Hierarchical Topological Inference on Planar Disc Maps

Abstract

Given a set V and three relations ⋈ _d , ⋈ _m and ⋈ _i , we wish to ask whether it is possible to draw the elements v ∈ V each as a closed disc homeomorph $\mathcal{D}_u$ in the plane in such a way that (1) $\mathcal{D}_u$ and $\mathcal{D}_w$ are disjoint for every (v,w) ∈⋈ _d , (2) $\mathcal{D}_u$ and $\mathcal{D}_w$ have disjoint interiors but share a point of their boundaries for every (v,w) ∈⋈ _m , and (3) $\mathcal{D}_u$ includes $\mathcal{D}_w$ as a sub-region for every (v,w) ∈⋈ _i . This problem arises from the study in geographic information systems. The problem is in NP but not known to be NP-hard or polynomial-time solvable. This paper shows that a nontrivial special case of the problem can be solved in almost linear time.