Annotating Simplices with a Homology Basis and Its Applications

  • Oleksiy Busaryev
  • Sergio Cabello
  • Chao Chen
  • Tamal K. Dey
  • Yusu Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

Abstract

Let \({\cal K}\) be a simplicial complex and g the rank of its p-th homology group \({\sf H}_{p}({\cal K})\) defined with ℤ2 coefficients. We show that we can compute a basis H of \({\sf H}_{p}({\cal K})\) and annotate each p-simplex of \({\cal K}\) with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(nω) time, where n is the size of \({\cal K}\) and ω < 2.376 is a quantity so that two n×n matrices can be multiplied in O(nω) time. The precomputed annotations permit answering queries about the independence or the triviality of p-cycles efficiently.

Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing an optimal basis of \({\sf H}_{1}({\cal K})\), we improve the previously known time complexity from O(n4) to O(nω + n2gω − 1). Here n denotes the size of the 2-skeleton of \({\cal K}\) and g the rank of \({\sf H}_{1}({\cal K})\). Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2O(g)nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(nω) + 2O(g)n2logn time using annotations.

Keywords

Simplicial complex topology homology basis optimal cycles matrix multiplication 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Oleksiy Busaryev
    • 1
  • Sergio Cabello
    • 2
  • Chao Chen
    • 3
  • Tamal K. Dey
    • 1
  • Yusu Wang
    • 1
  1. 1.Department of Computer Science and EngineeringThe Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsUniversity of LjubljanaSlovenia
  3. 3.Institute of Science and Technology AustriaKlosterneuburgAustria

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