Fully Dynamick-Center Clustering

Abstract

Static and dynamic clustering algorithms are a fundamental tool in any machine learning library. Most of the eorts in developing dynamic machine learning and data mining algorithms have been focusing on the sliding window model (where at any given point in time only the most recent data items are retained) or more simplistic models. However, in many real-world applications one might need to deal with arbitrary deletions and insertions. For example, one might need to remove data items that are not necessarily the oldest ones, because they have been agged as containing inappropriate content or due to privacy concerns. Clustering trajectory data might also require to deal with more general update operations.We develop a (2 + )-approximation algorithm for the k-center clustering problem with "small" amortized cost under the fully dynamic adversarial model. In such a model, points can be added or removed arbitrarily, provided that the adversary does not have access to the random choices of our algorithm. The amortized cost of our algorithm is poly-logarithmic when the ratio between the maximum and minimum distance between any two points in input is bounded by a polynomial, while k and are constant. Our theoretical results are complemented with an extensive experimental evaluation on dynamic data from Twitter, Flickr, as well as trajectory data, demonstrating the eectiveness of our approach.

On fully dynamic constant-factor approximation algorithms for clustering problems

“…For this we need to distinguish between an adaptive adversary that can choose each operation based on all the algorithm's query answers so far (e.g., which points are centers) and an oblivious adversary that knows the algorithm, but does not see the actual answers of the algorithm (which might depend on random bits). So far there are only dynamic O(1)-approximation algorithms known for k-center, k-median, and k-means with expected update times of Õ(k 2 /ǫ O (1) ) against an oblivious adversary [8,19] (where Õ suppresses factors that are polylogarithmic in n, k, and ∆). In particular, for none of the problems listed above there is a dynamic algorithm known that works against an adaptive adversary.…”

“…Already for the deletion-only case, no data structure was known that maintains a O(1)-approximation with update time k(log n) O (1) . Similarly as Chan et al [8], we initialize our data structure by selecting the first center c 1 uniformly at random among all points in P =: U 1 and assign all points to c 1 that are covered by c 1 . We say that they form a cluster with center c 1 .…”

“…We define U 2 to be the remaining points and continue similarly by selecting a point c 2 uniformly at random from U 2 , etc. In contrast to [8], we assign the computed centers into buckets such that for all points c i in the same bucket, the size of the corresponding set U i differs by at most a factor of 2. We stop if we have covered all points or if a bucket B * has 2k centers; hence, B * must contain the last 2k chosen centers.…”

“…We describe now how we maintain the solutions x, x, y, S, S, and S dynamically when points are inserted or deleted. Our strategy is similar to [8].…”

“…Indeed, already in 2004 Charikar et al [3] gave two insertions-only algorithms: a deterministic 8-approximation and a randomized 5.43-approximation, both with O(k log k) update time per operation, which was improved to an (almost optimal) insertion-only (2 + ǫ)-approximation algorithm with update time O((k log k))/ǫ 4 ) [28] for any small ǫ > 0. The best known fully dynamic algorithm was given by Chan et al [8]. It is randomized and achieves an approximation ratio of 2 + ǫ with an expected update time of O((k 2 log ∆)/ǫ) against an oblivious adversary.…”