Channel-Based Hierarchical Graph Drawing Framework
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Solution Overview
Problem
The Sugiyama framework for visualizing directed graphs is inefficient due to NP-hard problems in its phases, leading to unbounded computation time and increased complexity, with most heuristics providing unbounded results and influencing subsequent phases in a way that cannot be easily corrected.
Innovation Solution
A new framework that partitions graph vertices vertically into channels or paths, reducing visual complexity by only drawing necessary edges, with a two-step process involving cycle removal and channel decomposition, and uses path- or channel-based methods to draw hierarchical graphs without introducing dummy vertices, ensuring vertices are aligned and cross edges have at most one bend.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If the Sugiyama framework is used for visualizing directed graphs, then hierarchical drawings can be produced, but computation time becomes unbounded due to NP-hard problems
Solution Approach 1:
The patent segments the graph drawing problem into independent channel-based subproblems. By partitioning vertices into channels and handling each channel separately with polynomial-time algorithms, the overall computation remains polynomial while maintaining drawing quality.
Solution Approach 2:
Instead of following the traditional Sugiyama approach of layer-by-layer processing that leads to NP-hard problems, the patent inverts the approach by using channel-based processing from bottom to top, which enables polynomial-time solutions while achieving comparable or better drawing results.
2Manufacturing precision
If the Sugiyama framework is used, then hierarchical drawings can be produced, but complexity increases due to multiple phases and heuristics
Solution Approach 1:
The patent simplifies the complex multi-phase Sugiyama framework by segmenting the problem into independent channel-based processing steps. This reduces algorithmic complexity from NP-hard to polynomial time while maintaining drawing quality through focused, simpler operations on each channel.
Solution Approach 2:
The patent extracts and removes the problematic NP-hard elements from the traditional framework by replacing them with polynomial-time channel-based algorithms, thereby reducing overall algorithmic complexity while preserving essential drawing functionality.
3Loss of information
If traditional graph drawing methods are used, then complete graph information can be displayed, but visual complexity increases with many crossings
Solution Approach 1:
The patent segments the graph into channels and selectively draws only necessary edges within and between channels. This segmentation approach maintains complete reachability information while dramatically reducing visual complexity by eliminating redundant and unnecessary edge crossings.
Solution Approach 2:
The patent extracts and removes unnecessary edges from the drawing while preserving complete reachability information through the channel structure. This extraction process reduces visual complexity by eliminating redundant crossings that do not contribute to essential graph information.
Data Source
AI summary
Methods for visualizing directed graphs and their hierarchies are provided that completely depart from known approaches. The methods compute readable hierarchical visualizations that contain the complete reachability information of a graph. In one embodiment, only the necessary edges are drawn in the drawing, thus reducing the visual complexity of the resulting drawing. The methods require only polynomial time. In one aspect, the vertices of the graph are (vertically) partitioned into paths/channels, having cross edges that connect various vertices in different paths/channels. A corresponding graph drawing method is implemented in principally two steps: (a) a cycle removal step (if the graph contains cycles) and (b) a channel decomposition and hierarchical drawing step. A corresponding framework may offer a suite of solutions depending on requirements, does not introduce any dummy vertices, and keeps the vertices of a path/channel vertically aligned.