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A Fast Algorithm for Computing Zigzag Representatives

Tamal K. Dey, Tao Hou, Dmitriy Morozov

TL;DR

An O(m^2n) time algorithm is presented, where $n\leq m$ is the size of the largest complex in the filtration, where $n\leq m$ is the size of the largest complex in the filtration.

Abstract

Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features. Although one can locate a particular feature at any index in the filtration using existing algorithms, the resulting representatives may not be compatible with the zigzag: a representative cycle at one index may not map into a representative cycle at its neighbor. For this, one needs to compute compatible representative cycles along each bar in the barcode. Even though it is known that the barcode for a zigzag filtration with $m$ insertions and deletions can be computed in $O(m^ω)$ time, it is not known how to compute the compatible representatives so efficiently. For a non-zigzag filtration, the classical matrix-based algorithm provides representatives in $O(m^3)$ time, which can be improved to $O(m^ω)$. However, no known algorithm for zigzag filtrations computes the representatives with the $O(m^3)$ time bound. We present an $O(m^2n)$ time algorithm for this problem, where $n\leq m$ is the size of the largest complex in the filtration.

A Fast Algorithm for Computing Zigzag Representatives

TL;DR

An O(m^2n) time algorithm is presented, where is the size of the largest complex in the filtration, where is the size of the largest complex in the filtration.

Abstract

Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features. Although one can locate a particular feature at any index in the filtration using existing algorithms, the resulting representatives may not be compatible with the zigzag: a representative cycle at one index may not map into a representative cycle at its neighbor. For this, one needs to compute compatible representative cycles along each bar in the barcode. Even though it is known that the barcode for a zigzag filtration with insertions and deletions can be computed in time, it is not known how to compute the compatible representatives so efficiently. For a non-zigzag filtration, the classical matrix-based algorithm provides representatives in time, which can be improved to . However, no known algorithm for zigzag filtrations computes the representatives with the time bound. We present an time algorithm for this problem, where is the size of the largest complex in the filtration.

Paper Structure

This paper contains 17 sections, 7 theorems, 7 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Core definitions
  3. Representatives as wire bundles
  4. Algorithm
  5. Maintenance of pivoted matrices
  6. Detailed processing in iteration $i$ of WiredZigzag
  7. Experiments
  8. Proof of Proposition \ref{['prop:cyclebasis']}
  9. Proof of Proposition \ref{['prop:repsum']}
  10. Missing cases in the proof of Theorem \ref{['thm:bundle']}
  11. Implementation details
  12. Forward $K_i\xhookrightarrow{\;\sigma_{i}\;}K_{i+1}$
  13. $\psi^*_i$ is injective
  14. $\psi^*_i$ is surjective
  15. Backward $K_i\xhookleftarrow{\;\sigma_{i}\;}K_{i+1}$
  16. ...and 2 more sections

Key Result

Proposition 8

Let $z^B_1,\ldots,z^B_k$ be the cycles at index $j$ in representatives for all intervals of $\mathsf{Pers}^B(\mathcal{F}_i)$ containing $j$. Similarly, let $z_1^H,\ldots,z_{k'}^H$ be the cycles at index $j$ in representatives for all intervals of $\mathsf{Pers}^H(\mathcal{F}_i)$ containing $j$. Then

Figures (3)

  • Figure 2: Illustration of how summations of representatives for intervals respect the order '$\prec$' for the different cases in Definition \ref{['dfn:ind-order']}, with the double arrows indicating the directions of the summations. Boundary module intervals are shaded blue while homology module intervals are shaded orange.
  • Figure 3: Summing the two representatives generated by a single wire results in a new representative generated by a bundle containing the two wires.
  • Figure 4: Summing two representatives generated by the bundles $\{\omega_{2},\omega_{5},\omega_{7}\},\{\omega_{3},\omega_{7}\}$ respectively results in a new representative generated by the bundle $\{\omega_{2},\omega_{3},\omega_{5}\}$.

Theorems & Definitions (32)

  • Definition 1: Homology birth/death indices
  • Remark 2
  • Definition 3: Boundary birth/death indices
  • Definition 4: Homology representatives
  • Remark 5
  • Definition 6: Boundary representatives
  • Remark 7
  • Proposition 8
  • Definition 9: Total order on birth indices
  • Definition 10: Representative summation
  • ...and 22 more