Two applications of Min-Max-Jump distance

Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 21:17:54

Abstract: We explore two applications of Min-Max-Jump distance (MMJ distance): MMJ-based K-means and MMJ-based internal clustering evaluation index.  K-means and its variants are possibly the most popular clustering approach. A key drawback of K-means is that it cannot deal with data sets that are not the union of well-separated, spherical clusters. MMJ-based K-means proposed in this paper overcomes this demerit of K-means, so that it can handle irregularly shaped clusters. Evaluation (or "validation") of clustering results is fundamental to clustering and thus to machine learning. Popular internal clustering evaluation indices like Silhouette coefficient, Davies–Bouldin index, and Calinski-Harabasz index performs poorly in evaluating irregularly shaped clusters. MMJ-based internal clustering evaluation index uses MMJ distance and Semantic Center of Mass (SCOM) to revise the indices, so that it can evaluate irregularly shaped data. An experiment shows introducing MMJ distance to internal clustering evaluation index, can systematically improve the performance. We also devise two algorithms for calculating MMJ distance.

Main paper: 
 k-means: 
 K-means (7) and its variants (8; 9; 10) are possibly the most well-liked clustering approach. K-means divides the data into K groups, where K is a hyper-parameter to be optimized. It aims to reduce the within-cluster dissimilarity. While popular, K-means and its variants perform poorly for data sets that are not the union of well-separated, spherical clusters. MMJ-based K-means (MMJ-K-means) proposed in this paper overcomes this demerit of K-means, so that it can handle irregularly shaped clusters.Evaluation (or "validation") of clustering results is as difficult as the clustering itself (11) . Popular approaches involve "internal" evaluation and "external" evaluation. In internal evaluation, a clustering result is evaluated based on the data that was clustered itself. Popular internal evaluation indices are Davies-Bouldin index (12) , Silhouette coefficient (13) , Dunn index (14) , and Calinski-Harabasz index (15) etc. In external evaluation, the clustering result is compared to an existing "ground truth" classification, such as the Rand index (16) . However, knowledge of the ground truth classes is almost never available in practice.In Section 5.2, an experiment shows introducing Min-Max-Jump (MMJ) distance to internal clustering evaluation index, can systematically improve the performance.Euclidean distances are frequently used in machine learning and clustering methods to compare points. However, the distance is data-independent, and not tailored to the geometry of the data. Many metrics that are data-dependent have been devised, such as diffusion distances (17) and path-based distances (18; 19) . MMJ distance is a path-based distance.3 Definition of Min-Max-Jump Definition 1. Min-Max-Jump distance (MMJ distance)Ω is a set of points (at least one). For any pair of points p, q ∈ Ω, the distance between p and q is defined by a distance function d(p,q) (such as Euclidean distance). i, j ∈ Ω, Ψ (i,j,n,Ω) is a path from point i to point j, which has length of n points (see Table 1 ). Θ (i,j,Ω) is the set of all paths from point i to point j. Therefore, Ψ (i,j,n,Ω) ∈ Θ (i,j,Ω) . max_jump( Ψ (i,j,n,Ω) ) is the maximum jump in pathΨ (i,j,n,Ω) .The Min-Max-Jump distance between a pair of points i, j, which belong to Ω, is defined as:ii Ψ (i,j,n,Ω)Ψ (i,j,n,Ω) is a sequence from point i to point j, which has length of n points. All the points in the sequence must belong to set Ω. That is to say, it is a path starts from i, and ends with j.For convenience, the path is not allowed to have loops, unless the start and the end is the same point;d(i, j) d(i, j) is a distance metric between pair of points i and j, such as Euclidean distance;max_jump( Ψ (i,j,n,Ω) ) max_jump( Ψ (i,j,n,Ω) ) is the maximum jump in path Ψ (i,j,n,Ω) . A jump is the distance from two consecutive points p and q in the path;Θ (i,j,Ω) Θ (i,j,Ω)is the set of all paths from point i to point j. A path in Θ (i,j,Ω) can have arbitrary number of points (at least two). All the points in a path must belong to set Ω;M M J(i, j | Ω) M M J(i, j | Ω)is the MMJ distance between point i and j, where Ω is the Context of the MMJ distance;EQUATIONEQUATIONWhere ϵ is a path from point i to point j, max_jump(ϵ) is the maximum jump in path ϵ. Π is the set of all maximum jumps. min(Π) is the minimum of Set Π.Set Ω is called the Context of the Min-Max-Jump distance. It is easy to check M M J(i, i | Ω) = 0.In summary, Min-Max-Jump distance is the minimum of maximum jumps of all path between a pair of points, under the Context of a set of points.Similar distances have actually been studied in many places in the literature, including the maximum capacity path problem, the widest path problem, the bottleneck edge query problem, the minimax path problem, the bottleneck shortest path problem, and the longest-leg path distance (LLPD) (20; 21; 22; 23) .iii Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [No] Justification: We have used 145 datasets to test the models in the paper. Maybe it is not enough, we need more datasets to test the models. 
an example: 
 Suppose Set Ω is composed of the four points in Figure 1 . There are five (non-looped) paths from point a to point c in Figure 1 :1. a → c, the maximum jump is 28; To understand Min-Max-Jump distance, imagine someone is traveling by jumping in Ω. Suppose M M J(i, j | Ω) = δ. If the person wants to reach j from i, she must have the ability of jumping at least δ. Otherwise, j is unreachable from i for her. Whether the distance to a point is "far" or "near" is measured by how far (or how high) it requires a person to jump. If the requirement is large, then the point is "far", otherwise, it is "near."Theorem 1. Suppose i, j, p, q ∈ Ω ,EQUATIONEQUATIONthen,EQUATIONwhere d(x,y) is a distance function (Table 1 ).Proof. M M J(i, j | Ω) = δ is equivalent to ∃P ∈ Θ (i,j,Ω) , such that M (P ) = δ, and ∀T ∈ Θ (i,j,Ω) , M (T ) ≥ δ, where Θ (i,j,Ω) is the set of all paths from point i to point j under context Ω. M (P ) is the maximum jump in path P . We can assume M M J(p, q | Ω) > δ and M M J(p, q | Ω) < δ, then we will arrive to a contradiction in both cases. iv Theorem 2. Suppose r ∈ {1, 2, . . . , n},EQUATIONEQUATIONFor the meaning of Ω t , Ω r , Ω [1,n] , and Ω [1,n+1] , see Table 1 .Proof. There are n possibilities of the MMJ path from Ω n+1 to Ω r , under the context ofΩ [1,n+1] ,set X enumerate them all. Each element of X is the maximum jump of each possibility. Therefore,according to the definition of MMJ distance, M M J(Ω n+1 , Ω r | Ω [1,n+1] ) = min(X).Corollary 1. Suppose r ∈ {1, 2, . . . , N }, p / ∈ Ω,EQUATIONEQUATIONFor the meaning of Ω + p, see Table 1 .Proof. The proof follows the conclusion of Theorem 2.Theorem 3. Suppose i, j ∈ {1, 2, . . . , n},EQUATIONEQUATIONEQUATIONx 2 = max(t 1 , t 2 ) (16) then,EQUATIONProof. There are two possibilities of the MMJ path from Ω i to Ω j , under the context of Ω [1,n+1] :Ω n+1 is in the path or it is not in the path. x 2 is the min-max jump of the first possibility; x 1 is the min-max jump of the second possibility. Therefore, according to the definition of MMJ distance,M M J(Ω i , Ω j | Ω [1,n+1] ) = min(x 1 , x 2 ).Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? xii Answer: [Yes] Justification: Proofs of theoretical results have been provided in the paper. 
calculation of min-max-jump distance: 
 We propose two methods to calculate the pairwise Min-Max-Jump distance matrix of a dataset. There are other methods for calculating or estimating it, such as a modified SLINK algorithm (25) , or with Cartesian trees (26; 27) , or from a sequence of nearest neighbor graphs (23) , or a modified version of the Floyd-Warshall algorithm.The first method calculates M Ω by recursion. M Ω is the pairwise MMJ distance matrix of Ω (Table 1 ). M k,Ω [1,k] is the MMJ distance matrix of the first k points of Ω (Table 1 ). Note M 2,Ω [1, 2] is simpleto calculate. M Ω = M N,Ω [1,N ] .M Ω is a N × N symmetric matrix. Rows and columns of M Ω are indexed from 1 to N.Step 7 of Algorithm 1 can be calculated with the conclusion of Theorem 2;Step 12 of Algorithm 1 can be calculated with the conclusion of Theorem 3.Algorithm 1 has complexity of O(n 3 ), where n is the cardinality of Set Ω.v Algorithm 1 MMJ distance by recursionInput: Ω Output: M Ω 1: function MMJ_BY_RECURSION(Ω) 2:N ← length(Ω)3:Initialize M Ω with zeros4: Calculate M 2,Ω [1,2] , fill in M Ω [1, 2] and M Ω [2, 1] 5:for n ← 3 to N do 6:for r ← 1 to n -1 do7: Calculate M M J(Ω n , Ω r | Ω [1,n] ), fill in M Ω [n, r] and M Ω [r, n] 8:end for 9:for i ← 1 to n -1 do 10:for j ← 1 to n -1 do 11: if i < j then 12: Calculate M M J(Ω i , Ω j | Ω [1,n] ), update M Ω [i, j] and M Ω [j, i] 13:end if return M Ω 18: end functionAccording to the conclusion of Theorem 1, there are many duplicated values in M Ω . So in the second method we can calculate the MMJ distance value in one position and copy it to other positions inM Ω .A well-known fact about MMJ distance is: "the path between any two nodes in a minimum spanning tree (MST) is a minimax path." A minimax path in an undirected graph is a path between two vertices v, w that minimizes the maximum weight of the edges on the path. That is to say, it is a MMJ path.By utilizing this fact, we propose Algorithm 2.Input: Ω Output: M Ω 1: function MMJ_CALCULATION_AND_COPY(Ω) 2: Initialize M Ω with zeros 3: Construct a MST of Ω, noted T 4:Sort edges of T from large to small, generate a list, noted Lfor e in L do 6:Remove e from T . It will result in two connected sub-trees, T 1 and T 2 ; 7:Traverse T 1 and T 2 ;For all pair of nodes (p, q), where p ∈ T 1 , q ∈ T 2 . Fill in M Ω [p, q] and M Ω [q, p] with the weight of e. Unlike Algorithm 1, which accesses each cell of M Ω for O(n) times. The merit of the "Calculation and Copy" method is that it is easier to understand than using the Cartesian trees (26; 27) . Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: We have fully disclosed all the information needed to reproduce the main experimental results of the paper. 
mmj-based k-means: 
 K-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean (cluster center or centroid), serving as a prototype of the cluster (28) . Standard K-means uses Euclidean distance. We can revise K-means to use Min-Max-Jump distance, with the cluster centroid replaced by the Semantic Center of Mass (SCOM) (particularly, One-SCOM) of each cluster. For the definition of SCOM, see a previous paper (29) . One-SCOM is like medoid, but has some difference from medoid. Section 6.3 of (29) compares One-SCOM and medoid. In simple terms, the One-SCOM of a set of points, is the point which has the smallest sum of squared distances to all points in the set.Standard K-means usually cannot deal with non-spherical shaped data, such as the ones in Figure 2 .MMJ-based K-means (MMJ-K-means) can cluster such irregularly shaped data. Figure 2 compares Standard K-means and MMJ-K-means, on clustering three data which come from the scikit-learn project (30) . The Silhouette coefficient for a single sample is given as:s = b -a max(a, b)where a is the mean distance between a sample and all other points in the same class. b is the mean distance between a sample and all other points in the next nearest cluster. The Silhouette coefficient for a set of samples is given as the mean of Silhouette coefficient for each sample.We can also revise Silhouette coefficient to use Min-Max-Jump distance, forming a new internal clustering evaluation index called MMJ-based Silhouette coefficient (MMJ-SC). We tested the performance of MMJ-SC with the 145 datasets mentioned in another paper (31) . MMJ-SC obtained a good performance score compared with the other seven internal clustering evaluation indices mentioned in the paper (31) . Readers can check Table 2 and compare with Table 5 of Liu's paper (31) .were also tested. In calculation of these two indices, besides using MMJ distance, the center/centroid of a cluster is replaced by the One-SCOM of the cluster again, as in MMJ-K-means. It can be seen that MMJ distance systematically improves the three internal clustering evaluation indices (Table 2 ). The best performer is MMJ-CH, which achieves an accuracy of 90/145. The accuracy of an index is computed by evaluating the index's ability of recognizing the best partition of a dataset from hundreds of candidate partitions(31).The  
using pam: 
 Since One-SCOM is like medoid, in MMJ-K-means, we can also use the Partitioning Around Medoids (PAM) algorithm or its variants to find the One-SCOMs (32).There might be multiple One-SCOM points in a cluster, which have the same smallest sum of squared distances to all the points in the cluster. Usually they are not far from each other. We can arbitrarily choose one or keep them all. If we keep them all, then the One-SCOM of a cluster is not a point, but a set of points. If the One-SCOM is a set, when calculating a point's MMJ distance to the One-SCOM of a cluster, we can select the minimum of the point's MMJ distances to all the One-SCOM points.Border points defined in Definition 2 can further be differentiated as weak and strong border points.A point is defined to be a WBP if its nearest mean (center or One-SCOM) is not unique but less than K, where K is the number of clusters.A point is defined to be a SBP if its nearest mean (center or One-SCOM) is not unique and equals K, where K is the number of clusters.Then we can process different kinds of border points with different strategies. E.g., deeming the Strong Border Points as outliers and removing them. Guidelines:Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: Full details are provided with the code. 
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 Hamming distance is the number of values that are different between two vectors. It is typically used to compare two binary strings of equal length (1) .Manhattan distance is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates (2) .Chebyshev distance is defined as the greatest of difference between two vectors along any coordinate dimension (3) .Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance (4) .Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used for gauging the similarity and diversity of sample sets (5) .Haversine distance is the distance between two points on a sphere given their longitudes and latitudes.It is similar to Euclidean distance in that it calculates the shortest path between two points. The main difference is that there is no straight line, since the assumption is that the two points are on a sphere (6) . 
 
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Decision: The four reviewers are well-qualified and diverse, so the fact that none recommend acceptance is dispositive. The author has responded in detail, but has not changed the mind of any reviewer. Reviewer vcc5 is a relevant expert and is sympathetic. However, s/he still recommends rejection, like the other reviewers.

The author must understand that the writing changes needed are major. Employing an editor to improve the English in a surface way is likely not enough. Instead, the paper needs to follow the expository conventions of modern ML research, with more comprehensive, more precise, and more thoughtful qualitative discussion. For example, it is not sufficient simply to cite a web page for the meaning of the term "non-flat." And many previous papers have provided algorithms for clustering data such as that in Figures 2, 3, and 4, for example Manifold Clustering by Souvenir and Pless in Proceedings of the 10th International Conference on Computer Vision (ICCV 2005), pp. 648 – 653.

It is possible that this paper should be two separate submissions, one that is purely algorithmic for supposedly the fastest algorithm for computing all points path distances, and one for the application of this algorithm to clustering.

Reviews: review: paper_summary: paper_summary: Different distance metrics have been introduced in the literature for data analysis. In this paper the authors consider the min-max-jump distance and apply it in the context two applications, namely, k-means clustering and as an internal clustering evaluation index. They also present two algorithms for computing the min-max-jump distance.

Experimental comparisons reveal that min-max-jump based k-means clustering is better than standard k-means clustering. Also, the min-max-jump distance is shown to be a better internal clustering evaluation index.

This referee feels that this work is rather incremental. Also, experiments have been conducted only on very small datasets.
strength_weakness: strength_weakness: The work done is incremental with very limited novelty.
Extensive experiments are called for.
questions: questions: None
limitations: limitations: None
paper_summary: paper_summary: This paper presents the Min-Max-Jump (MMJ) distance concept and two calculation methods, focusing on path optimization in data analysis and clustering. The contributions include introducing MMJ distance, proposing efficient calculation methods, discussing its properties and applications, and offering a user-friendly approach for practical implementation. Overall, the paper introduces a new distance metric for path optimization and data analysis, providing useful tools and insights for various applications in the field.
strength_weakness: strength_weakness: W1: The paper's writing style deviates from academic norms, indicating a need for improvement in writing proficiency.

W2: The extremely brief Introduction lacks a detailed definition of the problem, its significance, and challenges. Moreover, it lacks citation support for the points presented. While Section 2.1 mentions methods like k-NN, UMAP, HDBSCAN, it fails to provide corresponding references, lacking essential academic backing.

W3: The overall structure of the paper lacks clarity, as it introduces different distance metrics in Section 2.1 but introduces a new distance measurement approach in Section 2.4, leading to disjointed logic.

W4: The presentation of various distance metrics in Section 2.1 appears disorganized and lacks coherence.

W5: The extensive definition provided towards the end of Section 6.3 disrupts the logical flow of the paper, suggesting a need to adjust the paper's structural coherence.
questions: questions: Q1:Can the authors provide more insights into the practical implications of their research findings and how they can be applied in real-world scenarios?
limitations: limitations: The paper does not discuss limitations. The authors seem to perceive their work as solely testing models with datasets without considering the shortcomings of the algorithms themselves.
paper_summary: paper_summary: This paper proposes a new metric, min-max jump distance. Effectively, say we are given a complete graph with vertex set $\Omega$ and edge weights $d(x,y)$ denoting the distance between $x$ and $y$, where $d$ is a metric. Then $MMJ(x,y|\Omega')$ is the minimum, over all paths between $x$ and $y$, of the maximum weight edge between $x$ and $y$ on the subgraph induced on the vertices $\Omega' \subseteq \Omega$. Explained nicely in the paper, if you started at vertex $x$ and wanted to get to $y$ and $d(\cdot,\cdot)$ denoted the distance required to ``jump'' from one vertex to another, what is the minimum distance you need to be able to jump to somehow traverse from $x$ to $y$?

This is a nice intuitive metric, and has strong connections to the minimum spanning tree. In fact, I suspect there is more literature to draw from minimum spanning tree research that could yield conclusions about MMJ. The MST is also nice in clustering since oddly shaped clusters (non-convex, for instance) can have small MSTs. This is the idea of MMJ: use it as a metric for K-means so that it can identify non-spherical clusters.

The paper proves some notable theory about the properties of MMJ. Mostly, they show how: 1) When adding a new vertex $p$ to a set $\Omega$, $p$'s MMJ within the context of $\Omega+p$ can be computed knowing all pairwise MMJ's within $\Omega$ within the context of $\Omega$. This effectively adds a new point and evaluates it within the complete, updated context. 2) Given the MMJs for this additional point $p$ within the context of $\Omega +p$, expand the MMJ context of all other pairs in $\Omega$ to the context of $\Omega+p$.

This can then be used in a very dynamic programming-like manner to start with just two points, add new points $p$ and find the context of $p$ and all other points, and then update all known existing MMJs to the new context. This is their algorithm 1, requiring $O(n^3)$ time. They also use properties of the MST to bypass unnecessary calculation to yield algorithm 2, which takes only $O(n^2)$ time (to find the MST).

They then evaluate the performance of algorithms using the MMJ measure on irregular shaped clusters to verify that MMJ helps identify these. This makes sense, since they likely have small MSTs, but not small average/sum/max/etc distances within clusters. Notably, they show how MMJ improves K-means.
strength_weakness: strength_weakness: There are a few notable downsides. In terms of the result, I'm not entirely convinced of its novelty. How much of this is actually a re-iteration of MST-based methods already understood? Is this really better than other MST-based algorithms on irregular clusters (think single linkage)? I know that there is a lot of literature that explores irregular shaped clusters, but I am not an expert in this area and so I cannot place this work in the context of existing results. I wish the authors would explain that. Though even if these algorithms aren't entirely better than state of the art, the novelty of the nice formulation of MMJ is certainly appreciated.

However, the biggest flaw in the paper is the writing quality. There are places where the paper is nice and concise, but most of the time it just lacks exposition to understand the higher level of things or adequate details to fully understand what is happening. Formal proofs are contained in the paper, but the jumps in some of the proofs are too large. Theorems and proofs are placed back to back with no high level explanation. Algorithms are written and pseudocode with only the briefest justifications, and no thorough explanations. This is not an acceptable paper for NeurIPS, and I think these issues are too extensive to simply ask the authors to revise. Though if other commenters disagree, I am amenable to changing my opinion.

And one of the disappointing things about this is how natural this work is and how much it lends itself to nice intuitive explanations and visual depictions! For instance, you could do some very nice visualizations of Algorithm 1, where you depict a matrix and show which indices have been calculated to what context $\Omega_n$ at each time point. This clarifies the different purpose of the two loops.

I hope to see this paper submitted again later in a more cleaned up state!
questions: questions: 1. What exactly is a path-based distance?

2. When you define MMJ, you do not specify exactly what a path is. All you introduce is a point set and a distance function. Are we to assume that a "path" is done on the complete graph with edge weights equal to the distances between points? Please clarify in the paper. I read the paper with this assumption.

3. How would MMJ compare to the similarly defined max-min jump measure? I would think it would be more akin to a similarity measure than a distance.

4. Line 114: what is M(P)? Also, how do you arrive at the contradiction in this proof? This needs more explanation!

5. Line 122: What do you mean there are n possibilities? Aren't there something like n! candidate paths, since you check all paths of all lengths with a fixed set of endpoints?

6. Thm 2 proof: You say X enumerates them all - what about the direct path that just traverses the single edge between the points?

7. Is Corollary 1 actually any different from Thm 2? It seems like in both you just have a set $\Omega_{[1,n]}$ and then you have a point outside of the set that you want to get the MMJ to (with respect to some reference point). I think Corollary 1 is just a cleaner way to state Theorem 2.

8. In Theorem 3, is it actually true that $x_2$ is the MMJ if $\Omega_{n+1}$ is in the path? It doesn't break your argument, but if the solution path doesn't go through $\Omega_{n+1}$, then it's possible that the MMJ path from i to n+1 and from j to n+1 share an edge. So the MMJ path from i to j that goes through n+1 could actually be worse since we aren't allowed to traverse that edge twice. Again, the proof still works, but I think that claim only holds if the MMJ path from i to j does, in fact, go through n+1. Otherwise, we might have something larger, which is ignored by taking the minimum.

9. Section 4.1: Why does k appear twice in the matrix subscripts? Couldn't it just be something like $M_k$?

10. How does MMJ-K-means perform on spherical data compared to K-means? Presumably, worse. It would be nice to quantify how much worse.
limitations: limitations: None notable
paper_summary: paper_summary: The paper proposes to use a new distance, Min-Max-Jump, which is the minimum largest distance on any path between two points, to be used in k-means clustering to learn clusters.
strength_weakness: strength_weakness: The distance in the paper is, in fact, related to single linkage clustering that assign give a pair of points a distance at which the pair is joint to one cluster. This need to be analyzed to relate to previous work as well as to compute pairwise distances efficiently. 

Theoretical property of the distance is poor. The paper should review many other density-based distance functions to put this work into the correct context. 

There would be a lot of problems using this distance as many of pairs of nodes would share the same distance. There is no analysis on the  metric property of this distance. 

On evaluation, the methods need to compare to single-linkage clustering (SSL) at the very least as all the advantages of using this distance with k-means are available in SLL in its simplest form.

Presentation-wise, it is hardly up the standard. There are methods/algorithms/concepts that are mentioned as "a is like b with a difference" without a formal definition. This mixes up definitions and properties.
questions: questions: N/A
limitations: limitations: The paper uses a new distance without theoretical justifications. It learns nonconvex clusters by using a nonconvex clustering-based distance (without explicitly mentioning it), which is hardly a novelty.

Comments: comment: I thank the authors for the rebuttal. As also in the rebuttal, the paper lacks much justifications to be accepted in this conference. I'd like to keep my score.
I appreciate the authors' responses, however the writing quality issue still stands and I believe the changes required are too great to allow acceptance.
Rebuttal has been read.