 
Summary: The Cost of Compatible Refinement of Simplex
Decomposition Trees
F. Betul Atalay1
and David M. Mount2
1
Mathematics and Computer Science Department, Saint Joseph's University,
Philadelphia, PA. fatalay@sju.edu
2
Department of Computer Science and Institute for Advanced Computer Studies,
University of Maryland, College Park, MD. mount@cs.umd.edu
Summary. A hierarchical simplicial mesh is a recursive decomposition of space into
cells that are simplices. Such a mesh is compatible if pairs of neighboring cells meet
along a single common face. Compatibility condition is important in many applica
tions where the mesh serves as a discretization of a function. Enforcing compatibility
involves refining the simplices of the mesh further, thus generates a larger mesh. We
show that the size of a simplicial mesh grows by no more than a constant factor
when compatibly refined. We prove a tight upper bound on the expansion factor for
2dimensional meshes, and we sketch upper bounds for ddimensional meshes.
1 Introduction
Hierarchical data structures based on repeated subdivision of space have been
