Small Stretch Spanners on Dynamic Graphs

Ausiello, Giorgio; Franciosa, Paolo Giulio; Italiano, Giuseppe F.

doi:10.7155/jgaa.00133

Cited by 24 publications

(22 citation statements)

References 16 publications

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“…In [6], a 3-spanner and a 5-spanner of an unweighted graph is maintained under an intermixed sequence of (n) edge insertions and deletions in O( ) amortized time per operation, where is the maximum degree of the original graph. The maintained 3-spanner has O(n 3/2 ) edges, while the 5-spanner has O(n 4/3 ) edges.…”

Section: Introductionmentioning

confidence: 99%

Graph Spanners in the Streaming Model: An Experimental Study

et al. 2008

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This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (α, β)-spanner of a graph G is a subgraph S ⊆ G such that for each pair of vertices the distance in S is at most α times the distance in G plus β. To the best of our knowledge, this is the first computational study of graph spanner algorithms in a streaming setting. We compare experimentally the randomized algorithms proposed by Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and by Elkin (In: Algorithmica (2009) 55: 346-374 347 for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: ), designed for building small stretch spanners. All the algorithms we implemented work in a data streaming model where the input graph is given as a stream of edges in arbitrary order, and all of them need a single pass over the data. Differently from the algorithm in Ausiello et al., the algorithms in Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and Elkin (In: to know in advance the number of vertices in the graph.The results of our experimental investigation on several input families confirm that all these algorithms are very efficient in practice, finding spanners with stretch and size much smaller than the theoretical bounds and comparable to those obtainable by off-line algorithms. Moreover, our experimental findings confirm that small values of the stretch factor are the case of interest in practice, and that the algorithm by Ausiello et al. tends to produce spanners of better quality than the algorithms by Baswana and Elkin, while still using a comparable amount of time and space resources.

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Section: Introductionmentioning

confidence: 99%

Graph Spanners in the Streaming Model: An Experimental Study

et al. 2008

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“…Ausiello et al [3] 5 O(n 5/2 ) O(n) Ausiello et al [3] 2k − 1 O(kn 1+1/k ) expect. O(mn −1/k ) expect.…”

Section: Fully Dynamic Algorithms Stretchmentioning

confidence: 99%

“…Ausiello et al [3] are the first to design fully dynamic algorithms for graph spanners. They present a fully dynamic deterministic algorithm for maintaining spanners with stretch at most 5 only, and the update time guaranteed is O(n).…”

Section: Centralized Algorithmsmentioning

confidence: 99%

Fully dynamic randomized algorithms for graph spanners

Baswana

Khurana

Sarkar

2012

ACM Trans. Algorithms

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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretch t ∈ N is a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a t-spanner of itself, the research as well as applications of spanners invariably deal with a t-spanner which has as small number of edges as possible.We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner.Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a t-spanner maintained at each stage by our algorithms matches, up to a poly-logarithmic factor, the worst case optimal size of a t-spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal. * The results of the preliminary version of this article appeared in ESA 2006 and SODA 2008

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“…Very recently Baswana [10] improved the result of Ausillo et al [3] and devised a fully dynamic algorithm for maintaining spanners of optimal size with stretch at most 6 with expected constant update time. Baswana [10] presented also a decremental algorithm for maintaining (2t − 1)-spanner of optimal size with expected update time of O(t 2 · log 2 n).…”

Section: # Passesmentioning

confidence: 91%

“…To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides non-trivial bounds for a wide range of stretch parameter t. The first algorithm of this type was devised recently by Ausillo et al [3]. This algorithm maintains 3-and 5-spanners of optimal size with O(n) amortized time per operation for an intermixed sequence of Ω(n) edge insertions and deletions.…”

Section: # Passesmentioning

confidence: 99%

Streaming and Fully Dynamic Centralized Algorithms for Constructing and Maintaining Sparse Spanners

Elkin

Automata, Languages and Programming

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Abstract. We present a streaming algorithm for constructing sparse spanners and show that our algorithm out-performs significantly the state-of-the-art algorithm for this task [20]. Specifically, the processing time-per-edge of our algorithm is drastically smaller than that of the algorithm of [20], and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters for the state-of-the-art algorithm.We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has a very small incremental update time, and a non-trivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides non-trivial bounds on both incremental and decremental update time for a wide range of stretch parameter t.

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Small Stretch Spanners on Dynamic Graphs

Cited by 24 publications

References 16 publications

Graph Spanners in the Streaming Model: An Experimental Study

Graph Spanners in the Streaming Model: An Experimental Study

Fully dynamic randomized algorithms for graph spanners

Streaming and Fully Dynamic Centralized Algorithms for Constructing and Maintaining Sparse Spanners

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