Author

# Ulrich Meyer

Other affiliations: Max Planck Society

Bio: Ulrich Meyer is an academic researcher from Goethe University Frankfurt. The author has contributed to research in topic(s): Shortest path problem & Time complexity. The author has an hindex of 27, co-authored 137 publication(s) receiving 3036 citation(s). Previous affiliations of Ulrich Meyer include Max Planck Society.

##### Papers published on a yearly basis

##### Papers

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TL;DR: A rather simple algorithm for the single source shortest path problem, which the authors call Delta-stepping, can be implemented very efficiently in sequential and parallel setting for a large class of graphs and achieves significant speedup on real machines.

Abstract: The single source shortest path problem for arbitrary directed graphs with n nodes, m edges and nonnegative edge weights can sequentially be solved using O(n ċ log n + m) operations. However, no work-efficient parallel algorithm is known that runs in sublinear time for arbitrary graphs. In this paper we present a rather simple algorithm for the single source shortest path problem. Our new algorithm, which we call Delta-stepping, can be implemented very efficiently in sequential and parallel setting for a large class of graphs. For random edge weights and arbitrary graphs with maximum node degree d, sequential Δ-stepping needs O(n + m + d ċ L) total average-case time, where L denotes the maximum shortest path weight from the source node s to any node reachable from s. For example, this means linear time on directed graphs with constant maximum degree. Our best parallel version for a PRAM takes O(d ċ L ċ log n + log2 n) time and O(n + m + d ċ L ċ log n) work on average. For random graphs, even O(log2 n) time and O(n + m) work on average can be achieved. We also discuss how the algorithm can be adapted to work with nonrandom edge weights and how it can be implemented on distributed memory machines. Experiments indicate that already a simple implementation of the algorithm achieves significant speedup on real machines.

238 citations

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24 Aug 1998TL;DR: A PRAM algorithm is given based on Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel.

Abstract: The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously work-efficient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random edge weights uniformly distributed in [0,1]. We use the G (n, d/n) model: the graph consists of n nodes and each edge is chosen with probability d/n. Our PRAM algorithm needs O(n 1/3 log n) log n) time and O (n log n+dn) work with high probability (whp). We also give extensions to external memory computation. Simulations show the applicability of our approach even on non-random graphs.

161 citations

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TL;DR: In this article, the authors present a very simple distributed algorithm for computing a small CDS with an approximation factor of at most 6.91, improving upon the previous best-known approximation of 8 due to Wan et al. [2002].

Abstract: Several routing schemes in ad hoc networks first establish a virtual backbone and then route messages via backbone nodes. One common way of constructing such a backbone is based on the construction of a connected dominating set (CDS). In this article we present a very simple distributed algorithm for computing a small CDS. Our algorithm has an approximation factor of at most 6.91, improving upon the previous best-known approximation factor of 8 due to Wan et al. [2002]. The improvement relies on a refined analysis of the relationship between the size of a maximal independent set and a minimum CDS in a unit disk graph. This subresult also implies improved approximation factors for many existing algorithm.

154 citations

01 Jan 2005

TL;DR: This paper presents a very simple distributed algorithm for computing a small CDS, improving upon the previous best known approximation factor of 8 and implying improved approximation factors for many existing algorithm.

Abstract: Several routing schemes in ad hoc networks first establish a virtual backbone and then route messages via back-bone nodes. One common way of constructing such a backbone is based on the construction of a minimum connected dominating set (CDS). In this paper we present a very simple distributed algorithm for computing a small CDS. Our algorithm has an approximation factor of at most 6.91, improving upon the previous best known approximation factor of 8 due to Wan et al. [INFOCOM'02], The improvement relies on a refined analysis of the relationship between the size of a maximal independent set and a minimum CDS in a unit disk graph. This subresult also implies improved approximation factors for many existing algorithm.

151 citations

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01 Jan 2003

TL;DR: A rather simple algorithm for the single source shortest path problem, which the authors call Delta-stepping, can be implemented very efficiently in sequential and parallel setting for a large class of graphs and achieves significant speedup on real machines.

Abstract: The single source shortest path problem for arbitrary directed graphs with n nodes, m edges and nonnegative edge weights can sequentially be solved using O(n.log n + m) operations. However, no work-efficient parallel algorithm is known that runs in sublinear time for arbitrary graphs. In this paper we present a rather simple algorithm for the single source shortest path problem. Our new algorithm, which we call Delta-stepping, can be implemented very efficiently in sequential and parallel setting for a large class of graphs. For random edge weights and arbitrary graphs with maximum node degree d, sequential Δ-stepping needs O(n + m + d.L) total average-case time, where L denotes the maximum shortest path weight from the source node s to any node reachable from s. For example, this means linear time on directed graphs with constant maximum degree. Our best parallel version for a PRAM takes O(d.L.log n + log 2 n) time and O(n + m + d L.log n) work on average. For random graphs, even O(log 2 n) time and O(n + m) work on average can be achieved. We also discuss how the algorithm can be adapted to work with nonrandom edge weights and how it can be implemented on distributed memory machines. Experiments indicate that already a simple implementation of the algorithm achieves significant speedup on real machines.

140 citations

##### Cited by

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TL;DR: Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis.

Abstract: Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

12,323 citations

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TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

6,328 citations

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^{1}TL;DR: A model for processing large graphs that has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier.

Abstract: Many practical computing problems concern large graphs. Standard examples include the Web graph and various social networks. The scale of these graphs - in some cases billions of vertices, trillions of edges - poses challenges to their efficient processing. In this paper we present a computational model suitable for this task. Programs are expressed as a sequence of iterations, in each of which a vertex can receive messages sent in the previous iteration, send messages to other vertices, and modify its own state and that of its outgoing edges or mutate graph topology. This vertex-centric approach is flexible enough to express a broad set of algorithms. The model has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier. Distribution-related details are hidden behind an abstract API. The result is a framework for processing large graphs that is expressive and easy to program.

3,556 citations

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TL;DR: It is shown that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent, which made it possible to formulate a variational principle for the force-free magnetic fields.

Abstract: where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

1,739 citations