Title: Improved Guarantees for Fully Dynamic $k$-Center Clustering with Outliers in General Metric Spaces
Venue: NeurIPS 2024
Accepted: yes
Keywords: $k$-center with outliers

fully dynamic model

metric spaces
Abstract: The metric $k$-center clustering problem with $z$ outliers, also known as $(k,z)$-center clustering, 
involves clustering a given point set $P$ in a metric space $(M,d)$ using at most $k$ balls, 
minimizing the maximum ball radius while excluding up to $z$ points from the clustering. 
This problem holds fundamental significance in various domains such as machine learning, 
data mining, and database systems.

This paper addresses the fully dynamic version of the problem, where the point set undergoes continuous updates (insertions and deletions) over time. The objective is to maintain an approximate $(k,z)$-center clustering with efficient update times. 
We propose a novel fully dynamic algorithm that maintains a $(4+\epsilon)$-approximate 
solution to the $(k,z)$-center clustering problem that covers 
all but at most $(1+\epsilon)z$ points at any time in the sequence with probability $1-k/e^{\Omega(\log k)}$. 
The algorithm achieves an expected amortized update time of $\mathcal{O}(\epsilon^{-2} k^6\log(k) \log(\Delta))$, and is applicable to general metric spaces. 
Our dynamic algorithm presents a significant improvement over the recent dynamic $(14+\epsilon)$-approximation algorithm by Chan, Lattanzi, Sozio, and Wang for this problem.

Improved Guarantees for Fully Dynamic k -Center Clustering with Outliers in General Metric Spaces

Abstract

The metric k -center clustering problem with z outliers, also known as ( k, z ) -center clustering, involves clustering a given point set P in a metric space ( M,d ) using at most k balls, minimizing the maximum ball radius while excluding up to z points from the clustering. This problem holds fundamental significance in various domains, such as machine learning, data mining, and database systems.

This paper addresses the fully dynamic version of the problem, where the point set undergoes continuous updates (insertions and deletions) over time. The objective is to maintain an approximate ( k, z ) -center clustering with efficient update times. We propose a novel fully dynamic algorithm that maintains a (4 + ϵ ) -approximate solution to the ( k, z ) -center clustering problem that covers all but at most (1 + ϵ ) z points at any time in the sequence with probability 1 -k/e Ω(log k ) . The algorithm achieves an expected amortized update time of O ( ϵ -3 k 6 log( k ) log(∆)) , and is applicable to general metric spaces. Our dynamic algorithm presents a significant improvement over the recent dynamic (14 + ϵ ) -approximation algorithm by Chan, Lattanzi, Sozio, and Wang [5] for this problem.

1 Introduction

Clustering problems and algorithms play an important role across a multitude of fields, helping researchers and practitioners in the analysis of data and identification of patterns. These techniques find extensive application in diverse domains, including machine learning, where they help in categorizing and understanding complex datasets. In data mining, clustering methods are utilized to uncover hidden structures and relationships within large datasets, facilitating better decision-making and insight generation. Moreover, in image and signal processing, clustering algorithms assist in segmenting and classifying data, enabling tasks such as image recognition and signal denoising.

In bioinformatics, clustering techniques are essential for organizing biological data and identifying patterns in genetic sequences, protein structures, and gene expression profiles. Similarly, in anomaly

detection, clustering methods are employed to identify unusual or unexpected patterns in data, which may indicate potential anomalies or security breaches. Furthermore, in social network analysis, clustering algorithms help in understanding the structure and dynamics of social networks by identifying communities and influential nodes.

The k -center problem is known as one of the fundamental clustering problems. Given a set of points P in a metric space and a number k , the aim of the k -center problem is to find k centers such that the maximum distance between any point and its closest center is minimized. This can also be equivalently formulated as finding a minimum radius r and centers c 1 , . . . , c k such that the balls ∪ k i =1 B P ( c i , r ) cover the point set. The k -center problem can be 2 -approximated, and this is the best possible approximation guarantee [13]. In the last decade, the focus has shifted to analyzing the problem under various complications that arise in applications.

One line of research is to study the k -center problem (and other clustering problems) in different computational models like streaming or for dynamic point sets. In the first case, points arrive sequentially, and only a summary can be stored in memory. In the second case, the point set is maintained by insertion queries and deletion queries for single points, and algorithms have to update their solution after any such query.

Another line of research is to study clustering under constraints. For example, capacitated clustering is very popular, i.e., limiting the number of points per cluster. However, lower bounds on cluster sizes have also been studied in the context of anonymity, and newer works have also considered constraints that model societal concerns like fair or diverse composition of clusters. In this paper, we study clustering with outliers . Formulated as a constraint, the k -center problem with outliers allows k + z centers, but z of these have to be singletons, meaning no point may be assigned to them. We can intuitively formulate it with balls as follows: The k -center problem with outliers asks to find a minimum r and k centers c 1 , . . . , c k such that the balls ∪ k i =1 B P ( c i , r ) cover all but z points.

Solving clustering problems in the presence of outliers is a crucial task due to the common occurrence of measurement errors or other sources of significant deviation from the rest of the data in realworld datasets. Ignoring outliers can severely distort the results of clustering algorithms, leading to inaccurate groupings. To address this challenge, a common approach is to solve a clustering problem while excluding up to z data points considered as outliers.

The first approximation algorithm for the k -center problem with outliers is due to Charikar et al. [8]. The challenge when designing approximation algorithms in the presence of outliers is that one needs to show that enough points are covered by balls of bounded sizes around the approximate centers. It is not necessary to identify the outliers of an optimal solution exactly as long as the number of uncovered points remains small enough. Due to this, [8] and follow-up papers use charging arguments. Points covered by the solution of the algorithm are mapped to points in optimum clusters, and it is then shown that the number of uncharged points is small enough.

We explore the k -center clustering problem with z outliers within the fully dynamic model, where the point set experiences continuous updates through insertions and deletions over time. Quite some work on this problem has been done for inputs from metric spaces with bounded doubling dimension [2, 3, 19]. This setting allows for geometric data structures that allow easier navigation through the data set and also bounding of the number of changes that can occur due to a query by dimension-dependent volume arguments.

We study the general metric setting. General metric spaces represent a key objective for clustering algorithms due to their broad range of distance functions, ensuring applicability to any data type. Chan et al. recently showed that in general metric spaces, any dynamic O (1) -approximation algorithm for k -center clustering excluding at most z outliers has an amortized update time of Ω( z ) [5]. For real-world applications, the fraction of outliers of the data could be arbitrarily large. Therefore, we allow for (1 + ϵ ) z outliers to be excluded to avoid the dependence on z of the update time.

The only previous work in this setting is the algorithm by Chan et al. [5], which returns a solution with at most (1 + ϵ ) z outliers with probability at least 1 -δ , for 0 < δ ≤ 1 k and ϵ > 0 . It works by maintaining a clustering with t := k · ⌈ log 1+ ϵ k δ ⌉ clusters of radius 2 r and using this to derive a final clustering with k clusters and radius 14 r . They maintain such a clustering for all r ∈ Γ := { (1 + τ ) i : d min ≤ (1 + τ ) i ≤ (1 + τ ) · d max , i ∈ N } , with d min and d max the minimum and maximum distances, respectively, between any two points ever inserted. It is then shown that there exists an instance r ∈ Γ that will approximate the optimal radius to within a factor (14 + τ ) ,

while allowing for (1 + ϵ ) z outliers, with probability at least 1 -δ . The amortized time per update is O ( | Γ | · k 2 ϵ 2 log 2 1 δ ) , with | Γ | = (log d max d min ) /τ . The total memory requirement is O ( | Γ | · | n | ) where n is the number of points in the current set.

1.1 Our contribution

We introduce a novel fully dynamic (4 + ϵ ) -approximation algorithm designed to maintain a k -center clustering while allowing for at most (1 + ϵ ) z outliers at any time in the sequence.

The expected amortized update time is O ( ϵ -3 k 6 log( k ) log(∆)) per operation (insertion or deletion). This is independent of z and of the number of points currently present in the sequence. This characteristic makes the algorithm applicable to real-world scenarios.

Notice that our data structure is continually storing an actual solution, so we can produce this solution at any time and do not need to specify additional query times (like, for example [9, 2, 18]).

Our main technical contribution is a novel combination of the sampling-based level data structure by Chan et al. [5] and the original greedy strategy by Charikar et al. [8] that enables us to achieve a much improved approximation guarantee compared to [5].

Theorem 1.1. Let ( M,d ) be a metric space and ϵ > 0 be an accuracy parameter. The spread ratio ∆ = d max d min of all points ever inserted is assumed to be bounded. There exists a randomized fully dynamic algorithm that maintains a k -center solution that allows up to (1 + ϵ ) z many outliers on the current set of points. At every point in time, the current clustering is a (4 + ϵ ) -approximation to an optimal solution for the ( k, z ) -center problem with high probability. Upon insertion or deletion of a point, the data structure is updated in amortized update time O ( ϵ -3 k 6 log( k ) log(∆)) .

To achieve this, we make use of a data structure that is described in Section 2.1 and maintained by the respective algorithms handling updates to the point set. The algorithm handling insertions is specified in Section 2.2, and the deletion algorithm is described in Section 2.3. Similar to the approach by Chan et al. [5], we maintain a hierarchical structure consisting of levels, each containing one cluster. Unlike their work, however, we only maintain ≤ k levels. For the correct radius guess, the union of these levels directly gives the desired (4 + ϵ ) -approximation. If a level violates certain properties, this and all higher levels are reclustered. As opposed to [5], we do not need to recluster every time a center gets deleted but only when a cluster does not contain enough points anymore. The algorithm handling the reclustering is described in Section 2.4.

Our analysis of the approximation ratio follows a similar argument as the proof of the 3-approximation for static k -center with z outliers as given by Charikar et al. [8] and works by charging points from chosen clusters to points in optimal clusters. We follow the same iterative charging method but extend their approach by maintaining a set of artificial outliers in order to adjust the argument to our algorithm specifically.

Note that we assume that a center can be any point from the underlying metric space, if the point was present in the data set once. Requiring that centers can only be placed at currently active points, we can achieve a 6-approximation. We formalize this in Lemma G.1.

1.2 Further Related work

The k -center problem is very well understood; two 2 -approximation algorithms for it are known [13, 15] and a matching lower bound is also known [16]. The k -center problem with outliers was first studied and 3 -approximated in [8]. In 2016, [4] gave a 2 -approximation for this problem, but the algorithm is rather complex and less amenable to practical implementation. Charikar et al. [7] developed an elegant algorithm to maintain an 8 -approximation for the vanilla k -center problem in the streaming model. Streaming can be seen as a dynamic insertion-only model. McCutchen and Khuller [17] improved this algorithm to a (2 + ϵ ) -approximation and also extended it to a (4 + ϵ ) -approximation for k -center with outliers. The k -center problem with outliers has also been studied in the sliding window model [10, 18] that also bears some similarity with our setting. For bounded doubling dimension, [18] gave a (3 + ϵ ) -approximation, but the algorithm for the general metric case is only stated as a O (1) -approximation.

The fully dynamic k -center problem without outliers in general metrics was studied by [1, 6, 11], and the best-known result is a (2+ ϵ ) -approximation with an amortized update time of O ( k polylog( n, ∆)) by Bateni et al. [1]. The same paper also shows that any algorithm that provably satisfies a non-trivial approximation guarantee needs Ω( nk ) queries to the distance function, i.e., the amortized update time is in Ω( k ) , making their algorithm close to tight. The fully dynamic k -center problem without outliers was also studied for metrics with bounded doubling dimension [14, 19, 12] and in Euclidean space [20].

1.3 Preliminaries

In the preliminaries section, we provide an introduction to the ( k, z ) -center problem and discuss the dynamic model used in our paper.

We study the k -center clustering problem with z outliers, formally defined as follows.

Definition 1.2 ( ( k, z ) -center clustering) . Let P be a point set in a metric space ( M,d ) and let k, z ∈ N be two parameters. The goal is to compute a set C ⊆ M of size at most k , such that the maximum distance of all but at most z points to their nearest c ∈ C is minimized. That is, find C such that min Z ⊆ P , | Z |≤ z max x ∈ P \ Z min c ∈ C d ( x, c ) , with | C | ≤ k .

We define d min and d max as the minimum and maximum distances, respectively, between any two points ever inserted. The ratio ∆ = d max d min is referred to as the spread ratio, and is assumed to be bounded. Let r OPT be the optimal radius for the ( k, z ) -center clustering problem of a point set P . We also define the ball B P ( c, r ) = { p ∈ P : d ( c, p ) ≤ r } to be the set of points in P that are within distance r from c . If P is clear from the context, we may drop P from the definition B P ( c, r ) . For a non-negative integer m , we denote { 1 , · · · , m } by [ m ] .

In the version of the ( k, z ) -center clustering problem that we consider, we allow (1 + ϵ ) z outliers instead of strictly z . Furthermore, we require centers of clusters to be in the metric space ( M,d ) , but they need not be currently present in P . This is referred to as the non-discrete version of the problem. Lemma G.1 in the Appendix shows how our data structure can also support the discrete version of the problem and provides a 6 -approximation solution for it.

Fully dynamic model. We consider the ( k, z ) -center clustering problem in the fully dynamic model against an adaptive adversary . In this model, we start with an empty point set, P = ∅ , and process a sequence of operations determined by the adversary. We assume the adversary does not know the random bits chosen by our algorithm; however, it can observe the algorithm's output and adapt its responses in real time (unlike an oblivious adversary , which fixes a sequence of operations in advance). Each operation can be either an insertion, where a point from the metric space ( M,d ) is added to P , or a deletion, where a point currently in P is removed. We assume only points currently in P may be deleted. Let P t represent the point set P after t operations. In other words, P t consists of all points in ( M,d ) that have been inserted but not deleted after t operations.

2 Algorithm

To explain the main ideas of our algorithm, we start by describing an iterative offline algorithm, which gives a (4 + ϵ ) approximation for the ( k, z ) -center clustering problem.

Offline algorithm. We start with a point set P 1 = P , ϵ > 0 and parameters k, z, r ∈ N . Here, r is a fixed guess for the optimal radius. For each iteration i , we sample a set of S i ⊂ P i of | S i | = ψϵ -1 k 2 log k points with replacement, uniformly at random. Here, ψ ≥ 6 β is a constant, where β > α ≥ 1 are constants. We find a point c i ∈ S , such that B ( c i , 2 r ) covers at least ϕ points from P i , where ϕ is some threshold to be defined later. We then create a new cluster C i = B P i ( c i , 4 r ) , let P i +1 = P i \ C i , and continue to the next iteration. We stop once k clusters have been created or P i = ∅ . Let λ be a random variable representing the number of iterations we complete, where λ can be at most k . After we have done λ iterations we have computed clusters C 1 , C 2 , ..., C λ with corresponding centers c 1 , c 2 , ..., c λ . The points that are not covered by the union of these clusters will be the set of outliers that our algorithm reports. If the set of outliers is at most (1 + ϵ ) z points, we can report a solution for the ( k, z ) -center clustering problem. This offline algorithm will be used as a

sub-routine for the fully dynamic algorithm, which will be explained in Section 2.4. The definition of ϕ and the pseudocode of the algorithm will also be given in that section.

Guesses for unknown r OPT. Since the optimal r is usually not known, we can run the algorithm for all r ∈ R = { (1 + ϵ ) i : d min ≤ (1 + ϵ ) i ≤ (1 + ϵ ) · d max , i ∈ N } . We then choose the smallest r ∈ R such that all but at most (1 + ϵ ) z points are covered. We will show that this gives a (4 + ϵ ) -approximation.

Level 1

Level 2

Levels 3,...,

λ

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1

Level

λ

Outliers

c

1

c

2

c

λ

C

1

C

2

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Leveling. We can visualize the output of this algorithm as (at most) λ +1 levels, where the first λ ≤ k levels each represent a cluster, and the last level represents the outliers (See Figure 1). More precisely, level i represents cluster C i with center c i , which was created in the i th iteration. Note that the construction of level i only considers the points in P i , which does not include points in the lower levels { 1 , 2 , ..., i -1 } .

The main idea for the fully dynamic algorithm is to maintain each level i for i ∈ [ λ +1] under an arbitrary number of insertions and deletions by updating the cluster C i when necessary. In the rest of the paper, we use subscript i to refer to the level and superscript t to refer to the time. For example, P t i refers to the points present in level i at time t , and r t OPT refers to the optimal radius at time t . If the level or time is clear from the context, these may be omitted. For instance, if we define a situation at time t ′ , we will not repeat the superscript t ′ on every term.

2.1 Data structure and invariants

We will construct a clustering C 1 , . . . , C λ of the current set of points P t that we store in the data structure D r introduced by [6]. If t is clear from the context, we will just write P for P t . As we do not know the current optimal radius, we will maintain one data structure D r for every choice of r ∈ R = { (1 + ϵ ) i : d min ≤ (1 + ϵ ) i ≤ d max , i ∈ N } . Every data structure consists of up to λ ≤ k cluster-levels, with each level i containing a cluster C i . In level i , we keep track of the set of the points P i currently not covered, i. e., P i = P \ ⋃ j<i C j . Level λ +1 contains the outliers, i.e., Z r = P \ ∪ i ≤ λ C i . We denote n i = | P i | . By B ( p, r ) we denote the ball with radius r centered at p , and B A ( p, r ) = B ( p, r ) ∩ A for A ⊆ P .

Let α ≥ 1 . The data structure D r consists of the following components:

① A list F r = { c 1 , c 2 , ..., c λ } of λ ≤ k

③ A set Z r = P \ ∪ i ∈ [ λ ] C i of outliers such that for all i ∈ [ λ ] and x ∈ Z r , d ( x, c i ) > 4 r .

Let α ≥ 1 be a constant. We have the following invariants for D r .

① Level invariant: for all i , P i +1 = P i \ C i , with C i = B P i ( c i , 4 r ) .

② Dense invariant: for all c i ∈ F r , |B P i ( c i , 2 r ) | ≥ min ( z +1 , n i -z k -i +1 -ϵz αk ) .

2.2 Insertion

When a new point p is inserted at time t , Procedure 3 in the appendix is executed. As input, we have D r at time t , p ∈ ( M,d ) , ϵ > 0 and z, k ∈ N . First, we check whether p is inside one of the existing clusters, in which case we can add p to such a cluster. Otherwise, we add p to Z r . Next, we check if the dense invariant is maintained after the insertion of the new point. Generally, the dense invariant can be broken in two ways:

➊ If the insertion of a new point results in more than (1 + ϵ ) z outliers, then Lemma 3.7 shows that the dense invariant is no longer valid at some level i .

➋ If the addition of a point p to a cluster C i causes the dense invariant to be violated at some lower level j < i , that occurs because n t +1 j = n t j +1 .

If the dense invariant is broken for some level i , we recluster levels i, ..., λ with λ ≤ k by invoking Procedure 5 as a sub-routine. Observe that if there are multiple levels i where the dense invariant is broken, we choose the lowest one.

2.3 Deletion

For the deletion of a point p at time t , Procedure 4 in the appendix is executed. The input is D r at time t , p ∈ ( M,d ) , ϵ > 0 and z, k ∈ N . First we check if p is an outlier, in which case we remove p from Z r . If p is either a center or a point in a cluster, we find cluster C i which contains p . If cluster C i = B ( c i , 4 r ) contains at least min ( z +1 , | P i |-z k -i +1 -ϵz αk ) points, we do not re-cluster and simply remove p from C i . Note that the underlying point set exists in the metric space ( M,d ) . If the center c i of a cluster C i is deleted, provided that the dense invariant remains valid, we can continue to utilize c i as the center of cluster C i , given our knowledge that the point c i is located within the metric space ( M,d ) . In Lemma G.1, we explain how we can still obtain a 6 -approximation if centers have to come from the current point set. If | C i | < min ( z +1 , | P i |-z k -i +1 -ϵz αk ) after the deletion of p , then it also follows that |B ( c i , 2 r ) ∩ P i | < min ( z +1 , | P i |-z k -i +1 -ϵz αk ) , i.e., the dense invariant is violated on this level. In this case, we want to redistribute the points in P i such that the levels i, . . . , λ fulfill the dense invariant. Deletion of a point in level i does not violate the invariants at levels 1 , . . . , i -1 . We re-cluster the points in ( ∪ i ≤ j ≤ k C j ) ∪ Z r using Procedure 5. We finish by updating D r .

2.4 Clustering sub-routine

The clustering sub-routine is the offline algorithm that was described at the beginning of Section 2. The pseudocode of this sub-routine is shown in Procedure 5. We use it to iteratively build the levels of data structure D r . Two cases are distinguished based on whether z is small or large compared to the number of points in level i . In line 4, ψ is a constant with ψ ≥ 6 β , where β > α , and α is the constant used in the dense invariant. The threshold ϕ that was introduced in Section 2 is different depending on the size of z compared to n i . If z +1 ≤ n i -z 4( k -i +1) , then ϕ = n i -z 2( k -i +1) and if z +1 > n i -z 4( k -i +1) , then ϕ = n i -z k -i +1 -ϵz βk . If we find multiple points p ∗ in line 6 or 12 we choose one arbitrarily.

2.5 All aspects combined

The first step will be to initialize all D r such that F r , L r , Z r = ∅ . Then, the algorithm waits for an insertion or deletion operation. If a point is inserted, Procedure 3 is executed, and if a point is deleted, Procedure 4 is executed. The algorithm is ran simultaneously for all r ∈ R = { (1 + ϵ ) i : d min ≤ (1 + ϵ ) i ≤ (1 + ϵ ) · d max , i ∈ N } .

In the next sections, it will be shown that a (4 + ϵ ) -approximation is given by the clustering L r with r ∈ R the smallest r for which |Z r | ≤ (1 + ϵ ) z .

3 Analysis

We begin our analysis of the dynamic algorithm by introducing the concept of a dense cluster and specifying the criteria for a valid solution.

Definition 3.1 (Dense cluster) . Acluster B P i ( c i , 4 r ) is dense with respect to point set P i if it satisfies the dense invariant. That is, |B P i ( c i , 2 r ) | ≥ min ( z +1 , n i -z k -i +1 -ϵz αk ) .

Observe that for the dense invariant, we consider the ball B P i ( c i , 2 r ) with a radius of 2 r . However, a cluster (as seen in line 18 of Procedure5) corresponds to the points within the ball B P i ( c i , 4 r ) , which has a radius of 4 r .

Definition 3.2 (Valid solution) . A solution B P 1 ( c i , 4 r ) ∪ · · · ∪ B P λ ( c λ , 4 r ) with λ ≤ k for point set P 1 is valid if it covers all but at most (1 + ϵ ) z points from P 1 . We refer to each cluster B P j ( c j , 4 r ) for j ∈ { i, . . . , λ } of a valid solution as a valid cluster.

Lemma 3.3 (Running time offline algorithm) . The running time of the offline algorithm, shown in Procedure 5, is O ( nϵ -1 k 3 log( k )) , where n = | P | .

Proof. In each iteration of the while-loop, we need to compute |B ( p, 2 r ) | for all p ∈ S i . Since | S i | = O ( ϵ -1 k 2 log k ) , this takes O ( nϵ -1 k 2 log k ) time. There can be at most k iterations of the while-loop and hence the total running time is O ( nϵ -1 k 3 log k ) .

3.1 Maintaining invariants

In Supplementary Section B, we prove the following three lemmas. Lemmas 3.4 and 3.5 show that the dense and level invariants are maintained during the insertion or deletion of a point, respectively. Additionally, Lemma 3.6 shows that both invariants are maintained when invoking Procedure 5 as a subroutine upon the insertion or deletion of an arbitrary point with high probability if r ≥ r OPT. The case where r < r OPT is considered in Lemmas E.1 and E.2.

Lemma 3.4 (Procedure 3 maintains invariants) . Assume that at time t , we have point set P t , data structure D r = ( F r , L r , Z r ) . We assume that the level and dense invariants hold at time t and r ≥ r t +1 OPT . At the start of time t +1 , we insert point p using Procedure 3. After the insertion, the level and dense invariants still hold with probability 1 if Procedure 5 was not called and with the probability of at least 1 -2( k -i +1) e Ψlog k , where Ψ ≥ 1 if Procedure 5 was not called.

Lemma 3.5 (Procedure 4 maintains invariants) . Assume that at time t , we have point set P t , instance D r = ( F r , L r , Z r ) , parameters k, z ∈ N and ϵ > 0 . We assume that the level and dense invariants hold at time t and r ≥ r t +1 OPT . At the start of time t + 1 , we delete an arbitrary point p using Procedure 4. After the deletion, the level and dense invariants hold with probability 1 if Procedure 5 was not called, and with probability 1 -2( k -i +1) e Ψlog k with Ψ ≥ 1 if Procedure 5 was called.

Lemma 3.6 (Procedure 5 maintains invariants with high probability) . Suppose the level and dense invariants hold for all levels j < i and we call Procedure 5 on P i as the result of an insertion or deletion. Let λ ≤ k be a random variable representing the number of levels we have after completing Procedure 5. If r ≥ r OPT , Procedure 5 maintains the level and dense invariants for all levels j with i ≤ j ≤ λ with probability at least 1 -2( k -i +1) e Ψlog k , with Ψ ≥ 1 .

3.2 Approximation guarantee

Next, we prove that if the invariants hold, our data structure D r = ( F r , L r , Z r ) contains a valid solution to the k -center with z outliers problem. Furthermore, if r < (1 + ϵ ) r OPT , D r provides a (4 + ϵ ) -approximation. Without loss of generality, for simplicity we assume that D r contains k levels. The proof of Lemma 3.7 follows the structure of the proof given in [8]. This proof uses the greediness of the algorithm to argue that the balls in the solution cover enough points to charge to. Here, we have to modify the proof to work with the dense invariant.

Lemma 3.7 (Approximation guarantee) . Let P be the current point set. Assume that the level invariant and the dense invariant hold for all levels i ≤ k . Then we have the following guarantees:

① Valid solution: If r OPT ≤ r , then B ( c 1 , 4 r ) ∪ . . . ∪B ( c k , 4 r ) covers all but at most (1 + ϵ α ) z outliers in P .

② Approximate solution: If r OPT ≤ r < (1 + ϵ ) r OPT, then B ( c 1 , 4 r ) ∪ . . . ∪B ( c k , 4 r ) gives a (4 + ϵ ) -approximation for the k -center with z outliers problem on P .

Proof. Recall that the level invariant states that for all i , we have P i +1 = P i \B ( c i , 4 r ) and the dense invariant states that for all i , we have that |B ( c i , 2 r ) ∩ P i | ≥ min( z +1 , n i -z k -i +1 -ϵz αk ) .

Assume that in the optimal solution, we have balls O 1 , . . . , O k with radius ≤ r OPT. The union of these balls covers all but z points in P . In order to prove Part ① , we aim to charge all but at most ϵ α z points of the optimal solution to points in our solution B ( c 1 , 4 r ) ∪ . . . ∪ B ( c k , 4 r ) . We prove this by induction, and Part ② will then follow easily. In order to construct the charging argument, we need to argue that there are enough points in our solution to charge the points in the optimal balls. To this end, we construct modified optimal balls O ′ 1 , . . . , O ′ k , where O ′ i ⊆ O i for every i ≤ k .

For the base case, we have O ′ 1 = O 1 , · · · , O ′ k = O k . We will show that we can order the modified balls in such a way, that at the end of time step i , all but at most ϵz αk · i points from the first i modified balls are charged to distinct points in B ( c 1 , 4 r ) ∪ . . . ∪B ( c i , 4 r ) . This will allow us to prove that our solution covers at least as many points as the optimal solution.

Assume that all but at most ϵz αk · ( i -1) points in the first i -1 modified balls O ′ 1 ∪ O ′ 2 ∪ . . . ∪ O ′ i -1 have been charged to distinct points in B ( c 1 , 4 r ) ∪ . . . ∪ B ( c i -1 , 4 r ) and consider iteration i . We distinguish two cases, namely if B ( c 1 , 2 r ) ∪ . . . ∪ B ( c i , 2 r ) intersects one of the remaining modified balls, or if it does not. The charging argument for each case proceeds as follows:

Case 1. Case 1 is when B ( c 1 , 2 r ) ∪ . . . ∪ B ( c i , 2 r ) intersects a remaining modified ball, call this ball O ′ i . Note that O ′ i will be covered entirely by B ( c 1 , 4 r ) ∪ . . . ∪ B ( c i , 4 r ) , since r ≥ r OPT. Hence, we charge the points of O ′ i to themselves and mark these points as covered. (See case 1 in Figure 2.) We call this charging rule I . Since the modified balls are disjoint 1 , any point can be charged only once (to itself) by this rule. Next, we update O ′ 1 , O ′ 2 , . . . , O ′ k .

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We maintain two sets of credit points that we save and may use for future charging purposes. First, the set Z c i = B ( c i , 4 r ) \ ( O ′ 1 ∪ . . . ∪ O ′ k ) which is the set of points covered by B ( c i , 4 r ) that are not covered by O ′ 1 ∪ O ′ 2 ∪ . . . ∪ O ′ k . We let z c i = | Z c i | be the number of such points. In Figure 2, points p, q are such points. Observe that no point is charged to points in Z c i , allowing us to use them later. We refer to these points as credit points .

There may also be previous modified balls O ′ j , with j < i that were considered in case 2 and are still present in P i . More specifically, let Z d i be the set of (there may exist) points in O ′ j ∩ B ( c j , 4 r ) that have been charged to distinct points in B ( c j , 4 r ) (See the discussion of case 2, below). For example, points a, b, c in case 2 of Figure 2. Let z d i = | Z d i | be the number of such points. Since no points

are charged to points in Z d i , we save them as credit points for future charging purposes. We now update O ′ 1 , O ′ 2 , . . . , O ′ k as follows: O ′ 1 , . . . , O ′ i stay the same and we define z c i + z d i artificial outliers in ( O ′ i +1 ∪ . . . ∪ O ′ k ) ∩ P i +1 .

Case 2. Case 2 occurs if B ( c 1 , 2 r ) ∪ . . . ∪B ( c i , 2 r ) does not intersect any of the remaining modified balls. See Figure 2. Let O ′ i be one of the remaining modified balls covering ≤ n i -z k -i +1 points. Note that for finding O ′ i we do not count points that have been defined as artificial outliers, since these artificial outliers are already covered by balls of radius 4 r in previous levels and we do not cover them by the remaining modified balls. We prove in Lemma 3.8 that such a ball O ′ i exists.

Using the assumption of this lemma that the dense invariant holds, we show in Lemma 3.9 that |B ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz αk . In this way, we find an upper bound for the number of points in O ′ i and a lower bound for the number of points in B ( c i , 2 r ) . Now, we charge all but at most ϵz αk points of O ′ i to the points of B ( c i , 2 r ) and mark these points as charged. We call this charging rule II . Note the difference between charging argument in Case 1 and 2. In Case 1, we charge the points in O ′ i to themselves, but in Case 2, we charge them to points covered by B ( c i , 2 r ) . Recall that B ( c i , 2 r ) and O ′ i are disjoint.

Observe that points in balls O ′ i +1 , · · · , O ′ k will not be charged to points in B ( c i , 2 r ) . Indeed, points in balls O ′ i +1 , · · · , O ′ k are charged to the balls that our algorithms find either using rule I or rule II. However, this will not happen based on rule I since B ( c i , 2 r ) is disjoint from all balls O ′ i +1 , · · · , O ′ k . Moreover, rule II cannot also be applied since among balls O ′ i , O ′ i +1 , · · · , O ′ k , we already charged the points in O ′ i to B ( c i , 2 r ) and points in O ′ i +1 , · · · , O ′ k will be charged to points in balls B ( c i +1 , 2 r ) , · · · , B ( c λ , 2 r ) .

Next, we consider how we update O ′ 1 , O ′ 2 , . . . , O ′ k . Similar to case 1, we define the size of two sets of credits points. First, the variable z c i = |B ( c i , 4 r ) \ ( O ′ 1 ∪ . . . ∪ O ′ k ) | that corresponds to the number of points covered by B ( c i , 4 r ) that are not covered by O ′ 1 ∪ O ′ 2 ∪ . . . ∪ O ′ k . There are z c i -| O ′ i | points from B ( c i , 4 r ) that are free (i.e., no point is charged to) and can still be charged. We consider these points as credits that we save and may use for future charging purposes.

There may also be previous modified balls O ′ j , with j < i that were considered in case 2 and are still present in P i . More specifically, let Z d i be the set of (there may exist) points in O ′ j ∩ B ( c j , 4 r ) that have been charged to distinct points in B ( c j , 4 r ) . For example, points a, b, c in case 2 of Figure 2. Let z d i = | Z d i | be the number of such points. Since no points are charged to points in Z d i , we save them as credit points for future charging purposes. We now update O ′ 1 , O ′ 2 , . . . , O ′ k as follows: O ′ 1 , O ′ 2 , . . . , O ′ i stays the same and we define ( z c i - | O ′ i | ) + z d i artificial outliers in ( O ′ i +1 ∪ . . . ∪ O ′ k ) ∩ P i +1 .

Now, for both cases, we apply charging rule I to any points in the remaining modified balls O ′ i +1 ∪ . . . ∪ O ′ k that are inside B ( c i , 4 r ) . These points are then marked as covered. See Figure 3 for an example. Note that these points will not be in P i +1 = P i \B ( c i , 4 r ) . We assumed that O 1 ∪ O 2 ∪ . . . ∪ O k covers all but at most z points. After the charging has taken place and the modified clustering O ′ 1 ∪ O ′ 2 ∪ . . . ∪ O ′ k has been constructed, we have that | O ′ 1 ∪ O ′ 2 ∪ . . . ∪ O ′ k | = n -z -∑ j in case 1 ( z c j + z d j ) -∑ j in case 2 ( z c j -| O ′ j | + z d j ) .

Note that | O ′ j | refers to the number of points covered by the modified ball O ′ j at the time of iteration j . Then, by the way we have charged O ′ 1 ∪ . . . ∪ O ′ k to our solution B ( c 1 , 4 r ) ∪ . . . ∪ B ( c k , 4 r ) , we obtain |B ( c 1 , 4 r ) ∪ . . . ∪ B ( c k , 4 r ) | ≥ | O ′ 1 ∪ O ′ 2 ∪ . . . ∪ O ′ k | -ϵz α + ∑ j in case 1 ( z c j + z d j ) + ∑ j in case 2 ( z c j -| O ′ j | + z d j ) = n -(1 + ϵ α ) z . This concludes the proof of Part ① of the lemma. It follows easily that when r OPT ≤ r < (1 + ϵ ) r OPT, Part ② also holds, which completes the proof of this lemma.

Lemma 3.8. Let i be an iteration of the charging argument above such that we are in case 2. This means that B ( c 1 , 2 r ) ∪ . . . ∪ B ( c i , 2 r ) does not intersect any of the remaining modified balls. Then, there must be a remaining modified ball covering ≤ n i -z k -i +1 points.

For the proof of Lemma 3.8, see Supplementary Section C.

Lemma 3.9 (Coverage of B ( c i , 2 r ) ) . When we are in case 2 of the charging argument for some iteration i , we must have that |B ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz αk .

For the proof of Lemma 3.9, see Supplementary Section C.

3.3 Duration of dense clusters

After running Procedure 5 as a subroutine, we know that each cluster C i covers at least ϕ points. More specifically, for each level i , if z +1 ≤ n i -z 4( k -i +1) , then |B ( c i , 2 r ) | ≥ n i -z 2( k -i +1) and if z +1 > n i -z 4( k -i +1) , then |B ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz βk . The following two lemmas show that when C i satisfies one of these two constraints, it will be dense for a significant number of arbitrary operations. This will lead to the amortized update time being independent of n . The remaining lemmas are split into two cases, corresponding to the two cases in Procedure 5.

Lemma 3.10 (Duration of dense cluster for z is small) . Assume that we are currently at time t . Let us consider a level i in which z + 1 ≤ n t i -z 4( k -i +1) . Let p = arg max p ′ ∈ S i |B P i ( p ′ , 2 r ) | , and B max = B P i ( p, 2 r ) . Assume that n t i -z 2( k -i +1) ≤ |B max | . Then, we can add B P i ( p, 4 r ) as a cluster in our solution, and this cluster will be dense until time t ′ = t + t ∗ , with t ∗ ≥ n t i -z 4( k -i +1) .

For the proof of Lemma 3.10, see Supplementary Section D.

Lemma 3.11 (Duration of dense cluster for z is large) . Assume that we are currently at time t . Let z + 1 > n t i -z 4( k -i +1) for some level i . Let p = arg max p ′ ∈ S i |B P i ( p ′ , 2 r ) | , and B max = B P i ( p, 2 r ) . Assume that n t i -z k -i +1 -ϵz βk ≤ |B max | . Then, we can add B P i ( p, 4 r ) as a cluster in our solution, and this cluster will be dense until time t ′ = t + t ∗ , with t ∗ = Ω( ϵz ) .

For the proof of Lemma 3.11, see Supplementary Section D.

3.4 Small radius guesses

It has not yet been considered what will happen if Procedure 5 fails on some level i . That is, when there exists no p ∈ S i such that B P i ( p, 2 r ) covers sufficiently many points. Lemmas E.1 and E.2 show that if Procedure 5 fails at some level, then with high probability the guess of the optimal radius for that specific instance is too small ( r < r OPT). In this case, we postpone the construction of that level to a time t + t ∗ , referred to as t ′ r in the algorithm.

3.5 Computing update time

Finally, we prove the update time of our algorithm. Lemma 3.4, Lemma 3.5 and Lemma 3.6 prove that the algorithms maintain the invariants with high probability. Together with Lemma 3.7 and Lemma F.1 this implies Theorem 1.1.

3.6 Robustness to adversarial inputs

Our dynamic algorithm is robust against an adaptive adversary. Specifically, we do not assume that the adversary has predetermined the sequence of updates in advance, as with an oblivious adversary. Instead, the adversary can query the insertion and deletion updates in an online manner, with knowledge of our solution.

We only use randomization to generate the sample set S i in Procedure 5, which is then used to construct the levels. This randomness affects only the probability of failure in this procedure. After the insertion or deletion updates, we do not need to reconstruct a level until it no longer satisfies the invariants. The guarantees we provide for how long a level can remain valid are all in the worst case; therefore, they hold even when an adaptive adversary chooses the insertion and deletion operations online.

k

References

MohammadHossein Bateni, Hossein Esfandiari, Hendrik Fichtenberger, Monika Henzinger, Rajesh Jayaram, Vahab Mirrokni, and Andreas Wiese. Optimal fully dynamic k -center clustering for adaptive and oblivious adversaries. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 2677-2727. Society for Industrial and Applied Mathematics, 2023.

Leyla Biabani, Annika Hennes, Morteza Monemizadeh, and Melanie Schmidt. Faster query times for fully dynamic k -center clustering with outliers. Advances in Neural Information Processing Systems , 36, 2024.

Matteo Ceccarello, Andrea Pietracaprina, and Geppino Pucci. Solving k-center clustering (with outliers) in mapreduce and streaming, almost as accurately as sequentially. Proc. VLDB Endow. , 12(7):766-778, 2019.

Deeparnab Chakrabarty, Prachi Goyal, and Ravishankar Krishnaswamy. The non-uniform k -center problem. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP) , volume 55 of LIPIcs. Leibniz Int. Proc. Inform. , pages Art. No. 67, 15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016.

T-H Hubert Chan, Silvio Lattanzi, Mauro Sozio, and Bo Wang. Fully dynamic k-center clustering with outliers. Algorithmica , 86(1):171-193, 2024.

TH Hubert Chan, Arnaud Guerqin, and Mauro Sozio. Fully dynamic k-center clustering. In Proceedings of the 2018 World Wide Web Conference , pages 579-587, Lyon, France, 2018. International World Wide Web Conferences Steering Committee.

Moses Charikar, Chandra Chekuri, Tom´ as Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing , pages 626-635, 1997.

Moses Charikar, Samir Khuller, David M. Mount, and Giri Narasimhan. Algorithms for facility location problems with outliers. In S. Rao Kosaraju, editor, Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA , pages 642-651. ACM/SIAM, 2001.

Mark de Berg, Leyla Biabani, and Morteza Monemizadeh. k -center clustering with outliers in the MPC and streaming model. CoRR , abs/2302.12811, 2023.

Mark de Berg, Morteza Monemizadeh, and Yu Zhong. k-center clustering with outliers in the sliding-window model. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference) , volume 204 of LIPIcs , pages 13:1-13:13. Schloss Dagstuhl Leibniz-Zentrum f¨ ur Informatik, 2021.

Hendrik Fichtenberger, Monika Henzinger, and Andreas Wiese. On fully dynamic constantfactor approximation algorithms for clustering problems. CoRR , abs/2112.07217, 2021.

Jinxiang Gan and Mordecai J Golin. Fully dynamic k-center in low dimensions via approximate furthest neighbors. In 2024 Symposium on Simplicity in Algorithms (SOSA) , pages 269-278. SIAM, 2024.

Teofilo F Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical computer science , 38:293-306, 1985.

Gramoz Goranci, Monika Henzinger, Dariusz Leniowski, Christian Schulz, and Alexander Svozil. Fully dynamic k -center clustering in low dimensional metrics. In 2021 Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX) , pages 143-153. SIAM, 2021.

Dorit S Hochbaum and David B Shmoys. A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM (JACM) , 33(3):533-550, 1986.

Wen-Lian Hsu and George L Nemhauser. Easy and hard bottleneck location problems. Discrete Applied Mathematics , 1(3):209-215, 1979.

Richard Matthew McCutchen and Samir Khuller. Streaming algorithms for k -center clustering with outliers and with anonymity. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques , pages 165-178. Springer, 2008.

Paolo Pellizzoni, Andrea Pietracaprina, and Geppino Pucci. k-center clustering with outliers in sliding windows. Algorithms , 15(2):52, 2022.

Paolo Pellizzoni, Andrea Pietracaprina, and Geppino Pucci. Fully dynamic clustering and diversity maximization in doubling metrics. In Algorithms and Data Structures Symposium , pages 620-636. Springer, 2023.

Melanie Schmidt and Christian Sohler. Fully dynamic hierarchical diameter k -clustering and k -center. arXiv preprint arXiv:1908.02645 , 2019.

A Missing Pseudocode

In this section, we provide the missing pseudocodes. We start by Procedure INITIALIZATION, which is called to start the algorithm. This procedure defines the set R and then creates a counter for time t , and the elements of our data structure F r , L r , and Z r that we discussed before. It also creates the variable t ′ r , which is used when the radius r is smaller than r OPT. If r < r OPT, we may fail to find the p ∗ that we desired for a level i in Procedure OFFLINECLUSTER, and then have to stop the level construction. In this case, we set t ′ r to t + t ∗ to postpone the construction of that level to time t + t ∗ . See Lemmas E.1 and E.2 for the choice of t ∗ . If t ′ r is set to -1 , it means there is currently no construction to be postponed. We assume all the parameters and variables defined in Procedure INITIALIZATION are global, and the other procedures have access to them.

Then, we state Procedure UPDATE to handle the insertion or deletion of a point p . Depending on whether the query is an insertion or a deletion, it calls INSERT ( p, r ) or DELETE ( p, r ) for different values of r . Then it updates the counter for time. Next, if there is any level construction that was postponed to the current time t , it handles it.

Procedure 4 DELETE ( p, r

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B Missing proofs Section 3.1

Lemma 3.4 (Procedure 3 maintains invariants) . Assume that at time t , we have point set P t , data structure D r = ( F r , L r , Z r ) . We assume that the level and dense invariants hold at time t and r ≥ r t +1 OPT . At the start of time t +1 , we insert point p using Procedure 3. After the insertion, the level and dense invariants still hold with probability 1 if Procedure 5 was not called and with the probability of at least 1 -2( k -i +1) e Ψlog k , where Ψ ≥ 1 if Procedure 5 was not called.

Proof. If we enter the case in line 2, the new point p will be added to an existing cluster C i . Then, we have that p ∈ P j for all j ≤ i and p / ∈ P j for all j > i . If we enter the case in line 4, the new point p is added as an outlier. This is the highest level, so we have that p ∈ P i for all i . The level invariant is maintained by definition in these cases. If we enter the case in line 9 and recluster levels i, . . . , k , we recluster the points in P i as defined in line 7, and hence, none of the levels j < i are affected. Then by Lemma 3.6, the level invariant is maintained for all levels. The dense invariant is maintained because of the check we do in line 9 for all levels. Furthermore, we choose the lowest level i where the dense invariant does not hold and recluster from this level upwards. Hence, the

dense invariant will hold for all levels j < i . We call Procedure 5 on P i , so by Lemma 3.6, the dense invariant is maintained for levels j ≥ i with probability 1 -2( k -i +1) e Ψlog k , with Ψ ≥ 1 .

Lemma 3.5 (Procedure 4 maintains invariants) . Assume that at time t , we have point set P t , instance D r = ( F r , L r , Z r ) , parameters k, z ∈ N and ϵ > 0 . We assume that the level and dense invariants hold at time t and r ≥ r t +1 OPT . At the start of time t + 1 , we delete an arbitrary point p using Procedure 4. After the deletion, the level and dense invariants hold with probability 1 if Procedure 5 was not called, and with probability 1 -2( k -i +1) e Ψlog k with Ψ ≥ 1 if Procedure 5 was called.

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Proof. If we enter the case in line 2, where p is an outlier, it follows easily that the level and dense invariants are maintained. In the second case, starting in line 4, p is in some cluster C i , either as a center or as another point. If C i still covers sufficiently many points after the deletion of p as described in line 7, it follows easily that the level invariant is maintained. For the dense invariant, note that for all levels j < i , n j and with this n j -z k -j +1 -ϵz αk can only decrease. Furthermore, for all levels j > i , n j remains unchanged. This observation, combined with the fact that for any i = j , | C j | is unchanged, means that the dense invariant still holds for all levels.

If C i is no longer dense after deleting p , and we enter the case in line 9, the clusters in levels j < i remain unchanged. The level invariant will be maintained since P j is updated to P j \{ p } for all j < i . The dense invariant is maintained in levels j < i by the same reasoning as the previous case. On the remaining points, Procedure 5 is called in line 10. Using Lemma 3.6, the level and dense invariants are maintained for the remaining levels with probability at least 1 -2( k -i +1) e Ψlog k , where Ψ ≥ 1 . Crucial for maintaining the level invariant is line 6. This line ensures that we do not consider the points in levels j < i when constructing level i and any higher levels.

Lemma 3.6 (Procedure 5 maintains invariants with high probability) . Suppose the level and dense invariants hold for all levels j < i and we call Procedure 5 on P i as the result of an insertion or deletion. Let λ ≤ k be a random variable representing the number of levels we have after completing Procedure 5. If r ≥ r OPT , Procedure 5 maintains the level and dense invariants for all levels j with i ≤ j ≤ λ with probability at least 1 -2( k -i +1) e Ψlog k , with Ψ ≥ 1 .

Proof. By the while-loop structure in combination with line 18, the level invariant is maintained for all levels j ≥ i . For each newly constructed level j ≥ i , there are two cases within the while-loop.

For the first case where z +1 ≤ n j -z 4( k -j +1) , we find a ball B P j ( p ∗ , 2 r ) that covers ≥ n j -z 2( k -j +1) points with probability at least 1 -2 e Ψlog k if r ≥ r OPT. For the proof of this, we refer to Lemma E.1.

Then, the dense invariant holds for any such level since

In the second case where z +1 > n j -z 4( k -j +1) , we find a ball B P j ( p ∗ , 2 r ) that covers ≥ n j -z k -j +1 -ϵz βk points with probability 1 -2 e Ψlog k if r ≥ r OPT. For the proof of this, we refer to Lemma E.2. Since β > α (see Section 2.4), the dense invariant also holds for every level in this case. Then, the probability that the dense invariant holds for all levels j with i ≤ j ≤ λ is at least 1 -2( k -i +1) e Ψlog k . This is because λ -i +1 ≤ k -i +1 is the number of new levels we constructed during Procedure 5.

C Missing proofs Section 3.2

Lemma 3.8. Let i be an iteration of the charging argument above such that we are in case 2. This means that B ( c 1 , 2 r ) ∪ . . . ∪ B ( c i , 2 r ) does not intersect any of the remaining modified balls. Then, there must be a remaining modified ball covering ≤ n i -z k -i +1 points.

Proof. We prove this using strong induction on iterations in case 2. First, let us consider the base case. In the base case, we are in iteration i which is the first iteration to be in case 2 of the charging argument. There have been i -1 iterations before i which were charged according to case 1. For any

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This is because we assume that there are exactly z points outside our optimal solution O 1 ∪ . . . ∪ O k . These z points are also outside O ′ 1 ∪ . . . ∪ O ′ k since for 1 ≤ i ≤ k , O ′ i ⊆ O i . Of those z points, ( z -∑ i -1 j =1 ( z c j -x j )) have not yet been seen in an iteration. Hence, these points are definitely not covered by O ′ i ∪ . . . ∪ O ′ k . Furthermore, we know that ∑ i -1 j =1 ( z c j -x j ) artificial outliers have been defined in O ′ i ∪ . . . ∪ O ′ k in total. This is because all iterations j < i were in case 1, and any artificial outliers present in O ′ j will be propagated to ( O ′ j +1 ∪ . . . ∪ O ′ k ) ∩ P j +1 , as these are included in z c j . We subtract x j because these artificial outliers have already been counted by another z c j , and should not be counted again when computing the total amount of artificial outliers. Note that z d j = 0 for all iterations j < i as there were no previous iterations before j in case 2. Since at most n i -z points are covered by O ′ i ∪ . . . ∪ O ′ k , there must be at least one of the modified optimal ball which covers ≤ n i -z k -i +1 . This concludes the proof of the base case.

Now, consider the inductive step. We are in an iteration i that is in case 2. There have been i -1 previous iterations, of which an arbitrary number has been charged by case 2. Assume that for any such iteration j < i that was in case 2, we found an optimal ball O ′ j that was covering at most n j -z k -j +1 points. At the time of iteration j , the points of ball O ′ j were charged to distinct points in B ( c j , 2 r ) , but not necessarily covered. In the iterations between j and i , however, more points of the ball O ′ j may have been covered. Define O ′ j, unc as the points of O ′ j that are still uncovered at the time of iteration i . Now, for the remaining modified balls, we should have that:

Similar to the base case, ( z -∑ i -1 j =1 ( z c j -x j )) points will definitely be outside O ′ i ∪ . . . ∪ O ′ k . In the iterations before j < i in case 1, ( z c j + z d j ) points are stored as a credit. We subtract x j from this since we don't want to recount outliers from previous iterations. In each iteration j < i that was in case 2, ( z c j + z d j -| O ′ j | ) are stored as a credit. Even though B ( c j , 2 r ) is disjoint from the remaining modified balls in this case, B ( c j , 2 r ) can still cover some artificial outliers present in the remaining modified balls. We do not want to recount these for the total amount of outliers in O ′ i ∪ . . . ∪ O ′ k , and hence we subtract x j . Lastly, since the modified balls are disjoint, any points of O ′ j with j < i and iteration j in case 2 that are still in the point set n i are not covered by O ′ i ∪ . . . ∪ O ′ k . We can rewrite

this as follows:

This is because by the definition of z d j , ( ∑ j<i z d j + ∑ j<i in case 2 | O ′ j, unc | ) = ∑ j<i in case 2 | O ′ j | . Since at most n i -z points are covered by O ′ i ∪ . . . ∪ O ′ k , there must be at least one of the remaining modified balls which covers ≤ n i -z k -i +1 . This concludes the proof.

Lemma 3.9 (Coverage of B ( c i , 2 r ) ) . When we are in case 2 of the charging argument for some iteration i , we must have that |B ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz αk .

Proof. We prove the statement by induction on iterations i ≤ k in case 2 of the charging argument. For the base case, iteration i is the first iteration in case 2. This means that for all previous iterations j < i , O ′ j has been charged and covered by charging rule I. Hence, B ( c i , 2 r ) cannot intersect any such O ′ j since the points in O ′ j are not in P i . We know that B ( c i , 2 r ) also does not intersect any of the remaining modified balls by our assumption that iteration i is in case 2. Hence, B ( c i , 2 r ) does not intersect O ′ 1 ∪ . . . ∪ O ′ k . The ball B ( c i , 2 r ) can cover x ≤ ∑ i -1 j =1 ( z c j -x j ) artificial outliers, where x is the total number of artificial outliers present in O ′ i ∪ . . . ∪ O ′ k . The ball B ( c i , 2 r ) can cover at most z -∑ i -1 j =1 ( z c j -x j ) other points, since there are exactly z points outside O 1 ∪ . . . ∪ O k , of which ∑ i -1 j =1 ( z c j -x j ) have been seen in a previous ball B ( c j , 4 r ) , for j < i . Hence, |B ( c i , 2 r ) | ≤ z . So, in order to satisfy the dense invariant, |B ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz αk .

Now consider the induction case. We are in an arbitrary iteration i in case 2, and there have been an arbitrary number of iterations j < i in case 2. For iterations j < i in case 2, we assume that |B ( c j , 2 r ) | ≥ n j -z k -j +1 -ϵz αk . Even though B ( c i , 2 r ) is disjoint from the remaining modified balls, it can cover x artificial outliers and z -∑ i -1 j =1 ( z c j -x j ) other points. For x we have that:

∑ j<i in case 1 ( z c j + z d j -x j ) are the total number of artificial outliers from iterations in case 1, and ∑ j<i in case 2 ( z c j + z d j -| O ′ j | -x j ) are the total number of artificial outliers from case 2. Then, for the total amount of points in B ( c i , 2 r ) , we have:

This is because by the definition of z d j , we have that ∑ j<i z d j ≤ ∑ j<i in case 2 | O ′ j | . So, in order to satisfy the dense invariant, |B ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz αk . This concludes the proof.

D Missing proofs Section 3.3

Lemma 3.10 (Duration of dense cluster for z is small) . Assume that we are currently at time t . Let us consider a level i in which z + 1 ≤ n t i -z 4( k -i +1) . Let p = arg max p ′ ∈ S i |B P i ( p ′ , 2 r ) | , and B max = B P i ( p, 2 r ) . Assume that n t i -z 2( k -i +1) ≤ |B max | . Then, we can add B P i ( p, 4 r ) as a cluster in our solution, and this cluster will be dense until time t ′ = t + t ∗ , with t ∗ ≥ n t i -z 4( k -i +1) .

Proof. The subscript of n i will be omitted for simplicity. At time t , we know that n t -z 2( k -i +1) ≤ |B max | . The resulting cluster B P i ( p, 4 r ) will be dense as long as B max covers more than min ( z +1 , n t ′ -z k -i +1 -ϵz αk ) points at time t ′ . Hence, the dense invariant will not be broken as long as

B max is greater than z +1 . Let t ′ be the time at which |B max | < z +1 . We can lower bound the time t ∗ between t and t ′ using the change in size of B max. Then, we get the following lower bound:

Then, since z +1 ≤ n t -z 4( k -i +1) by our assumption, we have:

This concludes the proof.

Lemma 3.11 (Duration of dense cluster for z is large) . Assume that we are currently at time t . Let z + 1 > n t i -z 4( k -i +1) for some level i . Let p = arg max p ′ ∈ S i |B P i ( p ′ , 2 r ) | , and B max = B P i ( p, 2 r ) . Assume that n t i -z k -i +1 -ϵz βk ≤ |B max | . Then, we can add B P i ( p, 4 r ) as a cluster in our solution, and this cluster will be dense until time t ′ = t + t ∗ , with t ∗ = Ω( ϵz k ) .

Proof. The subscript of n i will be omitted for simplicity. At time t , we know that n t -z k -i +1 -ϵz βk ≤ |B max | . The resulting cluster B P i ( p, 4 r ) will be dense as long as B max covers more than min ( z +1 , n t ′ -z k -i +1 -ϵz αk ) points at time t ′ , with α a fixed constant smaller than β . Hence, the dense invariant will not be broken as long as B max covers at least n t ′ -z k -i +1 -ϵz αk points. Let t ′ be the time at which |B max | < n t ′ -z k -i +1 -ϵz αk . We can lower bound t ∗ by examining the change in size of B max. Using that n t ′ ≤ n t + t ∗ , we have:

Now, solving for t ∗ gives:

Since α and β are two fixed constants such that β > α , this gives t ∗ = Ω( ϵz k ) . This concludes the proof.

E Missing proofs Section 3.4

Lemma E.1 (Radius guess is small for z is small) . Assume that we are currently at time t . Let us consider a level i for which z +1 ≤ n t i -z 4( k -i +1) . Let p = arg max p ′ ∈ S i |B P i ( p ′ , 2 r ) | , where S i is the sample chosen in Algorithm 5, and B max = B P i ( p, 2 r ) . Assume that |B max | < n t i -z 2( k -i +1) . Then, with probability at least 1 -2 e Ψlog k , for Ψ ≥ 1 , we have that r < r OPT and until time t ′ = t + t ∗ we do not need to consider the instance for r , with t ∗ ≥ n t i -z 2( k -i +2) .

Proof. Let O 1 , . . . , O k be the optimal balls at time t . Consider the same charging argument used in Lemma 3.7, where we charged one of the optimal balls to B P i ( c i , 4 r ) in each level i . Using Equation 4 and the fact that O ′ i ⊆ O i for each of the remaining optimal balls, we know that | O i ∪ . . . ∪ O k | ≥ n t i -z . Then, there must be at least one of the remaining optimal balls covering ≥ n t i -z k -i +1 points. Let O i be the largest such ball. Using the Chernoff bound, we show that with high probability, one of the sampled points p ∈ S i is in O i . Define independent random variables X 1 , . . . , X | O i | , one for each point in O i . Each X j , corresponding to point j ∈ O i , will be 1 if j ∈ S i

and 0 otherwise. Define X = ∑ | O i | j =1 X j . We know that E [ X j ] = | S i | n t i . Then, using linearity of expectation, we find:

The condition z + 1 ≤ n t i -z 4( k -i +1) implies z ≤ n t i 4( k -i +1) ≤ n t i 4 . Then, since | S i | = ψϵ -1 k 2 log k , it follows that | S i | k (1 -z n t i ) ≥ 3 4 ψϵ -1 k log k . Given that ψ ≥ 6 β and β > α ≥ 1 , this is at least 3 ϵ -1 k log k ≥ 3Ψlog k for Ψ ≥ 1 .

Now, using the Chernoff bound, we find:

where Ψ ≥ 1 .

Hence, with probability of at least 1 -2 e Ψlog k , there will be at least 1 point from O i in S i . Let us call this point p . If r ≥ r OPT, B P i ( p, 2 r ) would cover all points of O i since p ∈ O i and O i has radius ≤ r OPT. However, since |B max | < n t i -z 2( k -i +1) , we must have that r < r OPT.

Left to prove is that until time t ′ = t + t ∗ we do not need this instance of r , where t ∗ ≥ n t i -z 2( k -i +2) . To this end, let B t max be B max at time t . Consider the situation at time t . We know that |B t max | < n t i -z 2( k -i +1) .

Let t ′ be some time after t such that |B t ′ max | ≥ n t ′ i -z k -i +1 . Hence, the instance for r becomes valid at time t ′ . We want to derive a lower bound for t ∗ = t ′ -t to complete the proof. By examining the change in the size of B max, we derive the following lower bound:

Using n t ′ i ≥ n t i -t ∗ and subsequently solving for t ∗ we find that:

This completes the proof.

Lemma E.2 (Radius guess is small for z is large) . Assume that we are currently at time t . Let us consider a level i in which z +1 > n t i -z 4( k -i +1) . Let p = arg max p ′ ∈ S i |B P i ( p ′ , 2 r ) | , where S i is the sample chosen in Algorithm 5, and B max = B P i ( p, 2 r ) . Assume that |B max | < n t i -z k -i +1 -ϵz βk . Then, with probability at least 1 -2 e Ψlog k , for Ψ ≥ 1 , we have that r < r OPT and until time t ′ = t + t ∗ we do not need to consider the instance for r , with t ∗ ≥ ϵz 2 βk .

Proof. Let O 1 , . . . , O k be the optimal balls at time t . As in Lemma E.1, let O i be the largest remaining optimal ball covering ≥ n t i -z k -i +1 points. Using the Chernoff bound, we show that with high probability, one of the sampled points p ∈ S i is in O i . Define independent random variables X 1 , . . . , X | O i | , one for each point in O i . Each X j , corresponding to point j ∈ O i , will be 1 if j ∈ S i and 0 otherwise. Define X = ∑ | O i | j =1 X j . We know that E [ X j ] = | S i | n t i . Then, using linearity of expectation, we find

Without loss of generality, we can assume n t i -z k -i +1 -ϵz βk ≥ 1 , otherwise there would not need to be any points in the ball B P i ( p, 2 r ) to satisfy the dense invariant, and level i would be trivial. Using this, Equation (15) can be simplified as follows:

We make a case distinction. Either z ≥ n t i 2 . In this case, Equation (15) further simplifies to

Recall that | S i | = ψϵ -1 k 2 log k . Given that ψ ≥ 6 β and β > α ≥ 1 , we have

for some Ψ ≥ 1 .

In the other case z < n t i 2 , Equation (15) simplifies to

for some Ψ ≥ 1 .

In both cases, using the Chernoff bound, we find that with probability at least 1 -2 e Ψlog k , there will be at least one point p from O i in X i . If r ≥ r OPT , B P i ( p, 2 r ) would cover all points of O i since p ∈ O i and O i has radius ≤ r OPT . However, since |B max | < n t i -z k -i +1 -ϵz βk < n t i -z k -i +1 ≤ | O i | , we must have that r < r OPT .

Left to prove is that until time t ′ = t + t ∗ , with t ∗ = Ω( ϵz k ) , we do not need this instance of r . To this end, let B t max be B max at time t . Consider the situation at time t . We know that |B t max | < n t i -z k -i +1 -ϵz βk .

Let t ′ be some time after t such that |B t ′ max | ≥ n t ′ i -z k -i +1 . The cluster B P i ( p, 2 r ) in level i is now covering sufficiently many points. We want to derive a lower bound for t ∗ = t ′ -t to complete the proof. By examining the change in the size of B max and using that n t ′ i ≥ n t i -t ∗ , we derive

as a lower bound for t ∗ . Solving for t ∗ gives

Since β is a constant, this is in Ω( )

ϵz k . This concludes the proof.

F Missing proofs from Section 3.5

Lemma F.1. The amortized update time of our dynamic algorithm is O ( ϵ -3 k 6 log( k ) log(∆)) .

Proof. Let us fix an arbitrary time t and assume that at time t , there are λ ≤ k clusters. Let time t ′ = t + t ∗ be the time at which we need to invoke Procedure 5 on an arbitrary level i ≤ k , due to the dense invariant being violated. We have two cases:

➊ z +1 ≤ n t i -z 4( k -i +1) : For this case, Lemma 3.10 shows the following: If we find a cluster such that |B P i ( c i , 2 r ) | ≥ n t i -z 2( k -i +1) at time t , then this cluster will remain dense for t ∗ ≥ n t i -z 4( k -i +1) update operations (either insert or delete).

➋ z +1 > n t i -z 4( k -i +1) : For this case, Lemma 3.11 proves that if we find a cluster such that |B P i ( c i , 2 r ) | ≥ n t i -z k -i +1 -ϵz βk at time t , then this cluster will remain dense for the next t ∗ ≥ ϵz 2 k ( 1 α -1 β ) time steps. Asymptotically, t ∗ = Ω( ϵz k ) . By substituting the lower bound of z in this formula, we obtain t ∗ = Ω( ϵn t i k 2 ) .

Observe that in both cases, we have a worst-case guarantee for the number of time steps during which the cluster B P i ( c i , 2 r ) remains dense. This, in turn, allows us to achieve a worst-case amortized update time, rather than the weaker notion of expected amortized update time. We analyze the update time of each case separately.

Let us start with the first case which is z +1 ≤ n t i -z 4( k -i +1) and suppose that we find a cluster such that |B P i ( c i , 2 r ) | ≥ n t i -z 2( k -i +1) at time t . We know that t ∗ ≥ n t i -z 4( k -i +1) . The cost of reclustering levels i, . . . , k according to Lemma 3.3 is O ( n t ′ i ϵ -1 · k 3 log k ) , with n t ′ i ≤ n t i + t ∗ . Then, the amortized update time of an arbitrary update operation is O ( 1 t ∗ ( n t i + t ∗ ) ϵ -1 k 3 log k ) = O ( n t i t ∗ ϵ -1 k 3 log k ) . Since z ≤ n t i 4( k -i +1) , we obtain n t i t ∗ ≤ n t i · 4( k -i +1) n t i -z ≤ 4 k · n t i n t i -z ≤ 4 k · n t i n t i (1 -1 4( k -i +1) ) = 4 k · 4( k -i +1) 4( k -i +1) -1 = O ( k ) . Thus, O ( n t i t ∗ ϵ -1 k 3 log k ) = O ( ϵ -1 k 4 log k ) .

Now, we consider the second case. Using a similar analysis as in the first case, we have t ∗ = Ω( ϵn t i k 2 ) . Then, the amortized update time of an arbitrary update operation is O ( n t i t ∗ ϵ -1 k 3 log k ) = O ( ϵ -2 k 5 log k ) .

After reclustering levels i, . . . , k , there is no longer a lower bound for t ∗ for levels j < i . Thus, it could happen that such a level j needs to be reclustered soon after time t ′ . Since j < k , this leads to an extra factor k such that the final amortized cost is O ( ϵ -2 k 6 log k ) .

Finally, we need to consider the case that offline clustering fails for a level i . That is, we fail to find a ball covering sufficiently many points in Procedure 5. If offline clustering fails on some level i in case 1, meaning we fail to find a ball such that |B P i ( c i , 2 r ) | ≥ n t i -z 2( k -i +1) , then Lemma E.1 proves that with probability 1 -1 e Ω(log k ) , the guess r is small (i.e., r < r OPT) and indeed, remains small and we do not need to consider this guess until a time t ′ = t + t ∗ where t ∗ ≥ n t i -z 2( k -i +2) .

At time t ′ , we recluster levels i, . . . , k . Then, since we are in the first case, we have z ≤ n t i 4( k -i +1) . Thus, the amortized update time of an arbitrary update operation in this case is O ( 1 t ∗ ( n t i + t ∗ ) ϵ -1 k 3 log k ) = O ( n t i t ∗ ϵ -1 k 3 log k ) = O ( ϵ -1 k 4 log k ) using Lemma 3.3.

Next, we consider the situation when the offline clustering fails in case 2. Specifically, this occurs when we fail to find a ball such that |B P i ( c i , 2 r ) | ≥ n i -z k -i +1 -ϵz βk . According to Lemma E.2, the guess r is small (i.e., r < r OPT) with probability 1 -1 e Ω(log k ) and indeed, remains small and we do not need to consider this guess until time t ′ = t + t ∗ where t ′ = Ω( ϵz k ) . This leads to an amortized update time of O ( ϵ -2 k 5 log k ) . As mentioned above, the running time needs to be multiplied by an additional factor k to account for the possible need to recluster lower levels.

Finally, we need to consider O ( log ∆ log(1+ ϵ ) ) guesses for the optimal radii. For small ϵ , this is in O ( log ∆ ϵ ) . Thus, we derive the final amortized update time of O ( ϵ -3 k 6 log k log ∆) .

G How to support discrete k -center clustering with outliers

Lemma G.1. Let ( M,d ) be a metric space and ϵ > 0 be an accuracy parameter. The spread ratio ∆ = d max d min of all points ever inserted is assumed to be bounded. There exists a randomized fully dynamic algorithm that maintains a discrete k -center solution that allows up to (1+ ϵ ) z many outliers on the current set of points. At every point in time t , the current clustering with centers c 1 , . . . , c λ is a (4 + ϵ ) -approximation to an optimal solution for the ( k, z ) -center problem with high probability and c i ∈ P t for all i ≤ λ . Upon insertion or deletion of a point, the data structure is updated in amortized update time O ( ϵ -3 k 6 log( k ) log(∆)) .

Proof. Let r and i be fixed, and c i be the center of cluster C i = B P i ( c i , 4 r ) . As long as c i is not deleted, we report c i as the i -th center. Note that any feasible solution for the discrete version is also a feasible solution for the non-discrete version. Therefore, the optimal radius for the discrete version is not smaller than the optimal radius for the non-discrete version.

After the deletion of the point c i , we consider two cases: min ( z +1 , n i -z k -i +1 -εz αk ) ≤ 0 or min ( z +1 , n i -z k -i +1 -εz αk ) > 0 . If min ( z +1 , n i -z k -i +1 -εz αk ) ≤ 0 , then n i < (1 + ε ) z as i ≥ 1 and α > 1 . Hence, we can report all the n i points as outliers and stop. We next consider the second case. The dense invariant states that |B P i ( c i , 2 r ) | ≥ min( z +1 , n i -z k -i +1 -εz αk ) . Therefore, |B P i ( c i , 2 r ) | > 0 holds in the second case, and there exists a point ˆ p ∈ B P i ( c i , 2 r ) . Then we report an arbitrary point ˆ p ∈ B P i ( c i , 2 r ) after the deletion of c i . Note that we do not replace c i by ˆ p in our data structure, and c i does not change in our data structure as long as the i -th level is not reconstructed. The point ˆ p is just the center that we report for the discrete version.

We next prove the 6 -approximation guarantee. To do this, we show that C i ⊆ B (ˆ p, 6 r ) . This implies that any feasible solution with radius 4 r for the non-discrete version can be used to report a feasible solution for the discrete version. Also, see Figure 4 for a visual representation. Let q ∈ C i . We show that q ∈ B (ˆ p, 6 r ) . Since q ∈ C i = B ( c i , 4 r ) , we have d ( q, c i ) ≤ 4 r . Moreover, d ( c i , ˆ p ) ≤ 2 r since ˆ p ∈ B ( c i , 2 r ) . Then by the triangle inequality, we have

To report a solution for the discrete version, it is enough to keep B ( c i , 2 r ) ∩ P i . Note that our data structure already stores C i ∩ P i , and since B ( c i , 2 r ) ∩ P i ⊆ C i ∩ P i , the space complexity and time complexity remain the same.

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Answer: [Yes]

Justification: The problem setting stated at the beginning of the abstract summarizes this paper's scope as formalized in Section 1.3. The main statements made in the abstract can be found in Theorem 1.1 in Section 1.1, which is a summarization of Lemma 3.7 in Section 3.2 and Lemma F.1 in Section 3.5.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a simple but effective method that can solve \"fully dynamic k center problem with outliers\" in general metric space. The fully dynamic setting requires the algorithm to adjust its output efficiently when deletion or insertion operations  occur. This paper uses a \"ball cover\" strategy to obtain a $(4+\\epsilon)$-approximate solution of k center problem with outliers. The radii of balls will be guessed close enough to the optimal k center radius after at most $O(\\log \\Delta)$ iterations. When a deletion or insertion occurs, the algorithm needs to re-cluster (offline algorithm) if necessary. However, the paper proves that the re-clustering operation will not happen frequently. As a result, even though the time complexity of re-clustering may be as high as $O(n\\epsilon^{-1}k^3\\log k \\log \\Delta)$ in worst case, the amortized time can be reduced to $\\epsilon^{-2}k^6\\log k \\log \\Delta$ for handling the insertion or deletion of a single point.\n\nSTRENGTHS:\n1. The approximate ratio of the proposed algorithm is $4+\\epsilon$, far better than previous paper like [5].\n2. The algorithm of this paper do not need extra assumption of metric space, *e.g.*, low doubling dimension assumption in [2], which makes the algorithm have strong applicability.\n3. The presentation is nice.\n\nWEAKNESSES:\n1. Time complexity. The amortized update time $\\epsilon^{-2}k^6\\log k \\log \\Delta$ is worse than the method in [5] (if $k$ is large), whose time complexity is $O(\\frac{k^2\\log \\Delta}{\\epsilon^2\\tau}\\log^2\\frac{1}{\\delta})$​.\n2. High update time complexity in the worst case. The proposed algorithm still need  $O(n\\epsilon^{-1}k^3\\log k \\log \\Delta)$  time to handle an insertion or deletion of a single point in the worst case.\n3. No experimental result. Although this work is mostly theoretical, it does not means that experiment is not necessary (especially for NeurIPS, not a purely theory venue). The previous work [5] provides the experimental results, so I strongly encourage the authors to compare the performance of both running time and cost to [5].\n\nQUESTIONS:\n1. About the time complexity of binary search of $r_{OPT}$, you have mentioned that the set of all possible $r$ is $R = \\{(1+\\epsilon)^i:d_{min}\\le (1+\\epsilon)^i\\le d_{max}, i\\in N\\}$. So $|R| = \\log_{1+\\epsilon}\\Delta = \\frac{\\log \\Delta}{\\log (1+\\epsilon)}$. So if you binary search in the set $R$, it seems like that you only need $O(\\log |R|) = O(\\log \\log \\Delta)$ steps. So can your time complexity improved to $O(\\epsilon^{-2}k^6\\log k \\log \\log \\Delta)$?\n2. If I want to delete some point which is also a center of the solution. I think this center point should also be deleted since this point has been removed from this metric space. But it seems that Procedure 2 in the paper do not consider this situation. How do your algorithm handle the deletion of the center points?\n3. What is the size of your memory usage?\n\nLIMITATIONS:\nPlease see weakness.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the k-center clustering problem with outliers in the fully dynamic setting. Specifically, given a metric space (M,d), in the (k,z)-clustering problem, the goal is to find at most k balls minimizing the maximum ball radius while excluding up to z points from the clustering. In the fully dynamic setting, the points are inserted or deleted from the underlying metric space, and the goal is to maintain the clustering faster than recomputing from scratch after each insertion or deletion.\n\nThe main contribution of this paper is a fully-dynamic, provable approximation algorithm for the (k,z) clustering problem that achieves (4+\\eps)-approximation ratio, and can support point updates in O(\\eps^{-2} k^6 \\log k \\log \\Delta), with the caveat that the algorithm covers all but at most (1+\\eps)z points (also known as a bi-criteria guarantee in the approximation algorithms literature). The algorithm is randomized, and works against an oblivious, online adversary. It also improves upon the approximation ratio of (14+\\eps) achieved in the recent work Chan, Lattanzi, Sozio, and Wang [5] for the same problem, at the cost of increasing the running time by a factor of k^4.\n\nFrom a technical point of view, the paper first introduces a static (4+\\epsilon) approximation algorithm for the problem. The algorithm is itself based on the idea of successive random sampling, which proceeds as follows:\n\n*  guess an optimal radius r (this is standard in (clustering) algorithms, and can be achieved by binary searching at the cost of paying (1+\\eps) in the approx. factor, and a factor that is log of aspect ratio in the underlying metric space in the runtime);\n* pick uniformly at random a subset of points S of the current point set, and pick a center c from S that covers a good fraction of the current points; \n* grow a ball of radius 4r around the center c; \n* remove the points belonging to the ball from the current point sets, and continue with the next iteration (level)\n* the number of levels is bounded by roughly k\n* the centers computed in each level are output as the solution the (k,z) clustering problem\n\nThe bulk of the effort is then to carefully set up a data-structure that maintains these levels (and the associated information) under point insertion and deletions. There are two main invariants: (i) the level invariant and (ii) the dense invariant (which is roughly controlling the number of points that remain unclustered in each level). The moral message of the paper is that when the invariants are broken at some level i, then you basically re-cluster everything from that level up.\n\nSTRENGTHS:\n*The k-center clustering is a fundamental problem in the clustering/unsupervised learning literature. The outlier version of the problem makes perfect sense, and it's of equal importance. The dynamic setting is very natural and has received increasing attention in the clustering literature in the last 5 years or so, with many papers trying to nail down the dynamic complexity of basic clustering objectives such as k-center, k-median and k-means. This work can be thought as a continuation of this line of work. \n\n* The algorithm is simple and elegant (it should also be very implementable). Appropriate effort is put in explaining high level ideas. The paper reads well.\n\nWEAKNESSES:\n* Personally, I wouldn't agree that this is an improvement over the work of Chan et al. [5]. Indeed, it does achieve a better approximation ratio, but at the cost of increasing the runtime. The paper indeed provides a new, interesting trade-off, but I believe the abstract we should discuss this more carefully.\n\n* Regarding techniques, while one can argue that idea of successive sampling is now folklore, one influential work that employs a very similar algorithm is due to Mettu-Plaxton \"Optimal Time Bounds for Approximate Clustering\". While the original algorithm is for k-median and k-means, there are follow up works that employ the same algorithm for k-center (albeit with worst approx guarantee than 2). I would encourage the authors to discuss differences/similarities between Mett-Plaxton and their work. In my opinion, this doesn't diminish the contribution of the paper at hand.\n\nQUESTIONS:\nMinor comments:\n\nLines 17-29, you should back up with citations the applications of clustering across many subfields of computer science and beyond\nLine 36, 'various complications' sounds a bit odd (maybe challenges would be a better fit here)\nLine 196, n_i, \\alpha never defined before in text\nLine 219, should your j start from 1 and not from i? -- similar comment for Definition 3.2\n\nMajor comment/question:\nLine 146 -- can you please elaborate whether non-discrete version of the (k,z) clustering you study here is less challenging? I imagine this helps you a lot with deletions; even if you delete a point in P, you can still leave the underlying point in M serve as cluster center, which doesn't invoke a re-build. Does the work of Chan et al. [5] have the some restriction?\n\nLIMITATIONS:\nYes,", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the fully-dynamic $k$-center with outliers problem in the metric space. In this setting, operations (including insertion and deletion) appear over time. The performance evaluation of an algorithm is based on its cost approximation and the (amortized) update time. However, previous research has shown that when exactly excluding $z$ outliers is required, any $O(1)$-approximation algorithm incurs an $\\Omega(z)$ update time. It is therefore reasonable to allow for a trade-off by permitting $(1+\\varepsilon)z$ outliers in order to achieve efficient update times. \n\nThe proposed algorithm can be summarized as follows: It partitions the dataset into into at most $k+1$ levels, where each level $i$ represents a cluster $C_i$ with center $c_i$. The algorithm dynamically maintains a data structure that adapts to the operations performed in real-time. At every time, the data structure should maintain the following two invariants:\n\n- level invariant: Each level $i$ has a cluster $C_i$ such that $P_{i+1} = P_i \\backslash C_i$.\n- dense invariant: For each center $c_i$ in $F_r$, $B(c_i, 2r)$ covers sufficient points in $P_i$.\n\nThe authors prove that it is a $(4+\\varepsilon)$-approximation algorithm with $O(\\varepsilon^{-2}k^6\\log k\\log \\Delta)$ expected amortized update time.\n\nSTRENGTHS:\n- The paper is very well written and easy to follow.\n- The algorithm provides a $(4+\\varepsilon)$-approximation algorithm for the fully-dynamic $k$-center with $z$ outliers problem, which is an improvement over previous work.\n- The paper includes a thorough theoretical analysis, proving the approximation guarantee and update time of the algorithm.\n\nWEAKNESSES:\n- the update time complexity $O(\\varepsilon^{-2}k^6\\log k\\log \\Delta)$ is not competitive, particularly for large $k$.\n- The paper does not provide empirical results or practical evaluations of the algorithm, which could demonstrate its performance in real-world scenarios.\n\nQUESTIONS:\n- can this method be slightly modified to support the change of $k$ and $z$. Alternatively, is there any negative results on handling the dynamic $k$ and $z$.\n- iDoes the analysis of the algorithm include a specific space complexity?  Additionally, i am curious about the hardness of the fully-dynamic clustering with outliers. Can you provide the lower bounds of the space and the update time in the context of the fully-dynamic setting?\n\nLIMITATIONS:\nthoretical paper", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

comments: SUMMARY:
The paper gives a new algorithm for the dynamic version of k-center clustering with outliers. The algorithm works in the fully dynamic model with both point insertions and deletions allowed. The points can belong to an arbitrary metric space, compared to some previous algorithms addressing low-dimensional metric spaces.

STRENGTHS:
This is an important problem and the paper significantly improves the approximation factor.

WEAKNESSES:
The algorithm is randomized and works only against oblivious adversaries. This is a shared characteristic with the previous paper on this topic.

The algorithm achieves a better approximation factor than the previous algorithms, but it comes at the cost of significantly higher dependency in the update time on the number of clusters, k. In particular, the factor of $k^6$ is probably prohibitive in practice.

Overall, even though this is a solid contribution to the study of the problem, this is not a breakthrough.

I also wish the authors did a better job highlighting the technical ideas that lead to the improvement over the previous algorithm.

QUESTIONS:
Is there a good reason why the algorithm is randomized? What are the obstacles to obtaining deterministic algorithms in this model?

The factor of $k^6$ in your upper bound is large. Is there a reason why in practice the algorithm would be significantly less costly?

LIMITATIONS:
No specific societal limitations to address. This is a theoretical work concerning a traditional computational problem.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-16