Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Minimum cuts and shortest homologous cycles
Proceedings of the twenty-fifth annual symposium on Computational geometry
Approximating loops in a shortest homology basis from point data
Proceedings of the twenty-sixth annual symposium on Computational geometry
Hardness Results for Homology Localization
Discrete & Computational Geometry
Improved algorithms for min cut and max flow in undirected planar graphs
Proceedings of the forty-third annual ACM symposium on Theory of computing
Optimal Homologous Cycles, Total Unimodularity, and Linear Programming
SIAM Journal on Computing
Minimum cuts and shortest non-separating cycles via homology covers
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Finding Cycles with Topological Properties in Embedded Graphs
SIAM Journal on Discrete Mathematics
An efficient computation of handle and tunnel loops via Reeb graphs
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
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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 ωn×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.