We introduce a new number of new techniques for the construction of low-distortion embeddings of a finite metric space. These include a generic Gluing Lemma which avoids the overhead typically incurred from the naïve concatenation of maps for different scales of a space. We also give a significantly improved and quantitatively optimal version of the main structural theorem of Arora, Rao, and Vazirani on separated sets in metrics of negative type. The latter result offers a simple hyperplane rounding algorithm for the computation of an O ( √ log n)-approximation to the Sparsest Cut problem with uniform demands, and has a number of other applications to embeddings and approximation algorithms. 1

The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2-approximation algorithm for the Bandwidth problem that takes worstcase O(1.9797 n) = O(3 0.6217n) time and uses polynomial space. This improves both the previous best 2- and 3-approximation algorithms of Cygan et al. which have an O ∗ (3 n) and O ∗ (2 n) worst-case time bounds, respectively. Our algorithm is based on constructing bucket decompositions of the input graph. A bucket decomposition partitions the vertex set of a graph into ordered sets (called buckets) of (almost) equal sizes such that all edges are either incident on vertices in the same bucket or on vertices in two consecutive buckets. The idea is to find the smallest bucket size for which there exists a bucket decomposition. The algorithm uses a simple divide-and-conquer strategy along with dynamic programming to achieve this improved time bound. 1

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Abstract—Analysis of large networks is a critical component of many of today’s application environments, including online social networks, protein interactions in biological networks, and Internet traffic analysis. The arrival of massive network graphs with hundreds of millions of nodes, e.g. social graphs, presents a unique challenge to graph analysis applications. Most of these applications rely on computing distances between node pairs, which for large graphs can take minutes to compute using traditional algorithms such as breadth-first-search (BFS). In this paper, we study ways to enable scalable graph processing for today’s massive networks. We explore the design space of graph coordinate systems, a new approach that accurately approximates node distances in constant time by embedding graphs into coordinate spaces. We show that a hyperbolic embedding produces relatively low distortion error, and propose Rigel, a hyperbolic graph coordinate system that lends itself to efficient parallelization across a compute cluster. Rigel produces significantly more accurate results than prior systems, and is naturally parallelizable across compute clusters, allowing it to provide accurate results for graphs up to 43 million nodes. Finally, we show that Rigel’s functionality can be easily extended to locate (near-) shortest paths between node pairs. After a onetime preprocessing cost, Rigel answers node-distance queries in 10’s of microseconds, and also produces shortest path results up to 18 times faster than prior shortest-path systems with similar levels of accuracy. I.

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