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15
Faulttolerant spanners for general graphs
 in STOC’09, 2009
"... The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected nvertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a kspanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u ..."
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The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected nvertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a kspanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u, v) denotes the distance between u and v in G Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k−1)spanner of size O(n1+1/k), and this sizestretch tradeoff is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [Levcopoulos et al., STOC’98]. A subgraph H is an fvertex fault tolerant kspanner of the graph G if for any set F ⊆ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F (u, v) ≤ k · δG\F (u, v). Levcopoulos et al. presented an efficient algorithm that given a set S of n points in Rd, constructs an fvertex fault tolerant geometric (1+)spanner for S, that is, a sparse graph H such that for every set F ⊆ S of size f and any pair of points u, v ∈ S \F, δH\F (u, v) ≤ (1+)uv, where uv  is the Euclidean distance between u and v. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [Czumaj & Zhao, SoCG’03]. This paper also raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an fvertex fault tolerant (2k−1)spanner of size
Dualfailure distance and connectivity oracles
 In Proc. of the 20th ACMSIAM Symposium On Discrete Algorithms (SODA
, 2009
"... Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its co ..."
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Cited by 14 (3 self)
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Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its connectivity and distance metric. In this paper we look at the problem of efficiently answering connectivity, distance, and shortest route queries in the presence of two node or link failures. Our data structure uses Õ(n2) space and answers queries in Õ(1) time, which is within a polylogarithmic factor of optimal and nearly matches the singlefailure distance oracles of Demestrescu et al. It may yet be possible to find distance/connectivity oracles capable of handling any fixed number of failures. However, the sheer complexity of our algorithm suggests that moving beyond dualfailures will require a fundamentally different approach to the problem. 1
Popular conjectures imply strong lower bounds for dynamic problems
 CoRR
"... Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one dete ..."
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Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 − ε)npoly n) time for some ε> 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3−ε) time for some ε> 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3−ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh’s problem defined in a recent paper by Pǎtraşcu[STOC 2010]. Index Terms—dynamic algorithms; all pairs shortest paths; 3SUM; lower bounds; I.
Dynamic connectivity: Connecting to networks and geometry
 In Proceedings 49th FOCS
, 2008
"... Dynamic connectivity is a wellstudied problem, but so far the most compelling progress has been confined to the edgeupdate model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fu ..."
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Dynamic connectivity is a wellstudied problem, but so far the most compelling progress has been confined to the edgeupdate model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems: Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by on vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in Õ(m2/3) amortized time, wheremdenotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC’02], which required fast matrix multiplication and had an update time of O(m 0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axisparallel line segments and rectangles. We provide similarly improved update times, Õ(n2/3), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublineartime range queries. In particular, we obtain the first sublinear update time for arbitrary 2D line segments: O ∗ (n9/10); for ddimensional simplices: O ∗ 1 1− (n d(2d+1)); and for ddimensional balls: O ∗ (n 1 − 1
Forbiddenset distance labels for graphs of bounded doubling dimension
 29th ACM Symp. on Principles of Distributed Computing (PODC
, 2010
"... The paper proposes a forbiddenset labeling scheme for the family of graphs with doubling dimension bounded by α. For an nvertex graph G in this family, and for any desired precision parameter ɛ> 0, the labeling scheme stores an O(1+ɛ −1) 2α log 2 nbit label at each vertex. Given the labels of ..."
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The paper proposes a forbiddenset labeling scheme for the family of graphs with doubling dimension bounded by α. For an nvertex graph G in this family, and for any desired precision parameter ɛ> 0, the labeling scheme stores an O(1+ɛ −1) 2α log 2 nbit label at each vertex. Given the labels of two endvertices s and t, and the labels of a set F of “forbidden ” vertices and/or edges, our scheme can compute, in time polynomial in the length of the labels, a 1+ɛ stretch approximation for the distance between s and t in the graph G\F. The labeling scheme can be extended into a forbiddenset labeled routing scheme with stretch 1 + ɛ for graphs of bounded doubling dimension.
Lower Bound Techniques for Data Structures
, 2008
"... We describe new techniques for proving lower bounds on datastructure problems, with the following broad consequences:
â¢ the first Î©(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989;
â¢ for static data structures, the first separation between linea ..."
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Cited by 7 (0 self)
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We describe new techniques for proving lower bounds on datastructure problems, with the following broad consequences:
â¢ the first Î©(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989;
â¢ for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show Î©(lg n/ lg lg n) bounds when the space is O(n Â· polylog n).
Using these techniques, we analyze a variety of central datastructure problems, and obtain improved lower bounds for the following:
â¢ the partialsums problem (a fundamental application of augmented binary search trees);
â¢ the predecessor problem (which is equivalent to IP lookup in Internet routers);
â¢ dynamic trees and dynamic connectivity;
â¢ orthogonal range stabbing;
â¢ orthogonal range counting, and orthogonal range reporting;
â¢ the partial match problem (searching with wildcards);
â¢ (1 + Îµ)approximate near neighbor on the hypercube;
â¢ approximate nearest neighbor in the lâ metric.
Our new techniques lead to surprisingly nontechnical proofs. For several problems, we obtain simpler proofs for bounds that were already known.
Connectivity Oracles for Failure Prone Graphs ∗
"... Dynamic graph connectivity algorithms have been studied for many years, but typically in the most general possible setting, where the graph can evolve in completely arbitrary ways. In this paper we consider a dynamic subgraph model. We assume there is some fixed, underlying graph that can be preproc ..."
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Cited by 6 (4 self)
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Dynamic graph connectivity algorithms have been studied for many years, but typically in the most general possible setting, where the graph can evolve in completely arbitrary ways. In this paper we consider a dynamic subgraph model. We assume there is some fixed, underlying graph that can be preprocessed ahead of time. The graph is subject only to vertices and edges flipping “off ” (failing) and “on ” (recovering), where queries naturally apply to the subgraph on edges/vertices currently flipped on. This model fits most real world scenarios, where the topology of the graph in question (say a router network or road network) is constantly evolving due to temporary failures but never deviates too far from the ideal failurefree state. We present the first efficient connectivity oracle for graphs susceptible to vertex failures. Given vertices u and v and a set D of d failed vertices, we can determine if there is a path from u to v avoiding D in time polynomial in d log n. There is a tradeoff in our oracle between the space, which is roughly mn ɛ, for 0 < ɛ ≤ 1, and the polynomial query time, which depends on ɛ. If one wanted to achieve the same functionality with existing data structures (based on edge failures or twin vertex failures) the resulting connectivity oracle would either need exorbitant space (Ω(n d)) or update time Ω(dn), that is, linear in the number of vertices. Our connectivity oracle is therefore the first of its kind. As a byproduct of our oracle for vertex failures we reduce the problem of constructing an edgefailure oracle to 2D range searching over the integers. We show there is an Õ(m)space oracle that processes any set of d failed edges in O(d 2 log log n) time and, thereafter, answers connectivity queries in O(log log n) time. Our update time is exponentially faster than a recent connectivity oracle of Pǎtra¸scu and Thorup for bounded d, but slower as a function of d.
New data structures for subgraph connectivity
 In Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP),pages 201–212
, 2010
"... Abstract. We study the “subgraph connectivity ” problem for undirected graphs with sublinear vertex update time. In this problem, we can make vertices active or inactive in a graph G, and answer the connectivity between two vertices in the subgraph of G induced by the active vertices. In this paper, ..."
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Abstract. We study the “subgraph connectivity ” problem for undirected graphs with sublinear vertex update time. In this problem, we can make vertices active or inactive in a graph G, and answer the connectivity between two vertices in the subgraph of G induced by the active vertices. In this paper, we solve two open problems in subgraph connectivity. We give the first subgraph connectivity structure with worstcase sublinear time bounds for both updates and queries. Our worstcase subgraph connectivity structure supports Õ(m4/5) update time, Õ(m1/5) query time and occupies Õ(m) space, where m is the number of edges in the whole graph G. In the second part of our paper, we describe another dynamic subgraph connectivity structure with amortized Õ(m2/3) update time, Õ(m1/3) query time and linear space, which improves the structure introduced by [Chan, Pǎtra¸scu, Roditty, FOCS’08] that takes Õ(m4/3) space. 1
Optimal Labeling for Connectivity Checking in Planar Networks with Obstacles
, 2009
"... We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such a way that this fact can be determined from the labels of x and y, the vertices in X and the ends of ..."
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Cited by 3 (3 self)
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We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such a way that this fact can be determined from the labels of x and y, the vertices in X and the ends of the edges of F. For a planar graph with n vertices, we construct labels of size O(log n). The problem is motivated by the need to quickly compute alternative routes in networks under node or edge failures.
Fully Dynamic Approximate Distance Oracles for Planar Graphs via ForbiddenSet Distance Labels
, 2012
"... This paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbiddenset labeling schemes for planar graphs. For a given nvertex planar graph G with edge weights drawn from [1,M]andparameterε>0, our forbiddenset labeling scheme uses labels of length λ = O(ε −1 log 2 n log (nM) · ( ..."
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This paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbiddenset labeling schemes for planar graphs. For a given nvertex planar graph G with edge weights drawn from [1,M]andparameterε>0, our forbiddenset labeling scheme uses labels of length λ = O(ε −1 log 2 n log (nM) · (ε −1 +logn)). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1 + ε), in O(F  2 λ)time. We then present a general method to transform (1 + ε) forbiddenset labeling schemas into a fully dynamic (1 + ε) distance oracle. Our fully dynamic (1 + ε) distanceoracle is of size O(n log n · (ε −1 +logn)) and has Õ(n1/2)query and update time, both the query and the update time are worst case. This improves on the best previously known (1+ε) dynamic distance oracle for planar graphs, which has worst case query time Õ(n2/3) and amortized update time of Õ(n2/3). Our (1 + ε) forbiddenset labeling scheme can also be extended into a forbiddenset labeled routing scheme with stretch (1 + ε).