Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems

Hansen, Mark D.

doi:10.1109/sfcs.1989.63542

Cited by 28 publications

(11 citation statements)

References 2 publications

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“…In the case when the points of P are arranged as a d-dimensional array in 480 d , Hansen [1989] has applied the algorithms described in Sections 3.1-3.3 to obtain an O(log 2 n)-times optimal approximation algorithm for D for any graph. A similar result is obtained with high probability in the case when the points of P are distributed uniformly in the unit sphere of ᑬ d .…”

Section: Minimum Cut Linear Arrangement One Of the Most Famous Np-hardmentioning

confidence: 99%

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton

Rao

1999

J. ACM

697

666

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Abstract. In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.

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Section: Minimum Cut Linear Arrangement One Of the Most Famous Np-hardmentioning

confidence: 99%

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton

Rao

1999

J. ACM

697

666

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“…Among other applications, this provides a O(log 2 n) approximation for Minimum Feedback Arc Set (in directed graphs) and a O(log 2 n)-approximation algorithm for Minimum Cut Linear Arrangement. Using the ideas of Leighton and Rao, Hanson [18] presented a O(log 2 n)-approximation algorithm for Minimum Linear Arrangement and even more generally for the problem of graph embeddings in d-dimensional meshes. Ravi et al [29] further extended these polylogarithmic approximation algorithms to Minimum Storage-Time Product and Minimum Containing Interval Graph.…”

Section: Related Workmentioning

confidence: 99%

ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

et al. 2008

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We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of O( √ log n log log n), O( √ log n log log n), and O( √ log T log log T ), respectively, the last running in time polynomial in T (T being the sum of execution times). The technical contribution of our paper is to introduce " 2 2 spreading metrics" (that can be computed by semidefinite programming) as relaxations for both undirected and directed "permutation metrics," which are induced by permutations of {1, 2, . . . , n}. The techniques introduced in the recent work of Arora, Rao and Vazirani (Proc. of 36th STOC, pp. 222-231, 2004) can be adapted to exploit the geometry of such 2 2 spreading metrics, giving a powerful tool for the design of divide-and-conquer algorithms. In addition to their applications to approximation algorithms, the study of such 2 2 spreading metrics as relaxations of permutation metrics is interesting in its own right. We show how our results imply that, in a certain sense we make precise, 2 2 spreading metrics approximate permutation metrics on n points to a factor of O( √ log n log log n).

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“…Part I examines the geometric embedding problem for many of the graphs which are important in the study of parallel computation, [33]. Given an undirected graph ( G with n vertices, and a set P of n points in the plane, the geometric embedding problem consists of finding a bijection from the vertices of G to the points in the plane which minimizes the sum total of edge lengths of the embedded graph.…”

Section: Results In Computational Geometrymentioning

confidence: 99%

Results in Computational Geometry: Geometric Embeddings and Query- Retrieval Problems

Hansen¹,

Leighton²

1990

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Unclassified'el. PERFORMING ORGANIZATION RIET NUMBERo7n. MONITORING ORGANIZATION REPORT NUMBER(S)MIT TITLE (Include Security Classification)Results in Computational Geometry: Geometric Embeddings and Query-Retrieval Problems. PERSONAL AUTHOR(S)Mark David Hansen Part I examines the geometric embedding problem for many of the graphs which are important in the study of parallel computation 133]. Given an undirected graph G with n vertices, and a set P of n points in the plane, the geometric embedding problem consists of finding a bijection from the vertices of G to the points in the plane which minimizes the sum total of edge lengths of the embedded graph. We give fast approximation algorithms for embedding d-dimensional grids in the plane which are within a factor of O(log n) times optimal cost for d > 2 and O(log 2 n) for d = 2. We also show that any embedding of a hypercube, butterfly. or shuffle-exchange graph must be within an 0(log n) factor of optimal cost. When the points of P are randomly distributed, or arranged in a grid, we give a polynomial time algorithm which can embed arbitrary weighted graphs in these points with cost within an 0(log 2 n) factor of optimal. Many of these results extend to higher dimensions when P C Rd.Aside from the intrinsic mathematical interest of these problems, they also have applications in the held of parallel processing. For example. we show how the algorithms which we develop for geometric embeddings can be used to give solutions which are within an 0(log 2 N) factor of optimal to problems of performance optimization for array-based paralMe processors in the following areas: communication load balancing, dynarmic allocatiou of jobs to processors, reconfiguring around faults.and simulating other architectures.Part II of this thesis examines query-retrieval problems concerning distributions of points in euclidean space. In this part, we describe a new technique for solving a variety of query-retrieval problems in optimal time with optimal or near-optimal space steps.) Depending on the problem being solved, the space required for the data structure is either linear or O(nlogn). Hence, the time bounds are optimal and the space bounds are optimal or near-optimal. Previously known data structures for these problems required a factor of fl(logvn(loglogn) 2 ) or fl(lognloglogn) more space and/or more time to answer each query.Our compaction technique incorporates planar separators, filtering search, and the probabilistic method for discrepancy problems. The fundamental idea is that kth -order Voronoi diagrams (and other suitable proximity diagrams) can be compacted from k°( 1 )n space to 0(n) space and still retain all the information that is essential for solving query problems. Many fundamental questions in computational geometry arise from the consideration of distributions of points in euclidean space. This thesis explores two important areas of computational geometry in this setting: geometric embeddings anr queryretrieval problems. Each area is addressed in a separate part...

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Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems

Cited by 28 publications

References 2 publications

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

Results in Computational Geometry: Geometric Embeddings and Query- Retrieval Problems

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