Upper bounds on the number of hidden nodes in sugiyama’s algorithm

Abstract

This paper analyzes the exact and asymptotic worst-case complexity of the simplification phase of Sugiyama’s algorithm [12] for drawing arbitrary directed graphs. The complexity of this phase is determined by the number of hidden nodes inserted. The best previously known upper bound for this number is (Formula presented). This paper establishes a relation between both partial results and gives upper bounds for many classes of graphs. This is achieved by constructing a worst-case example for every legal configuration C= (h, n, m) of the input hierarchy for the simplification phase. These results provide further insight into the worst-case runtime and space complexity of Sugiyama’s algorithm. Possible applications include their use as feasibility criteria, based on simply derived quantitative information on the graph.