., nearly complete graphs – this restriction is much too strong and no feasible solution can be found at all. The problem becomes obvious when thinking about orthogonal hyperfaces. As Figure 6.6 points out, there may exist vertices surrounded by

“too many” coltris. Due to the symmetrical distribution property of Simple–Podevsnef–

like drawing standards, some hyperfaces h_i are forced to have an opening angle of 90°.

Figure 6.6: Bend–on–End may not be able to generate valid solutions at all

Figure 6.7: The network of a hyperface for the Lazy Bend–on–End heuristics

Thus there is an upflow in every such h_i. If too many of these h_i have orthogonal hyperfaces that prohibit the upflows, no valid solution can be found.

6.3.2 Lazy Bend–on–End

To circumvent this problem and allow some more flexibility, we enhance the above heuris-tics to Lazy Bend–on–End by allowing bendable bundles: In addition to the properties described for the above heuristics, a hyperface can bend itself completely, without de-merging (cf. Remark 5.15).

This can be achieved by connecting the bend producing arcs of the subedges (except the last one) of each hyperedge directly to their counterparts on the bundle partner (Fig. 6.7).

This allows flow to travel orthogonally through a hyperface, but it still prohibits all downflows and any upflow except the one from the source node.

This heuristics is now capable of drawing all possible graphs, but, as Figure 6.8¹⁰

demon-10The hyperfaces obviously could be drawn without a demerge on their first bundle bend, using an appropriately modified precompaction algorithm

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Figure 6.8: Example of a graphdrawn by Lazy Bend–on–End. (124 bends)

strates, the results are quite poor: For the given non–planar graph with 10 nodes and 45 edges – the planarization step introduced 62 dummy nodes – the optimal SPED result requires 69bends. While classic Simple–Podevsnef generates an approximation using 77 bends, Lazy Bend–on–End introduces 124.

Also note the unaesthetic bumps that occur on the end faces of the hyperfaces, due to the forced right bend on the last subedge of their hyperedges.

6.3.3 Righteous Bend–on–End

We can refine the above ideas based on the following thoughts: Simple–Podevsnef has the property that for every two edges e₁ and e₂ that start parallel (e₂ being the right one), the first bend one₂ is a right bend. All other bends – whether one₁or e₂ – happen afterwards.

Coupled with the idea of bends to occur on the last hyperface and the necessity of bendable bundles, this leads to a heuristic with the following property: All hyperfaces have their bends collocated on the end face, except for right bends on the hyperedge and bends of the complete bundle.

This description can still be transformed into a min–cost–flow network, as Figure 6.9 demonstrates.

This heuristics leads to greatly improved results compared to Lazy Bend–on–End – Fig. 6.10 shows the same graph as before – but is still inferior to Simple–Podevsnef.

It is easy to see that the unaesthetic bumps mentioned above also remain only slightly mitigated in this heuristics.

6.3.4 Simple–Podevsnef with Simple Postprocessing

Since Simple–Podevsnef seems to be a quite efficient heuristics, it appears to be a good idea to generate even better solutions based on this scheme.

Figure 6.9: The network of a hyperface for the Righteous Bend–on–End heuristics

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Figure 6.10: Example of a graphdrawn by Righteous Bend–on–End. (102 bends)

Figure 6.11: Postprocessing after Simple–Podevsnef

We can use Simple–Podevsnef as a first step in generating a valid drawing. We know that such a solution will not introduce dummy merges and bundle crossings. In a second step – the postprocessing – we search all hyperfaces for “unnecessary” bends, only introduced by Simple–Podevsnef’s short–sighted demand for right bends on the first subedge (cf.

Fig. 6.11).

We seek for a situation with the following properties:

1. The hyperface f^H (with the hyperedge e^H) has an opening angle of 90°.

2. The subedge e₁ – the first subedge of the bundle partner of e^H – has at least one right bend.

3. The angle between e₁ and its counterclockwise successor in the incidence list of source(e₁) is 0°.

4. The shape of f^H remains valid (in terms of the definition of SPED), if it would start collapsed and e1 would have one right bend less.

Ad 4) This property can be easily checked by running throughe^H and its bundle partner:

e^H has to have at least one right bend, andf^H has to demerge before any otherwise invalid bend may occur.

If we find such a situation, we can safely reduce the opening angle of the hyperface to 0°, remove one right bend on e1, and increase the angle betweene1 and its successor to 90°.

Obviously the solution of this heuristics can never be inferior to Simple–Podevsnef. It even would optimally solve the example given in Corollary 6.1.

6.3.5 Simple–Podevsnef with Extended Postprocessing

We can enhance the above idea by the introduction of flow modifying postprocessing algo-rithms similar to the repair–function. These may perform operations related to downflows and left–FRFs based on concepts described in the Sections 5.6.3 and 5.6.4.

Such algorithms can broaden the field of hyperfaces that satisfy the fourth of the above properties (concerning the valid shapes of hyperfaces).