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Book ChapterDOI

Approximating Shortest Paths in Graphs

18 Feb 2009-pp 32-43
TL;DR: Some of the fundamental developments like spanners and distance oracles, their underlying constructions, as well as their applications to the approximate all-pairs shortest paths are traced.
Abstract: Computing all-pairs distances in a graph is a fundamental problem of computer science but there has been a status quo with respect to the general problem of weighted directed graphs. In contrast, there has been a growing interest in the area of algorithms for approximate shortest paths leading to many interesting variations of the original problem. In this article, we trace some of the fundamental developments like spanners and distance oracles, their underlying constructions, as well as their applications to the approximate all-pairs shortest paths.
Citations
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Journal ArticleDOI
TL;DR: This survey reviews selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time.
Abstract: We consider the point-to-point (approximate) shortest-path query problem, which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph). A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible.Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs, query processing on spatial networks, and reachability queries; and theoretical computer scientists analyze distance oracles and sparse spanners. Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well.In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.

249 citations

Posted Content
TL;DR: This survey studies and classifies shortest-path algorithms according to the proposed taxonomy and presents the challenges and proposed solutions associated with each category in the taxonomy.
Abstract: A shortest-path algorithm finds a path containing the minimal cost between two vertices in a graph. A plethora of shortest-path algorithms is studied in the literature that span across multiple disciplines. This paper presents a survey of shortest-path algorithms based on a taxonomy that is introduced in the paper. One dimension of this taxonomy is the various flavors of the shortest-path problem. There is no one general algorithm that is capable of solving all variants of the shortest-path problem due to the space and time complexities associated with each algorithm. Other important dimensions of the taxonomy include whether the shortest-path algorithm operates over a static or a dynamic graph, whether the shortest-path algorithm produces exact or approximate answers, and whether the objective of the shortest-path algorithm is to achieve time-dependence or is to only be goal directed. This survey studies and classifies shortest-path algorithms according to the proposed taxonomy. The survey also presents the challenges and proposed solutions associated with each category in the taxonomy.

78 citations

Proceedings ArticleDOI
17 Jan 2012
TL;DR: In this article, a (2k − 1)-approximate distance oracle for G of size O(kn 1+1/k) can be constructed in [EQUATION] time and answer queries in O(k) time.
Abstract: Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)-approximate distance oracle for G of size O(kn1+1/k) can be constructed in [EQUATION] time and can answer queries in O(k) time. We also give an oracle which is faster for smaller k. Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all k ≥ 6 and improve the O(kmn1/k) time bound of Thorup and Zwick except for very sparse graphs and small k. When m = [EQUATION] and k = O(1), our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erdos.

46 citations

Proceedings ArticleDOI
06 Jan 2013
TL;DR: In this paper, a (2k − 1)-approximate distance oracle with O(log k) query time was constructed in O(min{kmn1/k, √km + kn1+c/√k}) time for some constant c.
Abstract: Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 2, we show that a (2k − 1)-approximate distance oracle for G of size O(kn1+1/k) and with O(log k) query time can be constructed in O(min{kmn1/k, √km + kn1+c/√k}) time for some constant c. This improves the O(k) query time of Thorup and Zwick. Furthermore, for any 0 0 and k = O(log n/log log n).

38 citations

Dissertation
01 Jan 2010
TL;DR: This thesis investigates the problem of efficiently computing exact and approximate shortest paths in graphs, with the main focus being on shortest path query processing and proves that exploiting well-connected nodes yields efficient distance oracles for scale-free graphs.
Abstract: Computing shortest paths in graphs is one of the most fundamental and well-studied problems in combinatorial optimization. Numerous real-world applications have stimulated research investigations for more than 50 years. Finding routes in road and public transportation networks is a classical application motivating the study of the shortest path problem. Shortest paths are also sought by routing schemes for computer networks: the transmission time of messages is less when they are sent through a short sequence of routers. The problem is also relevant for social networks: one may more likely obtain a favor from a stranger by establishing contact through personal connections. This thesis investigates the problem of efficiently computing exact and approximate shortest paths in graphs, with the main focus being on shortest path query processing. Strategies for computing answers to shortest path queries may involve the use of pre-computed data structures (often called distance oracles) in order to improve the query time. Designing a shortest path query processing method raises questions such as: How can these data structures be computed efficiently? What amount of storage is necessary? How much improvement of the query time is possible? How good is the approximation quality of the query result? What are the tradeoffs between precomputation time, storage, query time, and approximation quality? For distance oracles applicable to general graphs, the quantitative tradeoff between the storage requirement and the approximation quality is known up to constant factors. For distance oracles that take advantage of the properties of certain classes of graphs, however, the tradeoff is less well understood: for some classes of sparse graphs such as planar graphs, there are data structures that enable query algorithms to efficiently compute distance estimates of much higher precision than what the tradeoff for general graphs would predict. The first main contribution of this thesis is a proof that such data structures cannot exist for all sparse graphs. We prove a space lower bound implying that distance oracles with good precision and very low query costs require large amounts of space. A second contribution consists of spaceand time-efficient data structures for a large family of complex networks. We prove that exploiting well-connected nodes yields efficient distance oracles for scale-free graphs. A third contribution is a practical method to compute approximate shortest paths. By means of random sampling and graph Voronoi duals, our method successfully accommodates both highly structured graphs stemming from transportation networks and less structured graphs stemming from complex networks such as social networks.

34 citations

References
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Book
01 Jan 1990
TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Abstract: From the Publisher: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects. In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage. As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds. Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.

21,651 citations

Book ChapterDOI
14 Jul 1980

4,755 citations

Journal ArticleDOI
TL;DR: In this article, a new method for accelerating matrix multiplication asymptotically is presented, based on the ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product.

2,454 citations


"Approximating Shortest Paths in Gra..." refers methods in this paper

  • ...The fastest known algorithm for matrix multiplication due to Coppersmith and Winograd [17] implies ω < 2....

    [...]

Journal ArticleDOI
TL;DR: A data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time and is easy to implement.
Abstract: A data structure for representing a set of n items from a umverse of m items, which uses space n + o(n) and accommodates membership queries m constant time is described. Both the data structure and the query algorithm are easy to ~mplement.

943 citations

Journal ArticleDOI
TL;DR: A new simulation technique, referred to as a synchronizer, which is a new, simple methodology for designing efficient distributed algorithms in asynchronous networks, is proposed and is proved to be within a constant factor of the lower bound.
Abstract: The problem of simulating a synchronous network by an asynchronous network is investigated. A new simulation technique, referred to as a synchronizer, which is a new, simple methodology for designing efficient distributed algorithms in asynchronous networks, is proposed. The synchronizer exhibits a trade-off between its communication and time complexities, which is proved to be within a constant factor of the lower bound.

762 citations


"Approximating Shortest Paths in Gra..." refers methods in this paper

  • ...The concept of spanners was defined formally by Peleg and Schäffer [26] though the associated notion was used implicitly by Awerbuch [4] in the context of network synchronizers....

    [...]