Faster and Better Solution to Embed Lp Metrics by Tree Metrics

Hierarchically Separated Tree (HST) is the most popular solution to embed a metric space into a tree metric. By using HSTs, many optimization problems, which are hard on defined metrics, become easier to get good approximation bounds with respect to the effectiveness, e.g., task assignment, trip planning, and facility location planning. Existing work focuses on constructing HSTs for arbitrary metric spaces, which makes a general-purpose algorithm take at least 𝑂 (𝑛 2 )-time to get tight distortion guarantees 𝑂 (log 𝑛). Here, distortion is a prevalent measurement of HSTs' effectiveness and usability. However, we observe that (1) in many applications that HSTs are applied, only 𝐿 𝑝 metrics are used (e.g., Euclidean space), (2) the state-of-the-art solution is still time-consuming to construct HSTs for large-scale data, and (3) distortions of existing algorithms are only satisfactory for high-dimensional data. Thus, in this paper, we are motivated to study the Embedding 𝐿 𝑝 metrics through Tree metrics (ELT) problem. We aim to design a faster algorithm than 𝑂 (𝑛 2 ) time to construct HSTs with not only 𝑂 (log 𝑛) distortion guarantees but also good and robust empirical results. Specifically, we first present a divide-and-conquer based general framework and prove that it has a distortion guarantee of 𝑂 (log 𝑛). To achieve a better time complexity than 𝑂 (𝑛 2 ), we next design two optimization techniques: reducing to nearest neighbor search (by indexing) and sampling. Finally, extensive experiments demonstrate that our algorithm DCsam outperforms the state-of-the-art algorithms by a large margin in terms of both distortion and running time.