Share this page Herbert Edelsbrunner; John L. Harer Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering.
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A theme that goes through this entire book is the exchange between discrete and continuous models of reality. Simple graphs. We draw the vertices as points or little circles and edges as line segments or curves connecting the points. For now, crossings between the curves are allowed.
Every simple graph with nvertices is a subgraph of the complete graph,Kn, that contains an edge for every pair of vertices; see Figure I. Figure I.
The length of this path is its number of edges, k. Vertices can repeat, allowing the path to cross itself or backtrack. Definition A. A simple graph is connected if there is a path between every pair of vertices. A connected component is a maximal subgraph that is connected. The small- est connected graphs are the trees, which are characterized by having a unique simple path between every pair of vertices.
Removing any one edge disconnects the tree. It has the same vertex set as the graph and uses a minimal set of edges necessary to be connected. This requires that the graph is connected to begin with. An alternative characterization of a connected graph can be based on the impossibility to cut it in two. Definition B. A simple graph is connected if it has no separation. Topological spaces. We now switch to a continuous model of reality, the topological space.
There are similarities to graphs, in particular if our interest is limited to questions of connectedness. Think of it as an abstraction of Euclidean space in which neighborhoods are open balls around points. The pair X,U is a called a topological space, but we will usually tacitly assume that Uis understood and refer to Xa topological space.
Computational Topology: an introduction
Major algorithms by subject area[ edit ] Algorithmic 3-manifold theory[ edit ] A large family of algorithms concerning 3-manifolds revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3-manifold theory into integer linear programming problems. This is an algorithm that takes as input a triangulated 3-manifold and determines whether or not the manifold is homeomorphic to the 3-sphere. It has exponential run-time in the number of tetrahedral simplexes in the initial 3-manifold, and also an exponential memory profile. Moreover, it is implemented in the software package Regina.
Additional Material for the Book