Results 1  10
of
13
Low Distortion Spanners
"... Abstract. A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an fspanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearl ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
(Show Context)
Abstract. A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an fspanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood. In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2 and 6spanners of Aingworth et al. and Baswana et al. and improve on the (1+ɛ, β)spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially. 1
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
(Show Context)
We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to loglogarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion. 1
On Pairwise Spanners
, 2013
"... Given an undirected nnode unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparses ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
Given an undirected nnode unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the uv distance only for pairs (u, v) in a given set P ⊆ V × V. Such Pspanners were studied before [Coppersmith,Elkin’05] only in the special case that f(·) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(·)). In more detail, for arbitrary P, we show that there exists a Pspanner of size O(n(P  log n) 1/4) with f(d) = d+4 log n. Alternatively, for any ε> 0, there exists a Pspanner of size O(nP  1/4 log n ε) with f(d) = (1 + ε)d + 4. We also consider the relevant special case that there is a critical set of nodes S ⊆ V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S × S)spanner of size O(n √ S) with f(d) = d + 2, and an (S × V)spanner of size O(n √ S  log n) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.
Sparse spanners vs. compact routing
 IN: SPAA
, 2011
"... Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of ˜ Θ (√n) bits 1 per node. The space lower bound is due to the existence of dense graphs with large girth. Dense graphs ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of ˜ Θ (√n) bits 1 per node. The space lower bound is due to the existence of dense graphs with large girth. Dense graphs can be sparsified to subgraphs, called spanners, with various stretch guarantees. There are spanners with additive stretch guarantees (some even have constant additive stretch) but only very few additive routing schemes are known. In this paper, we give reasons why routing in unweighted graphs with additive stretch is difficult in the form of space lower bounds for general graphs and for planar graphs. We prove that any routing scheme using routing tables of size µ bits per node and addresses of polylogarithmic length has additive stretch ˜ Ω ( p n/µ) for general graphs, and ˜ Ω ( √ n/µ) for planar graphs, respectively. Routing with tables of size Õ(n1/3) thus requires a polynomial additive stretch of ˜Ω(n 1/3), whereas spanners with average degree O(n 1/3) and constant additive stretch exist for all graphs. Spanners, however sparse they are, do not tell us how to route. These bounds provide the first separation of sparse spanner problems and compact routing problems. On the positive side, we give an almost tight upper bound: we present the first nontrivial compact routing scheme with o(lg 2 n)bit addresses, additive stretch Õ(n1/3), and table size Õ(n1/3) bits for all graphs with linear local treewidth such as planar, boundedgenus, and apexminorfree graphs.
On approximate distance labels and routing schemes with affine stretch
 IN INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING (DISC
, 2011
"... For every integral parameter k> 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, adistance label L(u) of size Õ(n2/(2k−1)). For any u, v ∈ G, givenL(u),L(v) we can return in time O(k) an affine approximation ˆ d(u, v) on the distance d(u, v) between u and v in ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
For every integral parameter k> 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, adistance label L(u) of size Õ(n2/(2k−1)). For any u, v ∈ G, givenL(u),L(v) we can return in time O(k) an affine approximation ˆ d(u, v) on the distance d(u, v) between u and v in G such that d(u, v) � ˆ d(u, v) � (2k − 2)d(u, v) +1. Hence we say that our distance label scheme has affine stretch of (2k − 2)d +1.Fork=2our construction is comparable to the O(n 5/3) size, 2d +1 affine stretch of the distance oracle of Pǎtraşcu and Roditty (FOCS ’10), it incurs a o(log n) storage overhead while providing the benefits of a distance label. For any k>1, givena restriction of o(n 1+1/(k−1) ) on the total size of the data structure, our construction provides distance labels with affine stretch of (2k − 2)d +1 which is better than the stretch (2k − 1)d scheme of Thorup and Zwick (J. ACM ’05). Our second contribution is a compact routing scheme with polylogarithmic addresses that provides affine stretch guarantees. With Õ(n 3/(3k−2))bit routing tables we obtain affine stretch of (4k − 6)d +1, for any k>1. Given a restriction of o(n 1/(k−1) ) on the table size, our routing scheme provides affine stretch which is better than the stretch (4k − 5)d routing scheme of Thorup and Zwick (SPAA ’01).
Additive spanners in nearly quadratic time
 In Proceedings of the 37th international colloquium conference on Automata, Languages and Programming (ICALP
, 2010
"... Abstract. We consider the problem of efficiently finding an additive Cspanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u, v, δH(u, v) ≤ δG(u, v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an addi ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We consider the problem of efficiently finding an additive Cspanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u, v, δH(u, v) ≤ δG(u, v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an additive 6spanner with O(n 4/3) edges in O(mn 2/3) time. It is unknown if there exists a constant C and an additive Cspanner with o(n 4/3) edges. Moreover, for C ≤ 5 all known constructions require Ω(n 3/2) edges. We give a significantly more efficient construction of an additive 6spanner. The number of edges in our spanner is n 4/3 polylog n, matching what was previously known up to a polylogarithmic factor, but we greatly improve the time for construction, from O(mn 2/3) to n 2 polylog n. Notice that mn 2/3 ≤ n 2 only if m ≤ n 4/3, but in this case G itself is a sparse spanner. We thus provide both the fastest and the sparsest (up to logarithmic factors) known construction of a spanner with constant additive distortion. We give similar improvements in the construction time of additive spanners under the assumption that the input graph has large girth, or more generally, the input graph has few edges on short cycles. 1
Fault tolerant approximate bfs structures
 In SODA
, 2014
"... A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a faulttolerant (α, β) approximate BFS structure (or FTABFS structure for short), namely, a subgraph H of th ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a faulttolerant (α, β) approximate BFS structure (or FTABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s, v,H \ F) ≤ α · dist(s, v,G \ F) + β for every v ∈ V. We first consider multiplicative (α, 0) FTABFS structures resilient to a failure of a single edge and present an algorithm that given an nvertex unweighted undirected graph G and a source s constructs a (3, 0) FTABFS structure rooted at s with at most 4n edges (improving by an O(log n) factor on the neartight result of [3] for the special case of edge failures). Assuming at most f edge failures, for constant integer f> 1, we prove that there exists a (polytime constructible) (3(f + 1), (f + 1) log n) FTABFS structure with O(fn) edges. We then consider additive (1, β) FTABFS structures. In contrast to the linear size of (α, 0) FTABFS structures, we show that for every β ∈ [1, O(log n)] there exists an nvertex graph G with a source s for which any (1, β) FTABFS structure rooted at s has Ω(n1+(β)) edges, for some function (β) ∈ (0, 1). In particular, (1, 3) FTABFS structures admit a lower bound of Ω(n5/4) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive
Efficient Distributed Source Detection with Limited Bandwidth
"... 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, or (S, d, k)detection task, requiring each node v to ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
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, or (S, d, k)detection task, requiring each node v to learn the (if for all of them d(s, v) ≤ d) or all entries (d(s, v), s) ∈ L (∞) v satisfying that d(s, v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadthfirst 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, nearoptimal 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. first k entries of L (∞)
unknown title
"... Des spanneurs aux spanneurs multichemins From spanners to multipath spanners ..."
Abstract
 Add to MetaCart
(Show Context)
Des spanneurs aux spanneurs multichemins From spanners to multipath spanners
Very Sparse Additive Spanners and Emulators
"... We obtain new upper bounds on the additive distortion for graph emulators and spanners on relatively few edges. We introduce a new subroutine called “strip creation, ” and we combine this subroutine with several other ideas to obtain the following results: 1. Every graph has a spanner on O(n1+) edge ..."
Abstract
 Add to MetaCart
(Show Context)
We obtain new upper bounds on the additive distortion for graph emulators and spanners on relatively few edges. We introduce a new subroutine called “strip creation, ” and we combine this subroutine with several other ideas to obtain the following results: 1. Every graph has a spanner on O(n1+) edges with Õ(n1/2−/2) additive distortion. 2. Every graph has an emulator on Õ(n1+) edges with Õ(n1/3−2/3) additive distortion whenever ∈ [0, 1 5 3. Every graph has a spanner on Õ(n1+) edges with Õ(n2/3−5/3) additive distortion whenever ∈ [0, 1 4 Our first spanner has the new best known asymptotic edgeerror tradeoff for additive spanners whenever ∈ [0, 1 7 Our second spanner has the new best tradeoff whenever