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Dynamically Maintaining the Persistent Homology of Time Series

Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, Monika Henzinger, Lara Ost

TL;DR

This work presents a dynamic data structure for maintaining the augmented persistent diagram of one-dimensional time-series data under a broad set of updates. The core idea is the banana tree, a path-decomposed binary tree that stores windows and their nested relationships in complementary structures for $f$ and $-f$, enabling $O(\log n + k)$ update times. The paper provides a linear-time construction, a suite of local maintenance operations (interschanges, cancellations, anti-cancellations), and topological operations (splits and glues) with correctness proofs and complexity guarantees. Collectively, these contributions enable efficient, dynamic tracking of persistent features in time-series data and open avenues for extending banana-tree ideas to broader topological-data-analysis problems.

Abstract

We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time $O(\log n + k)$, in which $n$ counts the critical items and $k$ the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.

Dynamically Maintaining the Persistent Homology of Time Series

TL;DR

This work presents a dynamic data structure for maintaining the augmented persistent diagram of one-dimensional time-series data under a broad set of updates. The core idea is the banana tree, a path-decomposed binary tree that stores windows and their nested relationships in complementary structures for and , enabling update times. The paper provides a linear-time construction, a suite of local maintenance operations (interschanges, cancellations, anti-cancellations), and topological operations (splits and glues) with correctness proofs and complexity guarantees. Collectively, these contributions enable efficient, dynamic tracking of persistent features in time-series data and open avenues for extending banana-tree ideas to broader topological-data-analysis problems.

Abstract

We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time , in which counts the critical items and the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.
Paper Structure (27 sections, 31 theorems, 3 equations, 11 figures)

This paper contains 27 sections, 31 theorems, 3 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Lists Viewed as Maps.
  4. Extended Persistent Homology.
  5. Windows and Augmented Persistence Diagram.
  6. Technical Overview
  7. Data Structures
  8. Dictionaries.
  9. Banana Trees.
  10. String and Spine.
  11. Algorithms
  12. Construction.
  13. Local Maintenance.
  14. Topological Maintenance.
  15. Discussion
  16. ...and 12 more sections

Key Result

Proposition 2.1

Let $f \colon [1,m] \to {\mathbb R}{\hbox{${\mathbb R}$}}$ be a generic piecewise linear map on a closed interval, and let $a, b$ be homological critical points of $f$, or of $-f$, with $f(a) = A$ and $f(b) = B$. Then

Figures (11)

  • Figure 1: Left: a real-valued map on a closed interval, $f$, with three minima and two maxima. Right: the map $-f$ drawn upside-down. Middle: the augmented persistence diagram with two (blue) points in the ordinary subdiagram $\sf Ord{\hbox{$\sf Ord$}}_{}{({f})}$ above the diagonal, three (pink) points in the relative subdiagram $\sf Rel{\hbox{$\sf Rel$}}_{}{({f})}$ below the diagonal, and one (green) point in the essential subdiagram $\sf Ess{\hbox{$\sf Ess$}}_{}{({f})}$.
  • Figure 2: The graph of a generic map on a closed interval. All windows shown are with simple wave, except for the leftmost window, whose wave is short. The global window as well as the (tiny) windows caused by the hooks (which will be introduced in \ref{['sec:3']}) are not shown. The light-blue shaded out-panels are part of the triple- but not of the double-panel windows.
  • Figure 3: The path-decomposed binary tree associated to the map in Figure \ref{['fig:maponly']} with special root, $\beta$, on the left, and the corresponding banana tree on the right.
  • Figure 4: Left: the path-decomposition of the binary tree for the map, $f$, displayed in Figure \ref{['fig:maponly']}. The nodes in its spine (to be defined shortly) are ${\tt {c}}{\hbox{${\tt {c}}$}}$, ${\tt {e}}{\hbox{${\tt {e}}$}}$, $\beta$, and ${\tt {o}}{\hbox{${\tt {o}}$}}$. Right: upside-down drawing of the path-decomposition of the binary tree for $-f$. The nodes in its spine are ${\tt {d}}{\hbox{${\tt {d}}$}}$, ${\tt {j}}{\hbox{${\tt {j}}$}}$, and $\beta$.
  • Figure 5: Left: the banana tree of the map in Figure \ref{['fig:maponly']}, with dotted curves showing the tree before splitting its paths; compare with the left drawing in Figure \ref{['fig:updowntrees']}. Right: upside-down drawing of the banana tree for the negated map; compare with the right drawing in Figure \ref{['fig:updowntrees']}.
  • ...and 6 more figures

Theorems & Definitions (34)

  • Proposition 2.1: Persistence in Terms of Windows BCES21
  • Lemma 4.1: Bananas and Windows
  • Lemma 4.2: Uniqueness of Banana Tree
  • Lemma 4.3: Spines and Windows
  • Theorem 5.1: Time to Construct
  • Lemma 5.1: Coupling of Interchanges
  • Theorem 5.2: Time to Adjust
  • Lemma 5.2: Sorted Stacks
  • Theorem 5.3: Time to Cut and Concatenate
  • Definition A .1: Contiguity
  • ...and 24 more