New Additive Spanners

Chechik, Shiri

doi:10.1137/1.9781611973105.36

Cited by 67 publications

(92 citation statements)

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“…Spanners and emulators with sublinear error were given in [57,51]. Spanners with purely additive error and lower bounds concerning them were given in [4,35,20,16,21,58,19, 1].…”

Section: Related Workmentioning

confidence: 99%

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Elkin

Neiman

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

155

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Miller et al. [43] devised a distributed 1 algorithm in the CONGEST model, that given a parameter k = 1, 2, . . ., constructs an O(k)-spanner of an input unweighted nvertex graph with O(n 1+1/k ) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k − 1)-spanner with O(n 1+1/k / ) edges, with probability 1 − , for any > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 − o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm.We also devise improved bounds (and algorithms realizing these bounds) for (1 + , β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any > 0, there exists a (1 + , ( log log n ) log log n )-emulator with O(n) edges. All previous constructions of (1 + , β)-spanners and emulators employ a superlinear number of edges, for all choices of parameters.Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs. * Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. 1 They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in [43].

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“…Spanners and emulators with sublinear error were given in [57,51]. Spanners with purely additive error and lower bounds concerning them were given in [4,35,20,16,21,58,19, 1].…”

Section: Related Workmentioning

confidence: 99%

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Elkin

Neiman

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

155

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“…An efficient construction for 6-additive spanners with O(n 4/3 ) edges was later presented in [5]. Recently, a construction for 4-additive spanners with O(n 7/5 ) edges was presented in [7]. In the distributed case, the construction for 2-additive spanners requires to concurrently build O( √ n) complete BFS trees.…”

Section: Related Workmentioning

confidence: 99%

Efficient distributed source detection with limited bandwidth

Lenzen

Peleg

2013

Proceedings of the 2013 ACM Symposium on Principles of Distributed Computing

140

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Given a simple graph G = (V, E) and a set of sources S ⊆ V , denote for each node v ∈ V by L (∞) v the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ∈ S. For integers d, k ∈ N∪{∞}, we consider the source detection,Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k = ∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d = ∞ and S = V requires constructing a partial BFS tree comprising at least k nodes from every node in V .In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d + k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions.

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“…However, progress has mysteriously halted at this n 4/3 threshold: it is currently open whether or not there exist spanners on O(n 4/3−δ ) edges, even if the additive error function can be as large as +n o(1) . Breaking this n 4/3 barrier is considered to be a major open question [DHZ96,BKMP05b,BKMP05a,Woo10,BW15,Knu14,Che13], but progress has proved quite difficult. In this sense, additive spanners do not yet enjoy a smooth tradeoff curve like multiplicative and mixed spanners do.…”

Section: Spannersmentioning

confidence: 99%

“…The first such spanner [BCE03] had +O(n 1−2ε ) error on O(n 1+ε ) edges for all ε ≥ 0. There were a series of works improving this error tradeoff: +O(n 1−3ε ) in [BKMP05b], +O(n 9/16−7ε/8 ) [Pet07], +Õ(n 1/2−3ε/2 ) with the restriction ε ≥ 3/17 [Che13], +Õ(n 1/2−ε/2 ) [BW15], and +Õ(n 2/3−5ε/3 ) [BW15]. Jointly, these last three spanners form the current state of the art beneath the n 4/3 threshold.…”

Section: Spannersmentioning

confidence: 99%

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Better Distance Preservers and Additive Spanners

Bodwin¹,

Williams²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We make improvements to the upper bounds on several popular types of distance preserving graph sketches. These sketches are all various restrictions of the additive pairwise spanner problem, in which one is given an undirected unweighted graph G, a set of node pairs P , and an error allowance +β, and one must construct a sparse subgraphThe first part of our paper concerns pairwise distance preservers, which make the restriction β = 0 (i.e. distances must be preserved exactly). Our main result here is an upper bound of |H| = O(n 2/3 |P | 2/3 + n|P | 1/3 ) when G is undirected and unweighted. This improves on existing bounds whenever |P | = ω(n 3/4 ), and it is the first such improvement in the last ten years.We then devise a new application of distance preservers to graph clustering algorithms, and we apply this algorithm to subset spanners, which require P = S × S for some node subset S, and (standard) spanners, which require P = V × V . For both of these objects, our construction generalizes the best known bounds when the error allowance is constant, and we obtain the strongest polynomial error/sparsity tradeoff that has yet been reported (in fact, for subset spanners, ours is the first nontrivial construction that enjoys improved sparsity from a polynomial error allowance).We leave open a conjecture that O(n 2/3 |P | 2/3 + n) pairwise distance preservers are possible for undirected unweighted graphs. Resolving this conjecture in the affirmative would improve and simplify our upper bounds for all the graph sketches mentioned above. * gbodwin@stanford.edu †

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New Additive Spanners

Cited by 67 publications

References 0 publications

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Efficient distributed source detection with limited bandwidth

Better Distance Preservers and Additive Spanners

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