Towards Faster Methods for Ensuring Summoning Circle Safety

Abstract: Summoning circles are common, useful tools for summoning beings and for teleportation. However, the dangers of unwanted interactions between summoning circle components have led magicians to stick to well-studied templates. While the fundamental interactions between glyphs are known, magicians continue to exert considerable effort to improve common templates, with one goal being to reduce the energy cost of a summoning circle by reducing the number of glyph types required. This paper uses mathematical analysis to present three contributions to this goal. First, we create a mapping between the glyphs of summoning circles and nodes in a corresponding graph, reducing summoning circle glyph filling from graph coloring and demonstrating that minimizing the number of glyph types in a given summoning circle pattern is NP-hard. Second, we show that, in a limited but common set of desired outcomes, summoning circle patterns can be rearranged to prevent edge crossings in the corresponding graph, thereby bounding the minimum glyph type count through the Four Color Theorem. Finally, we introduce the novel notion of piecewise sequentialization for modelling summoning circle interaction as sequences of magical reactions occuring in parallel, which enables the adoption of linear-scan techniques for register allocation to quickly compute guaranteed-safe assignments of glyph types to a summoning circle pattern, easing experimentation with new patterns.