Maintenance of a minimum spanning forest in a dynamic plane graph

Eppstein, David; Italiano, Giuseppe F.; Tamassia, Roberto; Tarjan, Robert E.; Westbrook, Jeffery

doi:10.1016/0196-6774(92)90004-v

Cited by 112 publications

(93 citation statements)

References 29 publications

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“…Finding and maintaining a minimum spanning tree (the (MST) problem) has been extensively studied in the literature (e.g. [2,3,10,12]). More recently, the problem of maintaining a small diameter was however solved in [16], and the distributed (MDST) problem was addressed in [5,6].…”

Section: Related Work and Resultsmentioning

confidence: 99%

A distributed algorithm for constructing a minimum diameter spanning tree

Bui

Butelle

Lavault

2004

Journal of Parallel and Distributed Computing

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We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph G = (V, E, ω). As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of G. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves O(|V |) time complexity and O (|V | |E|) message complexity.

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Section: Related Work and Resultsmentioning

confidence: 99%

A distributed algorithm for constructing a minimum diameter spanning tree

Bui

Butelle

Lavault

2004

Journal of Parallel and Distributed Computing

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“…For each subset Y E X we add the weighted edge set { ( x , y ) 1 y E Y } to the graph G (recall that the weight of each edge is the rectilinear distance between x and y ) . The MST of a planar weighted graph can be maintained using @log n) time per additiodinsertion of a point or edge [4]. Since (XI = O(1) and therefore 1 Y I = O( l), we can determine in O(1og n) time the MST cost savings for each candidate Steiner point.…”

Section: B a Batched Variantmentioning

confidence: 99%

“…In other words, for an original point set P , a set of already added candidate points S , and a new candidate x , add x to S if and only if A M S T ( P , { x } ) I AMST(P U S , {x}). Again, MST cost savings arising from the addition or deletion or a single point can be determined in time O(1og n) [4], bringing the total time for this entire step to 0 ( n 2 log n ) .…”

Section: B a Batched Variantmentioning

confidence: 99%

A new class of iterative Steiner tree heuristics with good performance

Kahng

Robins

1992

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

170

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Abstract-The minimum rectilinear Steiner tree (MRST) problem is very important for such aspects of physical layout as global routing and wiring estimation. Virtually all previous heuristics for computing rectilinear Steiner trees begin with a minimum spanning tree (MST) topology and rearrange edges to induce Steiner points. This paper gives a more direct approach which makes a significant departure from such spanning treebased strategies: we iteratively find optimal Steiner points to be added to the layout. Our method not only gives improved average-case performance, but also escapes the worst-case examples of existing approaches. We show that the performance ratio of our method can never be as bad as 3/2, and is in fact bounded by 4/3 on the entire class of instances where the c(MST)/c(MRST) cost ratio is exactly 3/2. Sophisticated computational geometry techniques allow efficient and practical implementation, and the method is naturally suited to technological regimes where, e.g., via costs can be high and the number of Steiner points should be limited. Extensive performance results show a 2% to 3% wire length reduction over the best previous heuristics. We describe a number of variants and extensions, and suggest directions for further research.

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“…A central fact that we use in this section and also in other parts of our work is the following observation [10]:…”

Section: Constructing a Cut Graphmentioning

confidence: 99%

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

2012

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We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu (2007) from planar graphs to bounded-genus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to bounded-genus graphs.

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Maintenance of a minimum spanning forest in a dynamic plane graph

Cited by 112 publications

References 29 publications

A distributed algorithm for constructing a minimum diameter spanning tree

A distributed algorithm for constructing a minimum diameter spanning tree

A new class of iterative Steiner tree heuristics with good performance

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

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