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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 7 (1 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
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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Cited by 4 (0 self)
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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
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 4 (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.
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 two ..."
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Cited by 3 (2 self)
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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.
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 2 (1 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.
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 1 (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.
Fully Dynamic Approximate Distance Oracles for Planar Graphs via ForbiddenSet Distance Labels
"... 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 − ..."
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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) dynamicdistanceoracleforplanargraphs,whichhas worst case query time Õ(n2/3)andamortizedupdatetime of Õ(n2/3). Our (1 + ε) forbiddenset labeling scheme can also be extended into a forbiddenset labeled routing scheme with stretch (1 + ε).
Optimal Labeling for Connectivity Checking in Planar Networks with Obstacles
"... 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 ..."
Abstract
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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