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An Exponential Time 2Approximation Algorithm for Bandwidth
"... 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 2approximation algorithm for the Bandwidth problem that takes worstcase O(1.9797 ..."
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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 2approximation 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 3approximation algorithms of Cygan et al. which have an O ∗ (3 n) and O ∗ (2 n) worstcase 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 divideandconquer strategy along with dynamic programming to achieve this improved time bound. 1
Distance scales, embeddings, and metrics of negative type
 Symposium on Discrete Algorithms (SODA
, 2005
"... We introduce a new number of new techniques for the construction of lowdistortion 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 ..."
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We introduce a new number of new techniques for the construction of lowdistortion 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
Efficient Shortest Paths on Massive Social Graphs
"... 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 ..."
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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 breadthfirstsearch (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 nodedistance queries in 10’s of microseconds, and also produces shortest path results up to 18 times faster than prior shortestpath systems with similar levels of accuracy. I.
Toward a Distance Oracle for BillionNode Graphs
"... The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadthfirst search) and offline precomputing pairwise shortest distances are ..."
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The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadthfirst search) and offline precomputing pairwise shortest distances are prohibitive in time or space complexity for billionnode graphs. In this paper, we study the feasibility of building distance oracles for billionnode graphs. A distance oracle provides approximate answers to shortest distance queries by using a precomputed data structure for the graph. Sketchbased distance oracles are good candidates because they assign each vertex a sketch of bounded size, which means they have linear space complexity. However, stateoftheart sketchbased distance oracles lack efficiency or accuracy when dealing with big graphs. In this paper, we address the scalability and accuracy issues by focusing on optimizing the three key factors that affect the performance of distance oracles: landmark selection, distributed BFS, and answer generation. We conduct extensive experiments on both real networks and synthetic networks to show that we can build distance oracles of affordable cost and efficiently answer shortest distance queries even for billionnode graphs. 1.