We explore the effectiveness of visualizing dense directed graphs by replacing individual edges with edges connected to 'modules'-or groups of nodes-such that the new edges imply aggregate connectivity. We only consider techniques that offer a lossless compression: that is, where the entire graph can still be read from the compressed version. The techniques considered are: a simple grouping of nodes with identical neighbor sets; Modular Decomposition which permits internal structure in modules and allows them to be nested; and Power Graph Analysis which further allows edges to cross module boundaries. These techniques all have the same goal--to compress the set of edges that need to be rendered to fully convey connectivity--but each successive relaxation of the module definition permits fewer edges to be drawn in the rendered graph. Each successive technique also, we hypothesize, requires a higher degree of mental effort to interpret. We test this hypothetical trade-off with two studies involving human participants. For Power Graph Analysis we propose a novel optimal technique based on constraint programming. This enables us to explore the parameter space for the technique more precisely than could be achieved with a heuristic. Although applicable to many domains, we are motivated by--and discuss in particular--the application to software dependency analysis.
a) Flat graph with 30 nodes and 65 links. (b) Power Graph rendering computed using the heuristic of Royer et al. [12]: 7 modules, 36 links. (c) Power Graph computed using Beam Search (see 5.1) finds 15 modules to reduce the link count to 25.Figure 1: Three renderings of a network of dependencies between methods, properties and fields in a software system. In the Power Graph renderings an edge between a node and a module implies the node is connected to every member of the module. An edge between two modules implies a bipartite clique. In this way the Power Graph shows the precise connectivity of the directed graph but with much less clutter.
Lazy Clause Generation is a powerful approach for reducing search in Constraint Programming. This is achieved by recording sets of domain restrictions that previously led to failure as new clausal propagators. Symmetry breaking approaches are also powerful methods for reducing search by avoiding the exploration of symmetric parts of the search space. In this paper, we show how we can successfully combine Symmetry Breaking During Search and Lazy Clause Generation to create a new symmetry breaking method which we call SBDS-1UIP. We show that the more precise nogoods generated by a lazy clause solver allow our combined approach to exploit symmetries that cannot be exploited by any previous symmetry breaking method. We also show that SBDS-1UIP can easily be modified to exploit almost symmetries very effectively.
Treatment plans for ten patients, initially treated with a conventional approach to low dose-rate brachytherapy (LDR, 145 Gy to entire prostate), were compared with plans for the same patients created with an inverse-optimisation planning process utilising a biologically-based objective. The 'biological optimisation' considered a non-uniform distribution of tumour cell density through the prostate based on known and expected locations of the tumour. Using dose planning-objectives derived from our previous biological-model validation study, the volume of the urethra receiving 125% of the conventional prescription (145 Gy) was reduced from a median value of 64% to less than 8% whilst maintaining high values of TCP. On average, the number of planned seeds was reduced from 85 to less than 75. The robustness of plans to random seed displacements needs to be carefully considered when using contemporary seed placement techniques. We conclude that an inverse planning approach to LDR treatments, based on a biological objective, has the potential to maintain high rates of tumour control whilst minimising dose to healthy tissue. In future, the radiobiological model will be informed using multi-parametric MRI to provide a personalised medicine approach.
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