On the locality of distributed sparse spanner construction
Cyril Gavoille2008, Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing - PODC '08
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Abstract
The paper presents a deterministic distributed algorithm that, given k 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k ) edges for every n-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O(kn 1+1/k ) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every > 0, constructs a (1 + , 2)-spanner of O( −1 n 3/2 ) edges in O( −1 ) rounds, without any prior knowledge on the graph.
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arXiv (Cornell University), 2023
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Lecture Notes in Computer Science, 2008
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Theoretical Computer Science, 2005
In this paper, we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n − 2 edges and an additive 3-spanner with at most O(n log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory 13 (1989) 99-116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(m log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)-spanner with at most 2n − 2 edges which is constructable in O(n k + m) time.
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Lecture Notes in Computer Science, 2011
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Faster Streaming algorithms for graph spanners
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On Additive Spanners in Weighted Graphs with Local Error
Graph-Theoretic Concepts in Computer Science, 2021
An additive +β spanner of a graph G is a subgraph which preserves distances up to an additive +β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β = cW , where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error +cW (s, t) for each vertex pair (s, t), where W (s, t) is the maximum edge weight along the shortest s-t path in G. These include pairwise +(2 + ε)W (•, •) and +(6 + ε)W (•, •) spanners over vertex pairs P ⊆ V × V on Oε(n|P| 1/3) and Oε(n|P| 1/4) edges for all ε > 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs +4W (•, •) spanner on O(n 7/5) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a +εW (•, •) spanner with Oε(n) lightness, and a +(4 + ε)W (•, •) spanner with Oε(n 2/3) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.
Graph Spanners in the Streaming Model: An Experimental Study
Algorithmica, 2009
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: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716–727, 9–13 July 2007) for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007), Engineering and Applications Track, Eilat, Israel, 8–10 October 2007. LNCS, vol. 4698, pp. 605–617, 2007), 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: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716–727, 9–13 July 2007) need 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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