Efficient Computation of Reeb Spaces and First Homology Groups

Sarah Percival, Purdue University

Abstract

This thesis studies problems in computational topology through the lens of semi-algebraic geometry. We first give an algorithm for computing a semi-algebraic basis for the first homology group, H1(S, F), with coefficients in a field F, of any given semi-algebraic set S ⊂ Rk defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. More precisely, if the given quantifier-free formula involves s polynomials whose degrees are bounded by d, the complexity of the algorithm is bounded by (sd)k O(1). This algorithm generalizes well known algorithms having singly exponential complexity for computing a semi-algebraic basis of the zero-th homology group of semi-algebraic sets, which is equivalent to the problem of computing a set of points meeting every semi-algebraically connected component of the given semi-algebraic set at a unique point. We then turn our attention to the Reeb graph, a tool from Morse theory which has recently found use in applied topology due to its ability to track the changes in connectivity of level sets of a function. The roadmap of a set, a construction that arises in semi-algebraic geometry, is a one-dimensional set that encodes information about the connected components of a set. In this thesis, we show that the Reeb graph and, more generally, the Reeb space, of a semi-algebraic set is homeomorphic to a semi-algebraic set, which opens up the algorithmic problem of computing a semi-algebraic description of the Reeb graph. We present an algorithm with singly-exponential complexity that realizes the Reeb graph of a function f : X → Y as a semi-algebraic quotient using the roadmap of X with respect to f.

Degree

Ph.D.

Advisors

Basu, Purdue University.

Subject Area

Mathematics

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