Revisiting Graph Persistence for Updates and Efficiency

TL;DR

This paper develops efficient algorithms for updating graph persistence, including ordinary and zigzag types, with significant improvements in update times for various cases, enhancing computational efficiency in topological data analysis.

Contribution

It introduces novel algorithms for local updates in graph persistence, achieving logarithmic and sublinear update times, and improves existing algorithms for zigzag persistence.

Findings

Switches in ordinary persistence can be done in O(log m) time.

Zigzag persistence computation is improved to O(m log m) time.

Various bar types in zigzag persistence can be updated efficiently in O(log m) or O(√m log m) time.

Abstract

It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has been shown that zigzag persistence on graphs also exhibits similar behavior. Motivated by these results, we revisit graph persistence and propose efficient algorithms especially for local updates on filtrations, similar to what is done in ordinary persistence for computing the vineyard. We show that, for a filtration of length $m$, (i) switches (transpositions) in ordinary graph persistence can be done in $O(\log m)$ time; (ii) zigzag persistence on graphs can be computed in $O(m\log m)$ time, which improves a recent $O(m\log^4n)$ time algorithm assuming $n$, the size of the union of all graphs in the filtration, satisfies $n\in\Omega({m^\varepsilon})$ for any fixed $0<\varepsilon<1$; (iii) open-closed, closed-open, and closed-closed bars in dimension $0$ for graph zigzag persistence can be updated in $O(\log m)$ time, whereas the open-open bars in dimension $0$ and closed-closed bars in dimension $1$ can be done in $O(\sqrt{m}\,\log m)$ time.

Tables

Table 1. Table 1: Mapping of different types of intervals for 𝖯𝖾𝗋𝗌 ∗ ​ ( 𝒰 ) subscript 𝖯𝖾𝗋𝗌 𝒰 \mathsf{Pers}_{*}({\mathcal{U}}) and 𝖯𝖾𝗋𝗌 ∗ ​ ( ℱ ) subscript 𝖯𝖾𝗋𝗌 ℱ \mathsf{Pers}_{*}(\mathcal{F})

$𝒰$		$ℱ$
${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{co}}\left(𝒰\right)$	$\leftrightarrow$	${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{co}}\left(ℱ\right)$
${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{oc}}\left(𝒰\right)$	$\leftrightarrow$	${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{oc}}\left(ℱ\right)$
${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{cc}}\left(𝒰\right)$	$\leftrightarrow$	${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{cc}}\left(ℱ\right)$
${\mathrm{𝖯𝖾𝗋𝗌}}_1^{\text{cc}}\left(𝒰\right)$	$\leftrightarrow$	${\mathrm{𝖯𝖾𝗋𝗌}}_0^{\text{oo}}\left(ℱ\right)\cup {\mathrm{𝖯𝖾𝗋𝗌}}_1^{\text{cc}}\left(ℱ\right)$

Equations

$$\mathcal{F}:G_0\leftrightarrow G_1\leftrightarrow \cdots \leftrightarrow G_m,$$

$${\mathsf{H}}_p(\mathcal{F}):{\mathsf{H}}_p(G_0)\leftrightarrow {\mathsf{H}}_p(G_1)\leftrightarrow \cdots \leftrightarrow {\mathsf{H}}_p(G_m),$$

$$\mathcal{F}:\varnothing =G_0\stackrel{{\sigma }_0}{\hookrightarrow }{G}_1\stackrel{{\sigma }_1}{\hookrightarrow }\cdots \cdots \stackrel{{\sigma }_{m-1}}{\hookrightarrow }{G}_m,$$

$$\mathcal{F}:\varnothing =G_0\hookrightarrow \cdots \hookrightarrow G_{i-1}\stackrel{\sigma }{\hookrightarrow }{G}_i\stackrel{\tau }{\hookrightarrow }{G}_{i+1}\hookrightarrow \cdots \hookrightarrow G_m$$

$$\mathcal{F}:G_0\hookrightarrow \cdots \hookrightarrow G_{i-1}\stackrel{e_1}{\hookrightarrow }{G}_i\stackrel{e_2}{\hookrightarrow }{G}_{i+1}\hookrightarrow \cdots \hookrightarrow G_m$$

$$\mathcal{F}:\varnothing =G_0\stackrel{{\sigma }_0}{\leftrightarrow }{G}_1\stackrel{{\sigma }_1}{\leftrightarrow }\cdots \stackrel{{\sigma }_{m-1}}{\leftrightarrow }{G}_m=\varnothing$$

$$\mathcal{U}:\varnothing ={\hat{G}}_0\stackrel{{\hat{\sigma }}_0}{\hookrightarrow }{\hat{G}}_1\stackrel{{\hat{\sigma }}_1}{\hookrightarrow }\cdots \stackrel{{\hat{\sigma }}_{k-1}}{\hookrightarrow }{\hat{G}}_k\stackrel{{\hat{\sigma }}_k}{\hookleftarrow }{\hat{G}}_{k+1}\stackrel{{\hat{\sigma }}_{k+1}}{\hookleftarrow }\cdots \stackrel{{\hat{\sigma }}_{m-1}}{\hookleftarrow }{\hat{G}}_m=\varnothing .$$

$$[2,2]\in {\mathsf{Pers}}_0^{\text{co}}(\mathcal{F})\leftrightarrow [2,2]\in {\mathsf{Pers}}_0^{\text{co}}(\mathcal{U}),[6,6]\in {\mathsf{Pers}}_0^{\text{oc}}(\mathcal{F})\leftrightarrow [6,6]\in {\mathsf{Pers}}_0^{\text{oc}}(\mathcal{U}),$$

$$[4,4]\in {\mathsf{Pers}}_0^{\text{oo}}(\mathcal{F})\leftrightarrow [4,4]\in {\mathsf{Pers}}_1^{\text{cc}}(\mathcal{U}),[1,7]\in {\mathsf{Pers}}_0^{\text{cc}}(\mathcal{F})\leftrightarrow [1,7]\in {\mathsf{Pers}}_1^{\text{cc}}(\mathcal{U}).$$

$$\begin{array}{rl}& {\mathcal{U}}_u:\varnothing ={\hat{G}}_0\stackrel{{\hat{\sigma }}_0}{\hookrightarrow }{\hat{G}}_1\stackrel{{\hat{\sigma }}_1}{\hookrightarrow }\cdots \stackrel{{\hat{\sigma }}_{k-1}}{\hookrightarrow }{\hat{G}}_k,\\ & {\mathcal{U}}_d:\varnothing ={\hat{G}}_m\stackrel{{\hat{\sigma }}_{m-1}}{\hookrightarrow }{\hat{G}}_{m-1}\stackrel{{\hat{\sigma }}_{m-2}}{\hookrightarrow }\cdots \stackrel{{\hat{\sigma }}_k}{\hookrightarrow }{\hat{G}}_k.\end{array}$$

$$\mathcal{F}:G_0\leftrightarrow \cdots \leftrightarrow G_{i-1}\stackrel{\sigma }{\hookrightarrow }{G}_i\stackrel{\tau }{\hookrightarrow }{G}_{i+1}\leftrightarrow \cdots \leftrightarrow G_m$$

$$\mathcal{F}:G_0\leftrightarrow \cdots \leftrightarrow G_{i-1}\stackrel{\sigma }{\hookleftarrow }{G}_i\stackrel{\tau }{\hookleftarrow }{G}_{i+1}\leftrightarrow \cdots \leftrightarrow G_m$$

$$\mathcal{F}:G_0\leftrightarrow \cdots \leftrightarrow G_{i-1}\stackrel{\sigma }{\hookrightarrow }{G}_i\stackrel{\tau }{\hookleftarrow }{G}_{i+1}\leftrightarrow \cdots \leftrightarrow G_m$$

$$\mathcal{U}:{\hat{G}}_0\hookrightarrow \cdots \hookrightarrow {\hat{G}}_{j-1}\stackrel{\hat{\sigma }}{\hookrightarrow }{\hat{G}}_j\stackrel{\hat{\tau }}{\hookrightarrow }{\hat{G}}_{j+1}\hookrightarrow \cdots \hookrightarrow {\hat{G}}_k\hookleftarrow \cdots \hookleftarrow {\hat{G}}_m$$

$$\mathcal{L}:\varnothing =H_0\stackrel{{\varsigma }_0}{\hookrightarrow }{H}_1\stackrel{{\varsigma }_1}{\hookrightarrow }\cdots \stackrel{{\varsigma }_{\ell -1}}{\hookrightarrow }{H}_{\ell }\stackrel{{\varsigma }_{\ell }}{\hookleftarrow }{H}_{\ell +1}\stackrel{{\varsigma }_{\ell +1}}{\hookleftarrow }\cdots \stackrel{{\varsigma }_{2\ell -1}}{\hookleftarrow }{H}_{2\ell }=\varnothing ,$$

$${\hat{G}}_m\subseteq {\hat{G}}_{m-\sqrt{m}}\subseteq {\hat{G}}_{m-2\sqrt{m}}\subseteq \cdots \subseteq {\hat{G}}_{m-\ell \sqrt{m}},$$

Full text

1 Introduction

Computing persistence for graphs has been a special focus within topological data analysis (TDA) [9, 10] because graphs are abundant in applications and they admit more efficient algorithms than general simplicial complexes. It is well known that the persistence algorithm on a graph filtration with $m$ additions can be implemented with a simple Union-Find data structure in $O(m\,\alpha(m))$ time, where $\alpha(m)$ is the inverse Ackermann’s function (see e.g. [9]). On the other hand, the general-purpose persistence algorithm on a simplicial filtration comprising $m$ simplices runs in $O(m^{\omega})$ time [18], where $\omega<2.373$ is the exponent for matrix multiplication. In a similar vein, Yan et al. [21] have recently shown that extended persistence [4] for graphs can also be computed more efficiently in $O(m^{2})$ time. The zigzag version [2] of the problem also exhibits similar behavior; see e.g. the survey [1]. Even though the general-purpose zigzag persistence algorithm runs in $O(m^{\omega})$ time on a zigzag filtration with $m$ additions and deletions [3, 8, 17, 18], a recent result in [6] shows that graph zigzag persistence can be computed in $O(m\log^{4}n)$ time using some appropriate dynamic data structures [14, 16] ($n$ is the size of the union of all graphs in the filtration).

Motivated by the above developments, we embark on revisiting the graph persistence and find more efficient algorithms using appropriate dynamic data structures, especially in the dynamic settings [5, 7]. In a dynamic setting, the graph filtration changes, and we are required to update the barcode (persistence diagram) accordingly. For general simplicial complexes as input, the vineyard algorithm of [5] updates the barcode in $O(m)$ time for a switch of two consecutive simplices (originally called a transposition in [5]). So, we ask if a similar update can be done more efficiently for a graph filtration. We show that, using some appropriate dynamic data structures, indeed we can execute such updates more efficiently. Specifically, we show the following:

In a standard (non-zigzag) graph filtration comprising $m$ additions, a switch can be implemented in $O(\log m)$ time with a preprocessing time of $O(m\log m)$. See Section 3. As a subroutine of the update algorithm for switches on graph filtrations, we propose an update on the merge trees (termed as merge forest in this paper) of the filtrations, whose complexity is also $O(\log m)$. 2. 2.

The barcode of a graph zigzag filtration comprising $m$ additions and deletions can be computed in $O(m\log m)$ time. Assuming $n\in\Omega(m^{\varepsilon})$ for any fixed positive $\varepsilon<1$, where $n$ is the size of the union of all graphs in the filtration, this is an improvement over the $O(m\log^{4}n)$ complexity of the algorithm in [6]. See Section 4. Also, our current algorithm using Link-Cut tree [20] is much easier to implement than the algorithm in [6] using the Dynamic Minimum Spanning Forest [16]. 3. 3.

For switches [7] on graph zigzag persistence, the closed-closed intervals in dimension 0 can be maintained in $O(1)$ time; the closed-open and open-closed intervals, which appear only in dimension [math], can be maintained in $O(\log m)$ time; the open-open intervals in dimension 0 and closed-closed intervals in dimension 1 can be maintained in $O(\sqrt{m}\,\log m)$ time. All these can be done with an $O(m^{1.5}\log m)$ preprocessing time. See Section 5.

2 Preliminaries

Graph zigzag persistence.

A graph zigzag filtration is a sequence of graphs

[TABLE]

in which each $G_{i}\leftrightarrow G_{i+1}$ is either a forward inclusion $G_{i}\hookrightarrow G_{i+1}$ or a backward inclusion $G_{i}\hookleftarrow G_{i+1}$. For computation, we only consider simplex-wise filtrations starting and ending with empty graphs in this paper, i.e., $G_{0}=G_{m}=\varnothing$ and each inclusion $G_{i}\leftrightarrow G_{i+1}$ is an addition or deletion of a single vertex or edge (both called a simplex). Such an inclusion is sometimes denoted as $G_{i}\xleftrightarrow{\;\sigma\;}G_{i+1}$ with $\sigma$ indicating the vertex or edge being added or deleted. The $p$-th homology functor ($p=0,1$) applied on $\mathcal{F}$ induces a zigzag module:

[TABLE]

in which each $\mathsf{H}_{p}(G_{i})\leftrightarrow\mathsf{H}_{p}(G_{i+1})$ is a linear map induced by inclusion. It is known [2, 13] that $\mathsf{H}_{p}(\mathcal{F})$ has a decomposition of the form $\mathsf{H}_{p}(\mathcal{F})\simeq\bigoplus_{k\in\Lambda}\mathcal{I}^{[b_{k},d_{k}]}$, in which each $\mathcal{I}^{[b_{k},d_{k}]}$ is an interval module over the interval $[b_{k},d_{k}]$. The multiset of intervals $\mathsf{Pers}_{p}(\mathcal{F}):=\{[b_{k},d_{k}]\,|\,k\in\Lambda\}$ is an invariant of $\mathcal{F}$ and is called the $p$-th barcode of $\mathcal{F}$. Each interval in $\mathsf{Pers}_{p}(\mathcal{F})$ is called a $p$-th persistence interval and is also said to be in dimension $p$. Frequently in this paper, we consider the barcode of $\mathcal{F}$ in all dimensions $\mathsf{Pers}_{*}(\mathcal{F}):=\bigsqcup_{p=0,1}\mathsf{Pers}_{p}(\mathcal{F})$.

Standard persistence and simplex pairing.

If all inclusions in Equation (1) are forward, we have a standard (non-zigzag) graph filtration. We also only consider standard graph filtrations that are simplex-wise and start with empty graphs. Let $\mathcal{F}$ be such a filtration. It is well-known [11] that $\mathsf{Pers}_{*}(\mathcal{F})$ is generated from a pairing of simplices in $\mathcal{F}$ s.t. for each pair $(\sigma,\tau)$ generating a $[b,d)\in\mathsf{Pers}_{*}(\mathcal{F})$, the simplex $\sigma$ creating $[b,d)$ is called positive and $\tau$ destroying $[b,d)$ is called negative. Notice that $d$ may equal $\infty$ for a $[b,d)\in\mathsf{Pers}_{*}(\mathcal{F})$, in which case $[b,d)$ is generated by an unpaired positive simplex. For a simplex $\sigma$ added from $G_{i}$ to $G_{i+1}$ in $\mathcal{F}$, we let its index be $i$ and denote it as $\mathrm{idx}_{\mathcal{F}}(\sigma):=i$. For another simplex $\tau$ added in $\mathcal{F}$, if $\mathrm{idx}_{\mathcal{F}}(\sigma)<\mathrm{idx}_{\mathcal{F}}(\tau)$, we say that $\sigma$ is older than $\tau$ and $\tau$ is younger than $\sigma$.

Merge forest.

Merge forests (more commonly called merge trees) encode the evolution of connected components in a standard graph filtration [10, 19]. We adopt merge forests as central constructs in our update algorithm for standard graph persistence (Algorithm 1). We rephrase its definition below:

Definition 1** (Merge forest).**

For a simplex-wise standard graph filtration

[TABLE]

its merge forest $\mathsf{MF}(\mathcal{F})$ is a forest (acyclic undirected graph) where the leaves correspond to vertices in $\mathcal{F}$ and the internal nodes correspond to negative edges in $\mathcal{F}$. Moreover, each node in $\mathsf{MF}(\mathcal{F})$ is associated with a level which is the index of its corresponding simplex in $\mathcal{F}$. Let $\mathsf{MF}^{i}(\mathcal{F})$ be the subgraph of $\mathsf{MF}(\mathcal{F})$ induced by nodes at levels less than $i$. Notice that trees in $\mathsf{MF}^{i}(\mathcal{F})$ bijectively correspond to connected components in $G_{i}$. We then constructively define $\mathsf{MF}^{i+1}(\mathcal{F})$ from $\mathsf{MF}^{i}(\mathcal{F})$, starting with $\mathsf{MF}^{0}(\mathcal{F})=\varnothing$ and ending with $\mathsf{MF}^{m}(\mathcal{F})=\mathsf{MF}(\mathcal{F})$. Specifically, for each $i=0,1,\ldots,m-1$, do the following:

$\sigma_{i}$** is a vertex:**

$\mathsf{MF}^{i+1}(\mathcal{F})$ equals $\mathsf{MF}^{i}(\mathcal{F})$ union an isolated leaf at level $i$ corresponding to $\sigma_{i}$.

$\sigma_{i}$** is a positive edge:**

Set $\mathsf{MF}^{i+1}(\mathcal{F})=\mathsf{MF}^{i}(\mathcal{F})$.

$\sigma_{i}$** is a negative edge:**

Let $\sigma_{i}=(u,v)$. Since $u$ and $v$ are in different connected components $C_{1}$ and $C_{2}$ in $G_{i}$, let $T_{1},T_{2}$ be the trees in $\mathsf{MF}^{i}(\mathcal{F})$ corresponding to $C_{1},C_{2}$ respectively. To form $\mathsf{MF}^{i+1}(\mathcal{F})$, we add an internal node at level $i$ (corresponding to $\sigma_{i}$) to $\mathsf{MF}^{i}(\mathcal{F})$ whose children are the roots of $T_{1}$ and $T_{2}$.

In this paper, we do not differentiate a vertex or edge in $\mathcal{F}$ and its corresponding node in $\mathsf{MF}(\mathcal{F})$.

3 Updating standard persistence on graphs

The switch operation originally proposed in [5] for general filtrations looks as follows on standard graph filtrations:

[TABLE]

In the above operation, the addition of two simplices $\sigma$ and $\tau$ are switched from $\mathcal{F}$ to $\mathcal{F}^{\prime}$. We also require that $\sigma\not\subseteq\tau$ [5] ($\sigma$ is not a vertex of the edge $\tau$) because otherwise $G^{\prime}_{i}$ is not a valid graph.

For a better presentation, we provide the idea at a high level for the updates in Algorithm 1; the full details are presented in Algorithm 2 in Section 3.1.

We also notice the following fact about the change on pairing caused by the switch in Equation (2) when $\sigma,\tau$ are both positive or both negative. Let $\sigma$ be paired with $\sigma^{\prime}$ and $\tau$ be paired with $\tau^{\prime}$ in $\mathcal{F}$. (If $\sigma$ or $\tau$ are unpaired, then let $\sigma^{\prime}$ or $\tau^{\prime}$ be $null$.) By the update algorithm for general complexes [5], either (i) the pairing for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ stays the same, or (ii) the only difference on the pairing is that $\sigma$ is paired with $\tau^{\prime}$ and $\tau$ is paired with $\sigma^{\prime}$ in $\mathcal{F}^{\prime}$.

Algorithm 1** (Update for switch on standard graph filtrations).**

For the switch operation in Equation (2), the algorithm maintains a merge forest $\mathbb{T}$ (which initially represents $\mathsf{MF}(\mathcal{F})$) and a pairing of simplices $\Pi$ (which initially corresponds to $\mathcal{F}$). The algorithm makes changes to $\mathbb{T}$ and $\Pi$ so that they correspond to $\mathcal{F}^{\prime}$ after the processing. For an overview of the algorithm, we describe the processing only for the cases (or sub-cases) where we need to make changes to $\mathbb{T}$ or $\Pi$:

A. The switch is a vertex-vertex switch:

First let $v_{1}:=\sigma$ and $v_{2}:=\tau$. As illustrated in Figure 1, the only situation where the pairing $\Pi$ changes in this case is that $v_{1},v_{2}$ are in the same tree in $\mathbb{T}$ and are both unpaired when $e$ is added in $\mathcal{F}$, where $e$ is the edge corresponding to the nearest common ancestor of $v_{1},v_{2}$ in $\mathbb{T}$. In this case, $v_{1}$,$v_{2}$ are leaves at the lowest levels in the subtrees $T_{1},T_{2}$ respectively (see Figure 1), so that $v_{2}$ is paired with $e$ and $v_{1}$ is the representative (the only unpaired vertex) in the merged connected component due to the addition of $e$. After the switch, $v_{1}$ is paired with $e$ due to being younger and $v_{2}$ becomes the representative of the merged component. Notice that the structure of $\mathbb{T}$ stays the same.

B. The switch is an edge-edge switch:

Let $e_{1}:=\sigma$ and $e_{2}:=\tau$. We have the following sub-cases:

B.1. $e_{1}$ is negative and $e_{2}$ is positive:

We need to make changes when $e_{1}$ is in a 1-cycle in $G_{i+1}$ (see Figure 2), which is equivalent to saying that $e_{1},e_{2}$ connect to the same two connected components in $G_{i-1}$. In this case, $e_{1}$ becomes positive and $e_{2}$ becomes negative after the switch, for which we pair $e_{2}$ with the vertex that $e_{1}$ previously pairs with. The node in $\mathbb{T}$ corresponding to $e_{1}$ should now correspond to $e_{2}$ after the switch.

B.2. $e_{1}$ and $e_{2}$ are both negative:

We need to make changes when the corresponding node of $e_{1}$ is a child of the corresponding node of $e_{2}$ in $\mathbb{T}$ (see Figure 3a). To further illustrate the situation, let $T_{1},T_{2}$ be the subtrees rooted at the two children of $e_{1}$ in $\mathbb{T}$, and let $T_{3}$ be the subtree rooted at the other child of $e_{2}$ that is not $e_{1}$ (as in Figure 3a). Moreover, let $u,v,w$ be the leaves at the lowest levels in $T_{1},T_{2},T_{3}$ respectively. Without loss of generality, assume that $\mathrm{idx}_{\mathcal{F}}(v)<\mathrm{idx}_{\mathcal{F}}(u)$. Since $T_{1},T_{2},T_{3}$ can be considered as trees in $\mathsf{MF}^{i-1}(\mathcal{F})$, let $C_{1},C_{2},C_{3}$ be the connected components of $G_{i-1}$ corresponding to $T_{1},T_{2},T_{3}$ respectively (see Definition 1).

We have that $C_{1},C_{2},C_{3}$ are connected by $e_{1},e_{2}$ in $G_{i+1}$ in the two different ways illustrated in Figure 4. For the two different connecting configurations, the structure of $\mathbb{T}$ after the switch is different, which is shown in Figure 3b and 3c. Furthermore, if $\mathrm{idx}_{\mathcal{F}}(w)<\mathrm{idx}_{\mathcal{F}}(u)$ and $e_{2}$ directly connects $C_{1},C_{3}$ as in Figure 4b, then we swap the paired vertices of $e_{1},e_{2}$ in $\Pi$. (See Section 3.1 for further details and justifications.)

In all cases, the algorithm also updates the levels of the leaves in $\mathbb{T}$ corresponding to $\sigma$ and $\tau$ (if such leaves exist) due to the change of indices for the vertices. Notice that the positivity/negativity of simplices can be easily read off from the simplex pairing $\Pi$.

Data structure for merge forests.

We use the Depth First Tour Tree (DFT-Tree) proposed by Farina and Laura [12] to implement the merge forest $\mathbb{T}$, which supports the following operations:

• •

$\textsc{root}(v)$: Returns the root of the tree containing node $v$.

• •

$\textsc{cut}(v)$: Deletes the edge connecting node $v$ to its parent.

• •

$\textsc{link}(u,v)$: Makes the root of the tree containing node $v$ be a child of node $u$.

• •

$\textsc{nca}(u,v)$: Returns the nearest common ancestor of two nodes $u,v$ in the same tree.

• •

$\textsc{change-val}(v,x)$: Assigns the value associated to a leaf $v$ to be $x$.

• •

$\textsc{subtree-min}(v)$: Returns the leaf with the minimum associated value in the subtree rooted at $v$.

Let $N$ be the number of nodes. All above operations in DFT-Tree take $O(\log N)$ time. Among the operations, root is used to determine whether two nodes in $\mathbb{T}$ are from the same tree; cut and link are used to make the structural changes as in Figure 3; change-val is used to record (and update during switches) the levels of leaves; subtree-min is used to return the leaf at the lowest level in a subtree.

Detecting cycles.

Algorithm 1 needs to check whether the edge $e_{1}$ resides in a 1-cycle in $G_{i+1}$ (see the edge-edge switch). We show how to check this by dynamically maintaining a minimum spanning forest (MSF) for $G:=G_{m}$ using the Link-Cut tree [20] in $O(\log m)$ time per switch. First, let the weight of each edge $e$ of $G$ be the index of $e$ in $\mathcal{F}$, i.e., if $e$ is added to a graph $G_{j}$ in $\mathcal{F}$, then $w(e)=j$. Moreover, for a path in a graph, define its bottleneck weight as the maximum weight of edges on the path. By representing the MSF of $G$ as a Link-Cut tree, we use the following procedure to check whether $e_{1}$ resides in a 1-cycle in $G_{i+1}$ when switching two edges $e_{1},e_{2}$ with $e_{1}$ negative and $e_{2}$ positive:

Let $e_{2}=(u,v)$. Retrieve the bottleneck weight $w_{\infty}$ of the path connecting $u,v$ in the MSF of $G$. This can be done in $O(\log m)$ time using the Link-Cut tree. 2. 2.

If $w_{\infty}$ equals $i-1$ (which is the weight of $e_{1}$), then $e_{1}$ is in a 1-cycle in $G_{i+1}$; otherwise, $e_{1}$ is not in a 1-cycle in $G_{i+1}$.

To see the correctness of the above procedure, first consider the case that $e_{1}$ is in a 1-cycle in $G_{i+1}$. We must have that $e_{1},e_{2}$ connect to the same two components as shown in Figure 2 (see the proof of Proposition 7). Consider running the Kruskal’s algorithm on the sequence of edges in $\mathcal{F}$: It must be true that $u,v$ first become connected after $e_{1}$ is added to the partial MSF maintained by the Kruskal’s algorithm. Therefore, the bottleneck weight of the path connecting $u,v$ in the MSF of $G$ equals $i-1$ which is the weight of $e_{1}$. Now consider the case that $e_{1}$ is not in a 1-cycle in $G_{i+1}$. The addition of $e_{2}$ creates a 1-cycle in $G^{\prime}_{i}$ because $e_{2}$ is positive in $\mathcal{F}^{\prime}$. So we have that $u,v$ are already connected in $G_{i-1}$. Consider running the Kruskal’s algorithm on the sequence of edges in $\mathcal{F}$. Before adding $e_{1}$ to the partial MSF in Kruskal’s algorithm, $u,v$ are already connected by a path in the partial MSF. Therefore, the bottleneck weight of the path connecting $u,v$ in the MSF of $G$ must be less than $i-1$.

For the switch from $\mathcal{F}$ to $\mathcal{F}^{\prime}$, we also need to update the MSF of $G$ due to the weight change on certain edges. We first notice that switching a vertex and an edge does not change the structure of the MSF and the update is easy. For a switch of two edges $e_{1},e_{2}$, we do the following in the different cases:

$e_{1},e_{2}$** are both positive:**

After the switch, $e_{1}$ and $e_{2}$ are still positive. Since only negative edges appear in the MSF of $G$ (see, e.g., the Kruskal’s algorithm), they are not in the MSF of $G$ before and after the switch and therefore the MSF stays the same.

$e_{1}$** is positive and $e_{2}$ is negative:**

Based on Proposition 7 in Section 3.1.4, $e_{1}$ stays positive and $e_{2}$ stays negative after the switch. Therefore, the fact that $e_{1}$ is not in the MSF and $e_{2}$ is in the MSF does not change before and after the switch. So we have that the structure of the MSF stays the same. Notice that due to the switch, the weight of $e_{1}$ in the MSF changes, whose update can be done by cutting $e_{1}$ first and then re-linking it.

$e_{1},e_{2}$** are both negative:**

After the switch, $e_{1}$ and $e_{2}$ are still negative. The structure of the MSF stays the same because $e_{1},e_{2}$ are both in the MSF before and after the switch. Notice that we also need to update the weights for the two edges in the Link-Cut tree as done previously.

$e_{1}$** is negative and $e_{2}$ is positive:**

As in Algorithm 1, we have the following subcases:

$e_{1}$** is in a 1-cycle in $G_{i+1}$:**

In this case, $e_{1}$ becomes positive and $e_{2}$ becomes negative after the switch (see Proposition 7). Therefore, after the switch, we should delete $e_{1}$ from the MSF and add $e_{2}$ to the MSF. This can be done by cutting $e_{1}$ and then linking $e_{2}$ in the Link-Cut tree.

$e_{1}$** is not in a 1-cycle in $G_{i+1}$:**

In this case, $e_{1}$ stays negative and $e_{2}$ stays positive after the switch (see Proposition 7). The structure of the MSF does not change and we only need to update the weight of $e_{1}$ as done previously.

Remark 1**.**

Updating the MSF for $G$ can be done in $O(\log^{4}m)$ amortized time by directly utilizing the Dynamic-MSF data structure proposed by Holm et al. [16]. However, according to the cases described above, our update on the MSF is not ‘fully-dynamic’ because it only entails simple operations such as changing the weights by one unit for one or two edges and possibly switching two edges whose weights are consecutively ordered. Therefore, we could achieve an $O(\log m)$ update in the worst case simply using the Link-Cut tree [20].

Determining different connecting configurations.

We also use the MSF of $G$ (represented by Link-Cut trees) to determine the different connecting configurations in Figure 4 for the relevant case in Algorithm 1. Specifically, if $u,w$ in Figure 4 are already connected in $G^{\prime}_{i}$, then the configuration in Figure 4b applies; otherwise, the the configuration in Figure 4a applies.

To see how to use the MSF to determine the connectivity of vertices in a filtration, we first rephrase Proposition 21 in the full version111https://arxiv.org/pdf/2103.07353.pdf of [6] as follows:

Proposition 2**.**

For a graph filtration $\mathcal{L}:H_{1}\hookrightarrow H_{2}\hookrightarrow\cdots\hookrightarrow H_{s}$, assign a weight $\mathrm{idx}_{\mathcal{L}}(e)$ to each edge $e$ in $H:=H_{s}$. For two vertices $x,y$ that are connected in $H$, let $\pi$ be the unique path connecting $x,y$ in the unique minimum spanning forest of $H$ and let $w_{\infty}$ be the bottleneck weight of $\pi$. Then, the index of the first graph in $\mathcal{L}$ where $x,y$ are connected equals $w_{\infty}+1$.

For two vertices $x,y$ in a graph filtration $\mathcal{L}$, checking whether $x,y$ are connected in a graph $H_{j}$ in $\mathcal{L}$ boils down to finding the first graph $H_{j_{*}}$ in $\mathcal{L}$ where $x,y$ are connected, and then checking whether $H_{j}$ is before or after $H_{j_{*}}$. This can be done in $O(\log m)$ time by finding the bottleneck weight of the path connecting $x,y$ in the MSF of $H$ based on Proposition 2.

Time complexity.

The costliest steps of Algorithm 1 are the operations on the DFT-Tree and Link-Cut tree. Hence, Algorithm 1 takes $O(\log m)$ time in the worst case.

Preprocessing.

In order to perform a sequence of updates on a given filtration, we also need to construct the data structures maintained by Algorithm 1 for the initial filtration. Constructing $\Pi$ is nothing but computing standard persistence pairs on graphs, which takes $O(m\,\alpha(m))$ time using Union-Find. Constructing $\mathbb{T}$ entails continuously performing the link operations on the DFT-Tree, which takes $O(m\log m)$ time. For constructing the MSF, we simply run the Kruskal’s algorithm, which can be done in $O(m\log m)$ time. Hence, the preprocessing takes $O(m\log m)$ time.

3.1 Full details and justifications for Algorithm 1

We present the full details of Algorithm 1 as follows:

Algorithm 2** (Full details of Algorithm 1).**

For the switch operation in Equation (2), the algorithm maintains a merge forest $\mathbb{T}$ (which initially represents $\mathsf{MF}(\mathcal{F})$) and a pairing of simplices $\Pi$ (which initially corresponds to $\mathcal{F}$). The algorithm makes changes to $\mathbb{T}$ and $\Pi$ so that they correspond to $\mathcal{F}^{\prime}$ after the processing. Specifically, it does the following according to different cases:

If the switch is a vertex-edge switch or an edge-vertex switch, then do nothing.

If the switch is a vertex-vertex switch, let $v_{1}:=\sigma$, $v_{2}:=\tau$. If $v_{1},v_{2}$ are in the same tree in $\mathbb{T}$, then do the following:

• •

Find the nearest common ancestor $x$ of $v_{1},v_{2}$ in $\mathbb{T}$ and let $e$ be the edge corresponding to $x$. If both of the following are true:

• –

$v_{1}$ is unpaired in $\Pi$ or $v_{1}$ is paired with an $e_{1}$ in $\Pi$ s.t. $\mathrm{idx}_{\mathcal{F}}(e_{1})\geq\mathrm{idx}_{\mathcal{F}}(e)$

• –

$v_{2}$ is unpaired in $\Pi$ or $v_{2}$ is paired with an $e_{2}$ in $\Pi$ s.t. $\mathrm{idx}_{\mathcal{F}}(e_{2})\geq\mathrm{idx}_{\mathcal{F}}(e)$

then swap the paired simplices of $v_{1},v_{2}$ in $\Pi$. Notice that $v_{1}$ or $v_{2}$ may be unpaired in $\Pi$, e.g., we could have that $v_{1}$ is paired with $e_{1}$ and $v_{2}$ is unpaired, in which case $v_{2}$ becomes paired with $e_{1}$ and $v_{1}$ becomes unpaired after the swap.

If the switch is an edge-edge switch, let $e_{1}:=\sigma$, $e_{2}:=\tau$. We have the following sub-cases:

$e_{1}$** and $e_{2}$ are both positive:**

Do nothing.

$e_{1}$** is positive and $e_{2}$ is negative:**

Do nothing.

$e_{1}$** is negative and $e_{2}$ is positive:**

If $e_{1}$ is in a 1-cycle in $G_{i+1}$, then: let the node corresponding to $e_{1}$ in $\mathbb{T}$ now correspond to $e_{2}$; let the vertex paired with $e_{1}$ in $\Pi$ now be paired with $e_{2}$; let $e_{1}$ be unpaired in $\Pi$.

$e_{1}$** and $e_{2}$ are both negative:**

If (the corresponding node of) $e_{1}$ is a child of (the corresponding node of) $e_{2}$ in $\mathbb{T}$, then do the following:

• •

Let $T_{1},T_{2}$ be the subtrees rooted at the two children of $e_{1}$ in $\mathbb{T}$. Furthermore, let $c\neq e_{1}$ be the other child of $e_{2}$, and let $T_{3}$ be the subtree rooted at $c$ (see Figure 3a). Since $T_{1},T_{2},T_{3}$ can be considered as trees in $\mathsf{MF}^{i-1}(\mathcal{F})$, let $C_{1},C_{2},C_{3}$ be the connected components of $G_{i-1}$ corresponding to $T_{1},T_{2},T_{3}$ respectively. We have that $C_{1},C_{2},C_{3}$ are connected by $e_{1},e_{2}$ in $G_{i+1}$ in the two different ways illustrated in Figure 4.

Let $u,v,w$ be the leaves at the lowest (smallest) levels in $T_{1},T_{2},T_{3}$ respectively. WLOG, assume that $\mathrm{idx}_{\mathcal{F}}(v)<\mathrm{idx}_{\mathcal{F}}(u)$. We further have the following cases:

• –

If $e_{2}$ directly connects $C_{2},C_{3}$ as in Figure 4a, then let the roots of $T_{2},T_{3}$ be the children of $e_{2}$, and let $e_{2}$ and the root of $T_{1}$ be the children of $e_{1}$ in $\mathbb{T}$ (see Figure 3b).

• –

If $e_{2}$ directly connects $C_{1},C_{3}$ as in Figure 4b, then let the roots of $T_{1},T_{3}$ be the children of $e_{2}$, and let $e_{2}$ and the root of $T_{2}$ be the children of $e_{1}$ in $\mathbb{T}$ (see Figure 3c). Moreover, if $\mathrm{idx}_{\mathcal{F}}(w)<\mathrm{idx}_{\mathcal{F}}(u)$, then swap the paired vertices of $e_{1},e_{2}$ in $\Pi$.

In all cases, the algorithm also updates the levels of the leaves in $\mathbb{T}$ corresponding to $\sigma$ and $\tau$ (if such leaves exist) due to the change of indices for the vertices. Notice that the positivity/negativity of simplices can be easily read off from the simplex pairing $\Pi$.

In the rest of the section, we justify the correctness of Algorithm 1 for all cases.

3.1.1 Justification for vertex-vertex switch

Proposition 3**.**

For the switch operation in Equation (2) where $v_{1}:=\sigma$ and $v_{2}:=\tau$ are vertices ($v_{1},v_{2}$ are thus both positive), the pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ change if and only if the following two conditions hold:

$v_{1}$* and $v_{2}$ are in the same tree in $\mathsf{MF}(\mathcal{F})$;* 2. 2.

$v_{1}$* and $v_{2}$ are both unpaired when $e$ is added in $\mathcal{F}$, where $e$ is the edge corresponding to the nearest common ancestor $x$ of $v_{1},v_{2}$ in $\mathsf{MF}(\mathcal{F})$.*

Proof.

Suppose that the two conditions hold. Let $j=\mathrm{idx}_{\mathcal{F}}(e)$, i.e., $e$ is added to $G_{j}$ to form $G_{j+1}$ and $x$ is at level $j$. Based on the definition of nearest common ancestors, we have that $v_{1},v_{2}$ are descendants of different children of $x$. Let $T_{1},T_{2}$ be the two trees rooted at the two children of $x$ in $\mathsf{MF}(\mathcal{F})$ respectively. WLOG, we can assume that $v_{1}$ is in $T_{1}$ and $v_{2}$ is in $T_{2}$ (see Figure 1). Since $x$ is at level $j$, we can view $T_{1},T_{2}$ as trees in $\mathsf{MF}^{j}(\mathcal{F})$. Let $C_{1},C_{2}$ be the connected components in $G_{j}$ corresponding to $T_{1},T_{2}$ respectively (see Definition 1). We have that $v_{1}\in C_{1}$ and $v_{2}\in C_{2}$. We then observe that as the simplices are added in a graph filtration, each connected component contains only one unpaired vertex which is the oldest one [11]. Since $v_{1}$, $v_{2}$ are both unpaired when $e$ is added to $\mathcal{F}$, we must have that $v_{1}$ is the oldest vertex of $C_{1}$ and $v_{2}$ is the oldest vertex of a $C_{2}$. Then, when $e$ is added in $\mathcal{F}$, $v_{2}$ must be paired with $e$ because $v_{2}$ is younger than $v_{1}$ (see the pairing in the persistence algorithm [11]). However, after the switch, $v_{1}$ must be paired with $e$ in $\mathcal{F}^{\prime}$ because $v_{1}$ is now younger than $v_{2}$. Therefore, the pairing changes after the switch and we have finished the proof of the ‘if’ part of the proposition.

We now prove the ‘only if’ part of the proposition. Suppose that the pairing changes after the switch. First, if $v_{1}$, $v_{2}$ are in different trees in $\mathsf{MF}(\mathcal{F})$, then the two vertices are in different connected components in $G:=G_{m}$. The pairings for $v_{1}$ and $v_{2}$ are completely independent in the filtrations and therefore cannot change due to the switch. Then, let $T$ be the subtree of $\mathsf{MF}(\mathcal{F})$ rooted at $x$ and let $j=\mathrm{idx}_{\mathcal{F}}(e)$. Similarly as before, we have that $v_{1}$ is in a connected component $C_{1}$ of $G_{j}$ and $v_{2}$ is in a connected component $C_{2}$ of $G_{j}$ for $C_{1}\neq C_{2}$. For contradiction, suppose instead that at least one of $v_{1}$, $v_{2}$ is paired when $e$ is added to $G_{j}$ in $\mathcal{F}$. If $v_{1}$ is paired when $e$ is added in $\mathcal{F}$, then let $u$ be the oldest vertex of $C_{1}$. Notice that $u\neq v_{1}$ because the oldest vertex of $C_{1}$ must be unpaired when $e$ is added in $\mathcal{F}$. We notice that the pairing for vertices in $C_{1}\setminus\{u\}$ only depends on the index order of these vertices in the filtrations, before and after the switch. Since the indices for simplices in a filtration are unique and $v_{2}\not\in C_{1}$, changing the index of $v_{1}$ from $i-1$ to $i$ does not change the index order of vertices in $C_{1}\setminus\{u\}$. Therefore, the pairing for vertices in $C_{1}\setminus\{u\}$ stays the same after the switch, which means that the pairing of $v_{1}$ does not change. This contradicts the assumption that the pairing for $v_{1}$, $v_{2}$ changes due to the switch. If $v_{2}$ is paired when $e$ is added in $\mathcal{F}$, we can reach a similar contradiction, and the proof is done. ∎

Proposition 4**.**

For the switch operation in Equation (2) where $v_{1}:=\sigma$ and $v_{2}:=\tau$ are both vertices, $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure, with the only difference being on the levels of $v_{1}$, $v_{2}$ due to the index change of simplices.

Proof.

The structure of the merge forest for a graph filtration only depends on how the edges merge different connected components for the filtration. Thus, switching two vertices does not alter the structure of the merge forest. ∎

3.1.2 Justification for vertex-edge switch

We have the following Proposition 5:

Proposition 5**.**

For the switch operation in Equation (2) where $v:=\sigma$ is a vertex and $e:=\tau$ is an edge, $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure, with the only difference being on the levels of $v$ and (possibly) $e$ due to the index change of simplices. Moreover, the pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ stay the same.

Remark**.**

Notice that $e$ may not correspond to a node in the merge forests in the above setting.

Proof.

The fact that the pairing stays the same follows from [5, Section 3], i.e., the ‘$R$ matrix’ is not reduced iff the two transposed simplices have the same dimension.

For the structure of the merge forests, we argue on the following cases:

$e$** is negative:**

Consider $\mathsf{MF}^{i+1}(\mathcal{F})$ as in Figure 5. Since $v$ is not a vertex of $e$, $v$ must be an isolated node in $\mathsf{MF}^{i+1}(\mathcal{F})$. From the figure, it is evident that the structure of $\mathsf{MF}^{i+1}(\mathcal{F})$ and $\mathsf{MF}^{i+1}(\mathcal{F}^{\prime})$ is the same where the only change is the levels of $v$ and $e$. Since the remaining construction of $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ from $\mathsf{MF}^{i+1}(\mathcal{F})$ and $\mathsf{MF}^{i+1}(\mathcal{F}^{\prime})$ (respectively) follows the same process, we have our conclusion.

$e$** is positive:**

Since adding $e$ does not alter the connected components in $\mathcal{F}$ and $\mathcal{F}^{\prime}$, $e$ does not correspond to a node in the merge forests and the proposition is obvious. ∎

3.1.3 Justification for edge-vertex switch

The behavior of an edge-vertex switch is symmetric to a vertex-edge switch and the justification is similar as done in Section 3.1.2.

3.1.4 Justification for edge-edge switch

For an edge-edge switch, rewrite the switch in Equation (2) as follows:

[TABLE]

Proposition 6**.**

For the switch operation in Equation (3) where $e_{1},e_{2}$ are positive in $\mathcal{F}$, $\mathsf{MF}(\mathcal{F}^{\prime})=\mathsf{MF}(\mathcal{F})$ and the pairings for the two filtrations stay the same.

Proof.

The edges $e_{1},e_{2}$ stay positive after the switch and so the pairing stays the same as positive edges are always unpaired. The merge forest stays the same because the positive edges cause no change to the connectivity in a filtration. ∎

Proposition 7**.**

For the switch operation in Equation (3) where $e_{1}$ is positive and $e_{2}$ is negative in $\mathcal{F}$, after the switch, $e_{1}$ stays positive and $e_{2}$ stays negative in $\mathcal{F}^{\prime}$. Moreover, $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure, with the only difference being on the level of $e_{2}$ due to the index change of simplices. The pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ also stay the same.

Proof.

First consider adding $e_{2}$ in $\mathcal{F}$, and suppose that $e_{2}=(u,v)$. Since $e_{2}$ is negative in $\mathcal{F}$, the vertices $u,v$ are in different connected components $C_{1},C_{2}$ of $G_{i}$ (see [11]). Moreover, since $e_{1}$ is positive in $\mathcal{F}$, the connectivity of $G_{i-1}$ and $G_{i}$ is the same. Then, when we add $e_{2}$ to $G_{i-1}$ in $\mathcal{F}^{\prime}$, we can also consider $C_{1},C_{2}$ as connected components of $G_{i-1}$ and consider $u,v$ to be vertices in $C_{1},C_{2}$ (respectively). Therefore, adding $e_{2}$ to $G_{i-1}$ in $\mathcal{F}^{\prime}$ connects the two components $C_{1},C_{2}$, which indicates that $e_{2}$ is still negative in $\mathcal{F}^{\prime}$. Notice that $e_{1}$ in $\mathcal{F}^{\prime}$ also creates a 1-cycle when added because $G_{i}\subseteq G_{i+1}$ (indicating that $e_{1}$ stays positive in $\mathcal{F}^{\prime}$), whose addition does not change the connectivity. Therefore, the variation of connected components in $\mathcal{F}$ and $\mathcal{F}^{\prime}$ is the same, and we have that $\mathsf{MF}(\mathcal{F}),\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure. The fact that the pairings of $\mathcal{F}$ and $\mathcal{F}^{\prime}$ stay the same follows from Case 4 of the algorithm presented in [5, Section 3]. ∎

Proposition 8**.**

For the switch operation in Equation (3) where $e_{1}$ is negative and $e_{2}$ is positive in $\mathcal{F}$, there are two different situations:

$e_{1}$** is in a 1-cycle in $G_{i+1}$:**

In this case, $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure, with the only difference that the node corresponding to $e_{1}$ in $\mathsf{MF}(\mathcal{F})$ now corresponds to $e_{2}$ in $\mathsf{MF}(\mathcal{F}^{\prime})$. Furthermore, the vertex paired with $e_{1}$ in $\mathcal{F}$ is now paired with $e_{2}$ in $\mathcal{F}^{\prime}$, and $e_{1}$ becomes positive (unpaired) in $\mathcal{F}^{\prime}$.

$e_{1}$** is not in a 1-cycle in $G_{i+1}$:**

In this case, $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure, with the only difference being on the levels of $e_{1}$ due to the index change of simplices. The pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ stay the same.

Proof.

First suppose that $e_{1}$ is in a 1-cycle in $G_{i+1}$. Then adding $e_{1}$ to $G^{\prime}_{i}$ in $\mathcal{F}^{\prime}$ does not change the connectivity of $G^{\prime}_{i}$ because $e_{1}$ is positive in $\mathcal{F}^{\prime}$. But we know that two connected components $C_{1},C_{2}$ merge into a single one from $G_{i-1}$ to $G_{i+1}$, following the assumptions on $\mathcal{F}$ (see Figure 2). So we must have that adding $e_{2}$ to $G_{i-1}$ in $\mathcal{F}^{\prime}$ causes the merge of $C_{1},C_{2}$. Hence, the change on the merges forests and pairings as described in the proposition is true. See Figure 2 for an illustration of the situation described above.

Now suppose that $e_{1}$ is not in a 1-cycle in $G_{i+1}$. Then, it is obvious that a 1-cycle $z\subseteq G_{i+1}$ created by the addition of $e_{2}$ in $\mathcal{F}$ does not contain $e_{1}$. Therefore, we have $z\subseteq G^{\prime}_{i}$ because the only difference of $G^{\prime}_{i}$ with $G_{i+1}$ is the missing of $e_{1}$. Then we have that $e_{2}$ is positive and $e_{1}$ is negative in $\mathcal{F}^{\prime}$. Since positive edges do not alter the connectivity of graphs, the second part of the proposition follows. ∎

Proposition 9 and 10 justify the case where $e_{1},e_{2}$ are both negative:

Proposition 9**.**

For the switch operation in Equation (3) where $e_{1},e_{2}$ are negative, if the corresponding node of $e_{1}$ is not a child of the corresponding node of $e_{2}$ in $\mathsf{MF}(\mathcal{F})$, then $\mathsf{MF}(\mathcal{F})$ and $\mathsf{MF}(\mathcal{F}^{\prime})$ have the same structure, with the only difference being on the levels of $e_{1},e_{2}$ due to the index change of simplices. The pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ also stay the same.

Proof.

Following the assumptions in the proposition, we have that the connected components $C_{1},C_{2}$ that $e_{1}$ merges and the connected components $C_{3},C_{4}$ that $e_{2}$ merges are all different and can all be considered as connected components in $G_{i-1}$. The proposition is then evident from this fact. ∎

Proposition 10**.**

For the switching of two edges $e_{1},e_{2}$ where $e_{1},e_{2}$ are both negative and the corresponding node of $e_{1}$ is a child of the corresponding node of $e_{2}$ in $\mathsf{MF}(\mathcal{F})$, Algorithm 1 makes the correct changes on the merge forest and the simplex pairing.

Proof.

From Figure 3 and 4, it is not hard to see that Algorithm 1 makes the correct changes on the merge forest for the situation in the proposition. Therefore, we only need to show that Algorithm 1 makes the correct changes on the pairing. Since indices of $u,v,w$ as defined in Algorithm 1 stay the same in $\mathcal{F}$ and $\mathcal{F}^{\prime}$, for these vertices, we use, e.g., $\mathrm{idx}(u)$ to denote the index of $u$ in the filtrations. We have the following cases (notice that $u$ is always paired with $e_{1}$ in $\mathcal{F}$):

$e_{2}$** directly connects $C_{2},C_{3}$ as in Figure 4a:**

Suppose that $\mathrm{idx}(w)<\mathrm{idx}(v)$. After the addition of $e_{2}$ in $\mathcal{F}^{\prime}$, $w$ is the representative (oldest vertex) for the merged component $C$ of $C_{2}$ and $C_{3}$ in $G^{\prime}_{i}$. Subsequently, $u$ is paired with $e_{1}$ in $\mathcal{F}^{\prime}$ due to the merge of $C$ and $C_{1}$ because $\mathrm{idx}(w)<\mathrm{idx}(v)<\mathrm{idx}(u)$. It is then evident that the pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ do not change with the current assumptions. If $\mathrm{idx}(v)<\mathrm{idx}(w)$, by similar arguments, we also have that the pairing does not change.

$e_{2}$** directly connects $C_{1},C_{3}$ as in Figure 4b:**

If $\mathrm{idx}(w)<\mathrm{idx}(u)$, when adding $e_{2}$ in $\mathcal{F}^{\prime}$, $u$ is paired with $e_{2}$ due to the merge of $C_{1}$ and $C_{3}$. Therefore, the pairings for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ change with the current assumptions. If $\mathrm{idx}(w)>\mathrm{idx}(u)$, by similar arguments, we have that the pairing does not change. ∎

From the justifications above, we conclude the following:

Theorem 11**.**

Algorithm 1 correctly updates the pairing and the merge forest for a switch operation in $O(\log m)$ time.

4 Computing graph zigzag persistence

In this section, we show how the usage of the Link-Cut Tree [20], can improve the computation of graph zigzag persistence. For this purpose, we combine two recent results:

• •

We show in [8] that a given zigzag filtration can be converted into a standard filtration for a fast computation of zigzag barcode.

• •

Yan et al. [21] show that the extended persistence of a given graph filtration can be computed by using operations only on trees.

Building on the work of [8], we first convert a given simplex-wise graph zigzag filtration into a cell-wise up-down filtration, where all insertions occur before deletions. Then, using the extended persistence algorithm of Yan et al. [21] on the up-down filtration with the Link-Cut Tree [20] data structure, we obtain an improved $O(m\log m)$ algorithm for computing graph zigzag persistence.

4.1 Converting to up-down filtration

First, we recall the necessary set up from [8] relevant to our purpose. The algorithm, called FastZigzag [8], builds filtrations on extensions of simplicial complexes called $\Delta$-complexes [15], whose building blocks are called cells or $\Delta$-cells. Notice that 1-dimensional $\Delta$-complexes are nothing but graphs with parallel edges (also termed as multi-edges) [8].

Assume a simplex-wise graph zigzag filtration

[TABLE]

consisting of simple graphs as input. We convert $\mathcal{F}$ into the following cell-wise up-down [3] filtration consisting of graphs with parallel edges:

[TABLE]

Cells $\hat{\sigma}_{0},\hat{\sigma}_{1},\ldots,\hat{\sigma}_{k-1}$ are uniquely identified copies of vertices and edges added in $\mathcal{F}$ with the addition order preserved (notice that the same vertex or edge could be repeatedly added and deleted in $\mathcal{F}$). Cells $\hat{\sigma}_{k},\hat{\sigma}_{k+1},\ldots,\hat{\sigma}_{m-1}$ are uniquely identified copies of vertices and edges deleted in $\mathcal{F}$, with the order also preserved. Notice that $m=2k$ because an added simplex must be eventually deleted in ${\mathcal{F}}$. \optshowWADSdelFigure 6 illustrates an example for converting an input graph zigzag filtration $\mathcal{F}$ into an up-down filtration ${\mathcal{U}}$ with parallel edges. In Figure 6, the edge $e$ is added twice in $\mathcal{F}$ in which the first addition corresponds to $\hat{e}_{1}$ in ${\mathcal{U}}$ and the second addition corresponds to $\hat{e}_{2}$ in ${\mathcal{U}}$.

\opt

showWADSdel

Definition 12**.**

In $\mathcal{F}$ or ${\mathcal{U}}$, let each addition or deletion be uniquely identified by its index in the filtration, e.g., index of $G_{i}\xleftrightarrow{\;\sigma_{i}\;}G_{i+1}$ in $\mathcal{F}$ is $i$. Then, the creator of an interval $[b,d]\in\mathsf{Pers}_{*}(\mathcal{F})\text{ or }\mathsf{Pers}_{*}({\mathcal{U}})$ is an addition/deletion indexed at $b-1$, and the destroyer of $[b,d]$ is an addition/deletion indexed at $d$.

As stated previously, each $\hat{\sigma}_{i}$ in ${\mathcal{U}}$ for $0\leq i<k$ corresponds to an addition in $\mathcal{F}$, and each $\hat{\sigma}_{i}$ for $k\leq i<m$ corresponds to a deletion in $\mathcal{F}$. This naturally defines a bijection $\phi$ from the additions and deletions in ${\mathcal{U}}$ to the additions and deletions in $\mathcal{F}$. Moreover, for simplicity, we let the domain and codomain of $\phi$ be the sets of indices for the additions and deletions. The interval mapping in [8] (which uses the Mayer-Vietoris Diamond [2, 3]) can be summarized as follows:

Theorem 13**.**

Given $\mathsf{Pers}_{*}({\mathcal{U}})$, one can retrieve $\mathsf{Pers}_{*}(\mathcal{F})$ using the following bijective mapping from $\mathsf{Pers}_{*}({\mathcal{U}})$ to $\mathsf{Pers}_{*}(\mathcal{F})$:* an interval $[b,d]\in\mathsf{Pers}_{p}({\mathcal{U}})$ with a creator indexed at $b-1$ and a destroyer indexed at $d$ is mapped to an interval $I\in\mathsf{Pers}_{*}(\mathcal{F})$ with the same creator and destroyer indexed at $\phi(b-1)$ and $\phi(d)$ respectively. Specifically,*

• •

If $\phi(b-1)<\phi(d)$, then $I=[\phi(b-1)+1,\phi(d)]\in\mathsf{Pers}_{p}(\mathcal{F})$, where $\phi(b-1)$ indexes the creator and $\phi(d)$ indexes the destroyer.

• •

Otherwise, $I=[\phi(d)+1,\phi(b-1)]\in\mathsf{Pers}_{p-1}(\mathcal{F})$, where $\phi(d)$ indexes the creator and $\phi(b-1)$ indexes the destroyer.

Notice the decrease in the dimension of the mapped interval in $\mathsf{Pers}_{*}(\mathcal{F})$ when $\phi(d)<\phi(b-1)$ (indicating a swap on the roles of the creator and destroyer).

While Theorem 13 suggests a simple mapping rule for $\mathsf{Pers}_{*}({\mathcal{U}})$ and $\mathsf{Pers}_{*}(\mathcal{F})$, we further interpret the mapping in terms of the different types of intervals in zigzag persistence. We define the following:

Definition 14**.**

Let $\mathcal{L}:\varnothing=K_{0}\leftrightarrow K_{1}\leftrightarrow\cdots\leftrightarrow K_{\ell}=\varnothing$ be a zigzag filtration. For any $[b,d]\in\mathsf{Pers}_{*}(\mathcal{L})$, the birth index $b$ is closed if $K_{b-1}\hookrightarrow K_{b}$ is a forward inclusion; otherwise, $b$ is open. Symmetrically, the death index $d$ is closed if $K_{d}\hookleftarrow K_{d+1}$ is a backward inclusion; otherwise, $d$ is open. The types of the birth/death ends classify intervals in $\mathsf{Pers}_{*}(\mathcal{L})$ into four types: closed-closed, closed-open, open-closed, and open-open.

Table 1 breaks down the bijection between $\mathsf{Pers}_{*}({\mathcal{U}})$ and $\mathsf{Pers}_{*}(\mathcal{F})$ into mappings for the different types, where $\mathsf{Pers}_{0}^{\text{co}}({\mathcal{U}})$ denotes the set of closed-open intervals in $\mathsf{Pers}_{0}({\mathcal{U}})$ (meanings of other symbols can be derived similarly).

We notice the following:

• •

$\mathsf{Pers}_{*}({\mathcal{U}})$ has no open-open intervals because there are no additions after deletions in ${\mathcal{U}}$.

• •

$\mathsf{Pers}_{1}({\mathcal{U}})$ and $\mathsf{Pers}_{1}(\mathcal{F})$ contain only closed-closed intervals because graph filtrations have no triangles.

• •

$[b,d]\in\mathsf{Pers}_{1}^{\text{cc}}({\mathcal{U}})$ is mapped to an interval in $\mathsf{Pers}_{0}^{\text{oo}}(\mathcal{F})$ when $\phi(d)<\phi(b-1)$.

\opt

showWADSdel

Example**.**

The interval mapping for the example in Figure 6 is as follows:

[TABLE]

4.2 Extended persistence algorithm for graphs

Yan et al. [21] present an extended persistence algorithm for graphs in a neural network setting (see also [22]) which runs in quadratic time. We adapt it to computing up-down zigzag persistence while improving its time complexity with a Link-Cut tree [20] data structure.

Definition 12 indicates that for the up-down filtration in Equation (4), $\mathsf{Pers}_{*}({\mathcal{U}})$ can be considered as generated from the cell pairs similar to the simplex pairs in standard persistence [11]. Specifically, an interval $[b,d]\in\mathsf{Pers}_{*}({\mathcal{U}})$ is generated from the pair $(\hat{\sigma}_{b-1},\hat{\sigma}_{d})$. While each cell appears twice in ${\mathcal{U}}$ (once added and once deleted), we notice that it should be clear from the context whether a cell in a pair refers to its addition or deletion. We then have the following:

Remark 2**.**

Every vertex-edge pair for a closed-open interval in $\mathsf{Pers}_{0}({\mathcal{U}})$ comes from the ascending part ${\mathcal{U}}_{u}$ of the filtration ${\mathcal{U}}$, and every edge-vertex pair for an open-closed interval in $\mathsf{Pers}_{0}({\mathcal{U}})$ comes from the descending part ${\mathcal{U}}_{d}$. These ascending and descending parts are as shown below:

[TABLE]

We first run the standard persistence algorithm with the Union-Find data structure on ${\mathcal{U}}_{u}$ and ${\mathcal{U}}_{d}$ to obtain all pairs between vertices and edges in $O(k\,\alpha(k))$ time, retrieving closed-open and open-closed intervals in $\mathsf{Pers}_{0}({\mathcal{U}})$. We also have the following:

Remark 3**.**

Each closed-closed interval in $\mathsf{Pers}_{0}({\mathcal{U}})$ is given by pairing the first vertex in ${\mathcal{U}}_{u}$, that comes from a connected component $C$ of $\hat{G}_{k}$, and the first vertex in ${\mathcal{U}}_{d}$ coming from $C$. There is no extra computation necessary for this type of pairing.

Remark 4**.**

Each closed-closed interval in $\mathsf{Pers}_{1}({\mathcal{U}})$ is given by an edge-edge pair in ${\mathcal{U}}$, in which one edge is a positive edge from the ascending filtration ${\mathcal{U}}_{u}$ and the other is a positive edge from the descending filtration ${\mathcal{U}}_{d}$.

To compute the edge-edge pairs, the algorithm scans ${\mathcal{U}}_{d}$ and keeps track of whether an edge is positive or negative. For every positive edge $e$ in ${\mathcal{U}}_{d}$, it finds the cycle $c$ that is created the earliest in ${\mathcal{U}}_{u}$ containing $e$ and then pairs $e$ with the youngest edge $e^{\prime}$ of $c$ added in ${\mathcal{U}}_{u}$, which creates $c$ in ${\mathcal{U}}_{u}$. To determine $c$ and $e^{\prime}$, we use the following procedure from [21]:

Algorithm 3**.**

Maintain a spanning forest $T$ of $\hat{G}_{k}$ while processing ${\mathcal{U}}_{d}$. Initially, $T$ consists of all vertices of $\hat{G}_{k}$ and all negative edges in ${\mathcal{U}}_{d}$. 3. 2.

For every positive edge $e$ in ${\mathcal{U}}_{d}$ (in the order of the filtration):

1. (a)

Add $e$ to $T$ and check the unique cycle $c$ formed by $e$ in $T$. 2. (b)

Determine the edge $e^{\prime}$ which is the youngest edge of $c$ with respect to the filtration ${\mathcal{U}}_{u}$. The edge $e^{\prime}$ has to be positive in ${\mathcal{U}}_{u}$. 3. (c)

Delete $e^{\prime}$ from $T$. This maintains $T$ to be a tree all along. 4. (d)

Pair the positive edge $e$ from ${\mathcal{U}}_{d}$ with the positive edge $e^{\prime}$ from ${\mathcal{U}}_{u}$.

We propose to implement the above algorithm by maintaining $T$ as a Link-Cut Tree [20], which is a dynamic data structure allowing the following operations in $O(\log N)$ time ($N$ is the number of nodes in the trees): (i) insert or delete a node or an edge from the Link-Cut Trees; (ii) find the maximum-weight edge on a path in the trees. Notice that for the edges in $T$, we let their weights equal to their indices in ${\mathcal{U}}_{u}$.

We build $T$ by first inserting all vertices of $\hat{G}_{k}$ and all negative edges in ${\mathcal{U}}_{d}$ into $T$ in $O(m\log m)$ time. Then, for every positive edge $e$ in ${\mathcal{U}}_{d}$, find the maximum-weight edge $\epsilon$ in the unique path in $T$ connecting the two endpoints of $e$ in $O(\log m)$ time. Let $e^{\prime}$ be the edge in $\{e,\epsilon\}$ whose index in ${\mathcal{U}}_{u}$ is greater (i.e., $e^{\prime}$ is the younger one in ${\mathcal{U}}_{u}$). Pair $e^{\prime}$ with $e$ to form a closed-closed interval in $\mathsf{Pers}_{1}({\mathcal{U}})$. After this, delete $e^{\prime}$ from $T$ and insert $e$ into $T$, which takes $O(\log m)$ time. Therefore, processing the entire filtration ${\mathcal{U}}_{d}$ and getting all closed-closed intervals in $\mathsf{Pers}_{1}({\mathcal{U}})$ takes $O(m\log m)$ time in total.

Theorem 15**.**

For a simplex-wise graph zigzag filtration $\mathcal{F}$ with $m$ additions and deletions, $\mathsf{Pers}_{*}(\mathcal{F})$ can be computed in $O(m\log m)$ time.

Proof.

We first convert $\mathcal{F}$ into the up-down filtration ${\mathcal{U}}$ in $O(m)$ time [8]. We then compute $\mathsf{Pers}_{*}({\mathcal{U}})$ in $O(m\log m)$ time using the algorithm described in this section. Finally, we convert $\mathsf{Pers}_{*}({\mathcal{U}})$ to $\mathsf{Pers}_{*}(\mathcal{F})$ using the process in Theorem 13, which takes $O(m)$ time. Therefore, computing $\mathsf{Pers}_{*}(\mathcal{F})$ takes $O(m\log m)$ time. ∎

5 Updating graph zigzag persistence

In this section, we describe the update of persistence for switches on graph zigzag filtrations. In [7], we considered the updates in zigzag filtration for general simplicial complexes. Here, we focus on the special case of graphs, for which we find more efficient algorithms for switches. In a similar vein to the switch operation on standard filtrations [5] (see also Section 3), a switch on a zigzag filtration swaps two consecutive simplex-wise inclusions. Based on the directions of the inclusions, we have the following four types of switches (as defined in [7]), where $\mathcal{F},\mathcal{F}^{\prime}$ are both simplex-wise graph zigzag filtrations starting and ending with empty graphs:

• •

Forward switch is the counterpart of the switch on standard filtrations, which swaps two forward inclusions (i.e., additions) and also requires $\sigma\nsubseteq\tau$:

[TABLE]

• •

Backward switch is the symmetric version of forward switch, requiring $\tau\not\subseteq\sigma$:

[TABLE]

• •

The remaining switches swap two inclusions of opposite directions:

[TABLE]

The switch from $\mathcal{F}$ to $\mathcal{F}^{\prime}$ is called an outward switch and the switch from $\mathcal{F}^{\prime}$ to $\mathcal{F}$ is called an inward switch. We also require $\sigma\neq\tau$ because if $\sigma=\tau$, e.g., for outward switch, we cannot delete $\tau$ from $G_{i-1}$ in $\mathcal{F}^{\prime}$ because $\tau\not\in G_{i-1}$.

5.1 Update algorithms

Instead of performing the updates in Equation (6 – 8) directly on the graph zigzag filtrations, our algorithms work on the corresponding up-down filtrations for $\mathcal{F}$ and $\mathcal{F}^{\prime}$, with the conversion described in Section 4.1. Specifically, we maintain a pairing of cells for the corresponding up-down filtration, and the pairing for the original graph zigzag filtration can be derived from the bijection $\phi$ as defined in Section 4.1.

For outward and inward switches, the corresponding up-down filtration before and after the switch is the same and hence the update takes $O(1)$ time. Moreover, the backward switch is a symmetric version of the forward switch and the algorithm is also symmetric. Hence, in this section we focus on how to perform the forward switch. The symmetric behavior for backward switch is mentioned only when necessary.

For the forward switch in Equation (6), let ${\mathcal{U}}$ and ${\mathcal{U}}^{\prime}$ be the corresponding up-down filtrations for $\mathcal{F}$ and $\mathcal{F}^{\prime}$ respectively. By the conversion in Section 4.1, there is also a forward switch (on the ascending part) from ${\mathcal{U}}$ to ${\mathcal{U}}^{\prime}$, where $\hat{\sigma},\hat{\tau}$ are $\Delta$-cells corresponding to $\sigma,\tau$ respectively:

[TABLE]

We observe that the update of the different types of intervals for up-down filtrations (see Table 1) can be done independently:

• •

To update the closed-open intervals for Equation (9) (which updates the closed-open intervals for Equation (6)), we run Algorithm 1 in Section 3 on the ascending part of the up-down filtration. This is based on descriptions in Section 4 (Table 1 and Remark 2).

• •

A backward switch in Equation (7) causes a backward switch on the descending parts of the up-down filtrations, which may change the open-closed intervals for the filtrations. For this, we run Algorithm 1 on the descending part of the up-down filtration. Our update algorithm hence maintains two sets of data structures needed by Algorithm 1, for the ascending and descending parts separately.

• •

Following Remark 3 in Section 4, to update the closed-closed intervals in dimension 0 for the switches, we only need to keep track of the oldest vertices in the ascending and descending parts for each connected component of $\hat{G}:=\hat{G}_{k}$ (defined in Equation (9)). Since indices of no more than two vertices can change in a switch, this can be done in constant time by a simple bookkeeping.

We are now left with the update of the closed-closed intervals in dimension 1 for the switch on up-down filtrations, which are generated from the edge-edge pairs (see Remark 4 in Section 4). To describe the update, we first present a high-level algorithm (Algorithm 4) by explicitly maintaining representatives (see Definition 16) for the edge-edge pairs. We then justify the correctness of Algorithm 4 by showing that the algorithm correctly maintains the representatives. After this, we propose an efficient implementation of Algorithm 4 by eliminating the maintenance of representatives. This implementation maintains MSF’s for graphs in $\sqrt{m}$ positions in the up-down filtration, achieving the claimed time complexity of $O(\sqrt{m}\,\log m)$ per update.

As mentioned, the maintenance of the edge-edge pairs is independent of the maintenance of pairs generating other types of intervals, i.e., when we perform update on one type of pairs (e.g., edge-edge pairs), we do not need to inform the data structure maintained for updating other types of pairs (e.g., vertex-edge pairs). One reason is that a switch involving a vertex, which affects other types of pairs (e.g., vertex-edge pairs), does not affect edge-edge pairs (see Algorithm 1 and 2). Also, it can be easily verified that for the different cases in an edge-edge switch, the update in Algorithm 1 and Algorithm 4 (presented below) can be conducted completely independently. Now define the following:

Definition 16**.**

For a cell-wise up-down filtration of graphs with parallel edges

[TABLE]

a representative cycle (or simply representative) for an edge-edge pair $(\varsigma_{b},\varsigma_{d})$ is a 1-cycle $z$ s.t. $\varsigma_{b}\in z\subseteq H_{b+1}$ and $\varsigma_{d}\in z\subseteq H_{d}$.

The following algorithm describes the general idea for the update on edge-edge pairs:

Algorithm 4**.**

We describe the algorithm for the forward switch in Equation (9). The procedure for a backward switch on an up-down filtration is symmetric. The algorithm maintains a set of edge-edge pairs $\Pi$ initially for ${\mathcal{U}}$. It also maintains a representative cycle for each edge-edge pair in $\Pi$. After the processing, edge-edge pairs in $\Pi$ and their representatives correspond to ${\mathcal{U}}^{\prime}$. As mentioned, a switch containing a vertex makes no changes to the edge-edge pairs (see Algorithm 2). So, suppose that the switch is an edge-edge switch and let $e_{1}:=\hat{\sigma}$, $e_{2}:=\hat{\tau}$. Moreover, let ${\mathcal{U}}_{u}$ be the ascending part of ${\mathcal{U}}$. We have the following cases:

A. $e_{1}$ and $e_{2}$ are both negative in ${\mathcal{U}}_{u}$:

Do nothing (negative edges are not in edge-edge pairs).

B. $e_{1}$ is positive and $e_{2}$ is negative in ${\mathcal{U}}_{u}$:

No pairing changes by this switch.

C. $e_{1}$ is negative and $e_{2}$ is positive in ${\mathcal{U}}_{u}$:

Let $z$ be the representative cycle for the pair $(e_{2},\epsilon)\in\Pi$. If $e_{1}\in z$, pair $e_{1}$ with $\epsilon$ in $\Pi$ with the same representative $z$ (notice that $e_{2}$ becomes unpaired).

D. $e_{1}$ and $e_{2}$ are both positive in ${\mathcal{U}}_{u}$:

Let $z,z^{\prime}$ be the representative cycles for the pairs $(e_{1},\epsilon),(e_{2},\epsilon^{\prime})\in\Pi$ respectively. Do the following according to different cases:

• •

If $e_{1}\in z^{\prime}$ and the deletion of $\epsilon^{\prime}$ is before the deletion of $\epsilon$ in ${\mathcal{U}}$: Let the representative for $(e_{2},\epsilon^{\prime})$ be $z+z^{\prime}$. The pairing does not change.

• •

If $e_{1}\in z^{\prime}$ and the deletion of $\epsilon^{\prime}$ is after the deletion of $\epsilon$ in ${\mathcal{U}}$: Pair $e_{1}$ and $\epsilon^{\prime}$ in $\Pi$ with the representative $z^{\prime}$; pair $e_{2}$ and $\epsilon$ in $\Pi$ with the representative $z+z^{\prime}$.

Remark**.**

Algorithm 4 has a time complexity of $O(m)$ dominated by the summation of 1-cycles.

Proposition 17 and Theorem 18 justify the correctness of Algorithm 4.

Proposition 17**.**

For $\mathcal{L}$ which is an up-down filtration as in Definition 16, let $E_{u}$ be the set of positive edges in the ascending part of $\mathcal{L}$ and $E_{d}$ be the set of positive edges in the descending part of $\mathcal{L}$. Formalize a pairing of $E_{u}$ and $E_{d}$ as a bijection $\pi:E_{u}\to E_{d}$. Then, the pairs $\{(e,\pi(e))\mid e\in E_{u}\}$ correctly generate $\mathsf{Pers}^{\mathrm{cc}}_{1}(\mathcal{L})$ if for each $e\in E_{u}$, $(e,\pi(e))$ admits a representative cycle in $\mathcal{L}$.

Proof.

The representative defined in Definition 16 is an adaption of the general representative for zigzag persistence defined in [6, 17]. Moreover, Proposition 9 in [6] says that if we can find a pairing for cells in $\mathcal{L}$ s.t. each pair admits a representative, then the pairing generates the barcode for $\mathcal{L}$. Since cells in $\mathcal{L}$ other than those in $E_{u}\cup E_{d}$ generate intervals in $\mathsf{Pers}_{0}(\mathcal{L})$, the corresponding cell pairs which generate $\mathsf{Pers}_{0}(\mathcal{L})$ must admit representatives. Combining Proposition 9 in [6] and the assumption that each pair $(e,\pi(e))$ admits a representative cycle, we can arrive at the conclusion. ∎

Theorem 18**.**

Algorithm 4 correctly updates the edge-edge pairs for the switch in Equation (9).

Proof.

By Proposition 17, the correctness of the updated edge-edge pairs in Algorithm 4 follows from the correctness of the updated representative cycles for these pairs. We omit the details for verifying the validity of these cycles, which are evident from the algorithm. ∎

As mentioned, to efficiently implement Algorithm 4, we eliminate maintaining the representatives in the algorithm. We notice that there are only two places in Algorithm 4 where the pairing changes:

• •

Case C when $e_{1}\in z$;

• •

Case D when $e_{1}\in z^{\prime}$ and the deletion of $\epsilon^{\prime}$ is after the deletion of $\epsilon$ in ${\mathcal{U}}$.

For Case C, we have the following fact:

Proposition 19**.**

In Case C of Algorithm 4, the edge $e_{1}$ is in $z$ if and only if $e_{1}$ is in a 1-cycle in $\hat{G}_{j+1}$ (refer to Equation (9)).

Proof.

By definition, $z$ is a 1-cycle in $\hat{G}_{j+1}$. So the fact that $e_{1}$ is in $z$ trivially implies that $e_{1}$ is in a 1-cycle in $\hat{G}_{j+1}$.

To prove the ‘if’ part of the proposition, suppose that $e_{1}$ is in a cycle $z^{\prime}\subseteq\hat{G}_{j+1}$ but $e_{1}\not\in z$. We have that the 1st betti number increases by 1 from $\hat{G}^{\prime}_{j}$ to $\hat{G}_{j+1}$ because $z^{\prime}$ is a 1-cycle created by the addition of $e_{1}$ in $\mathcal{U}^{\prime}$. By definition, $e_{2}\in z\subseteq\hat{G}_{j+1}$. Since $e_{1}\not\in z$, we have that $z\subseteq\hat{G}^{\prime}_{j}$. So the 1st betti number also increases by 1 from $\hat{G}_{j-1}$ to $\hat{G}^{\prime}_{j}$ because $z$ is a 1-cycle created by the addition of $e_{2}$ in $\mathcal{U}^{\prime}$. Therefore, the 1st betti number increases by 2 from $\hat{G}_{j-1}$ to $\hat{G}_{j+1}$. However, this is a contradiction because the fact that $e_{1}$ is negative and $e_{2}$ is positive in ${\mathcal{U}}_{u}$ implies that the 1st betti number increases by 1 from $\hat{G}_{j-1}$ to $\hat{G}_{j+1}$. ∎

The above proposition implies that to check whether $e_{1}\in z$, we only need to check whether $e_{1}$ is in a 1-cycle in $\hat{G}_{j+1}$. The checking can be similarly performed as in Algorithm 1 by maintaining a Link-Cut tree [20] for the MSF of $\hat{G}:=\hat{G}_{k}$, and then looking up the bottleneck weight of the path between $e_{2}$’s vertices in the MSF (see Section 3). Notice that edge weights in $\hat{G}$ are the edges’ indices in the ascending part of the up-down filtration. This also means that to perform the backward switch on the up-down filtration, we also need to maintain an MSF of $\hat{G}$ where the edge weights are indices in the descending part of the up-down filtration. Maintaining and querying the MSF is done in $O(\log m)$ time using the Link-Cut tree [20].

For Case D when the deletion of $\epsilon^{\prime}$ is after the deletion of $\epsilon$ in ${\mathcal{U}}$, we only need to to determine whether to keep the pairing (pairing $e_{1}$ with $\epsilon$ and $e_{2}$ with $\epsilon^{\prime}$) or change the pairing (pairing $e_{1}$ with $\epsilon^{\prime}$ and $e_{2}$ with $\epsilon$). Let the edge $\epsilon^{\prime}$ be deleted from $\hat{G}_{h}$ to form $\hat{G}_{h+1}$ in $\mathcal{U}$, i.e., the edge pair $(e_{2},\epsilon^{\prime})$ generates an interval $[j+1,h]\in\mathsf{Pers}_{1}(\mathcal{U})$. We have the following:

Observation 20**.**

For Case D of Algorithm 4 when the deletion of $\epsilon^{\prime}$ is after the deletion of $\epsilon$ in ${\mathcal{U}}$, if there is a cycle containing $e_{2}$ and $\epsilon^{\prime}$ in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$, then the pairing stays the same before and after the switch; otherwise, the pairing changes.

Remark**.**

Since we are focusing on the 1-cycles in graphs, for the graph intersection $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$, we only need to take the intersection of their edge sets. For convenience, we can assume $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$ to contain all vertices in $\hat{G}$.

To see the correctness of the observation, first suppose that there is a cycle $z^{\prime\prime}$ containing $e_{2}$ and $\epsilon^{\prime}$ in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$. Notice that $z^{\prime\prime}$ is nothing but a representative for the pair $(e_{2},\epsilon^{\prime})$ in $\mathcal{U}^{\prime}$. We also have that $z$ is still a representative for the pair $(e_{1},\epsilon)$ in $\mathcal{U}^{\prime}$. So the paring stays the same before and after the switch based on Proposition 17. If there is not a cycle containing $e_{2}$ and $\epsilon^{\prime}$ in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$, then there is no representative cycle for the pair $(e_{2},\epsilon^{\prime})$ in $\mathcal{U}^{\prime}$. So $e_{2}$ cannot be paired with $\epsilon^{\prime}$ in $\mathcal{U}^{\prime}$.

Based on Observation 20, in order to obtain the pairing in Case D, we only need to check whether there is a cycle containing both $e_{2}$ and $\epsilon^{\prime}$ in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$. This is equivalent to independently checking (i) whether there is a cycle $c_{1}$ in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$ containing $e_{2}$ and (ii) whether there is a cycle $c_{2}$ in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$ containing $\epsilon^{\prime}$. The reason is that, if $\epsilon^{\prime}\in c_{1}$ or $e_{2}\in c_{2}$, then either $c_{1}$ or $c_{2}$ would be a cycle in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$ containing both edges; otherwise, $c_{1}+c_{2}$ is a cycle in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h}$ containing both edges. We then reduce the checking of (i) and (ii) into a graph connectivity problem, e.g., checking (ii) is equivalent to checking whether the two vertices of $\epsilon^{\prime}$ is connected in $\hat{G}^{\prime}_{j}\cap\hat{G}_{h+1}$. We conclude:

Observation 21**.**

To determine the pairing in Case D of Algorithm 4, one only needs to perform two graph connectivity queries. Each query asks whether two vertices $u,v$ are connected in $H\cap H^{\prime}$, where $H$ is a graph in the ascending part of $\mathcal{U}^{\prime}$ and $H^{\prime}$ is a graph in the descending part of $\mathcal{U}^{\prime}$.

We now describe how to address the connectivity query as in Observation 21 for $\mathcal{U}$ (the query for $\mathcal{U}^{\prime}$ can be done similarly by first changing the data structures we maintain for $\mathcal{U}$ into ones for $\mathcal{U}^{\prime}$). One way to solve such a connectivity problem for $\mathcal{U}$ is to maintain an MSF for each $\hat{G}_{\lambda}$ in the descending part $\mathcal{U}_{d}$ of $\mathcal{U}$ ($\lambda\geq k$), where edges in $\hat{G}_{\lambda}$ are weighted by their indices in the ascending part $\mathcal{U}_{u}$. Then for a $\hat{G}_{\gamma}$ in $\mathcal{U}_{u}$ ($\gamma<k$), determining whether two vertices $u,v$ are connected in $\hat{G}_{\gamma}\cap\hat{G}_{\lambda}$ can be done by querying the bottleneck weight of the path connecting $u,v$ in the MSF of $\hat{G}_{\lambda}$. Representing the MSF’s as Link-Cut trees [20], such a query can be done in $O(\log m)$ time.

One problem with the previous approach is that whenever we perform a forward switch on $\mathcal{U}$, weights for $e_{1},e_{2}$ swap, which may induce a change on the MSF’s for $O(m)$ many graphs in $\mathcal{U}_{d}$. Since updating each MSF takes $O(\log m)$ time (see Section 3), the time complexity of the update on edge-edge pairs then becomes $O(m\log m)$.

To perform the update faster, we maintain MSF’s for $\sqrt{m}$ many graphs in $\mathcal{U}_{d}$, where the index difference of each two consecutive graphs for which MSF’s are maintained is $\sqrt{m}$. Formally, we always maintain MSF’s for the following graphs in $\mathcal{U}_{d}$:

[TABLE]

where $\ell=\lfloor k/\sqrt{m}\rfloor$. Notice that whenever we use ‘$\sqrt{m}$’ in the above, what we actually mean is either ‘$\lfloor\sqrt{m}\rfloor$’ or ‘$\lceil\sqrt{m}\rceil$’. We will continue abusing the notation in the rest of this section by using ‘$\sqrt{m}$’ in place of its ‘floor’ or ‘ceiling’.

Now, to obtain the MSF for an arbitrary $\hat{G}_{\lambda}$ in $\mathcal{U}_{d}$, we first find the graph $\hat{G}_{m-q\sqrt{m}}$ with the smallest index which is contained in $\hat{G}_{\lambda}$. Then, from the MSF of $\hat{G}_{m-q\sqrt{m}}$, we go over each edge $e$ between $\hat{G}_{m-q\sqrt{m}}$ and $\hat{G}_{\lambda}$ in $\mathcal{U}_{d}$ and see whether $e$ should be inserted into the MSF of $\hat{G}_{m-q\sqrt{m}}$. We can get the MSF for $\hat{G}_{\lambda}$ in $O(\sqrt{m}\,\log m)$ time using the above process where no more than $\sqrt{m}$ edges need to be processed. Notice that we can also understand the above process as building the MSF for $\hat{G}_{\lambda}$ using Kruskal’s algorithm starting from the ‘already built’ MSF for $\hat{G}_{m-q\sqrt{m}}$, where Link-Cut trees [20] are used for checking the connectivity.

Notice that after making queries on the MSF of $\hat{G}_{\lambda}$, we need to recover the MSF of $\hat{G}_{m-q\sqrt{m}}$ so that exactly the MSF’s of graphs in Sequence (10) are maintained. The MSF of $\hat{G}_{m-q\sqrt{m}}$ can be recovered by recording what edges are added to the original MSF of $\hat{G}_{m-q\sqrt{m}}$ to form the MSF of $\hat{G}_{\lambda}$, and then deleting those edges from the MSF of $\hat{G}_{\lambda}$ using the cut operation in Link-Cut trees [20]. Since no more than $\sqrt{m}$ edges need to be deleted, recovering the MSF of $\hat{G}_{m-q\sqrt{m}}$ takes $O(\sqrt{m}\,\log m)$ time.

To finish a forward switch, we also need to update the maintained MSF’s due to the weight change on edges, which takes $O(\sqrt{m}\,\log m)$ time because there are no more than $\sqrt{m}$ such MSF’s and updating each MSF takes $O(\log m)$ time. Notice that for a backward switch, instead of updating $\sqrt{m}$ many MSF’s maintained for $\mathcal{U}_{d}$, we need to reconstruct at most one such MSF. The reconstruction is the same as constructing the MSF for an arbitrary graph in $\mathcal{U}_{d}$, which takes $O(\sqrt{m}\,\log m)$ time. Notice that to do the reconstruction, we also need to record which edges are added to the MSF of $\hat{G}_{m-q\sqrt{m}}$ to form the MSF of $\hat{G}_{m-(q+1)\sqrt{m}}$, for any consecutive graphs $\hat{G}_{m-q\sqrt{m}},\hat{G}_{m-(q+1)\sqrt{m}}$ in Sequence (10). Therefore, updating the edge-edge pairs takes $O(\sqrt{m}\,\log m)$ time.

We now conclude the following:

Theorem 22**.**

For the switches on graph zigzag filtrations, the closed-closed intervals in dimension 0 can be maintained in $O(1)$ time; the closed-open and open-closed intervals, which appear only in dimension [math], can be maintained in $O(\log m)$ time; the open-open intervals in dimension 0 and closed-closed intervals in dimension 1 can be maintained in $O(\sqrt{m}\,\log m)$ time.

Preprocessing.

To perform the updates for a sequence of switches starting from a graph zigzag filtration, we need to construct the data structures maintained by the update algorithms. Given the initial filtration, we first compute its corresponding up-down filtration in $O(m)$ time [8]. For the ascending and descending parts of the initial up-down filtration, we run the preprocessing for Algorithm 1 in $O(m\log m)$ time. We then find the oldest vertices in the ascending and descending parts of the initial up-down filtration for each connected component of $\hat{G}$ in $O(m)$ time. To find the edge-edge pairs, we run Algorithm 3 in $O(m\log m)$ time. For updating the edge-edge pairs, we also construct the MSF for each graph in Sequence (10) individually. Since constructing each one is in $O(m\log m)$ time, constructing all the MSF’s is in $O(m^{1.5}\log m)$ time. Hence, the overall preprocessing takes $O(m^{1.5}\log m)$ time.

6 Discussion

We have designed $O(\log m)$ update algorithms for switches in standard graph persistence. Also, we have designed an $O(m\log m)$ algorithm for computing zigzag persistence on graphs. It remains open if computing graph zigzag persistence can be further improved to run in $O(m\log n)$ time. Similarly, for updates in graph zigzag persistence, we have designed algorithms for all types of intervals. Among them, the updates for closed-closed intervals in dimension 1 and open-open intervals in dimension 0 take more than logarithmic time though still admitting a sub-linear algorithm ($O(\sqrt{m}\,\log m)$). It would be especially interesting to see if the $O(\sqrt{m}\,\log m)$ complexity can be improved.

Furthermore, for all updates, we currently only considered switches, where the length of the filtrations is always fixed. How about those operations which change the length of the filtration, e.g., ones that insert/delete some inclusion arrows into/from the filtration? Can these operations be done in logarithmic time, or at least in sub-linear time?