  We introduce a theoretical and computational framework to use discrete Morse
theory as an efficient preprocessing in order to compute zigzag persistent
homology.
  From a zigzag filtration of complexes $(K_i)$, we introduce a zigzag Morse
filtration whose complexes $(A_i)$ are Morse reductions of the original
complexes $(K_i)$, and we prove that they both have same persistent homology.
This zigzag Morse filtration generalizes the filtered Morse complex of
Mischaikow and Nanda, defined for standard persistence.
  The maps in the zigzag Morse filtration are forward and backward inclusions,
as is standard in zigzag persistence, as well as a new type of map inducing non
trivial changes in the boundary operator of the Morse complex. We study in
details this last map, and design algorithms to compute the update both at the
complex level and at the homology matrix level when computing zigzag
persistence. We deduce an algorithm to compute the zigzag persistence of a
filtration that depends mostly on the number of critical cells of the
complexes, and show experimentally that it performs better in practice.
