Zigzag persistent homology in matrix multiplication time

We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our algorithm runs in O(M(n) + n2 log2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n2.376 ), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n3 ) time in the worst case. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems; G.2.1 [Discrete Mathematics]: Combinatorics General Terms Algorithms, Theory Keywords Zigzag persistent homology, matrix multiplication.

• Type: Journal
• Year: 2011
• Where: COMPGEOM
• Authors: Nikola Milosavljevic, Dmitriy Morozov, Primoz Skraba
• Keywords: Analysis Of Algorithms, COMPGEOM 2011, Discrete Geometry, Matrix Multiplication, Problem Complexity