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Approximate shortest paths avoiding a failed vertex : optimal data structures for unweighted graphs
"... Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be tapproximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data struct ..."
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Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be tapproximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t> 1. report tapproximate shortest path between u and v when vertex z fails We present data structures for the single source as well allpairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size nearly match the size of their best static counterparts. 1.
Additive spanners in nearly quadratic time
 IN PROCEEDINGS OF THE 37TH INTERNATIONAL COLLOQUIUM CONFERENCE ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP
, 2010
"... 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 6s ..."
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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.
Improved Approximation for the Directed Spanner Problem
, 2011
"... We present an O ( √ n log n)approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonne ..."
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We present an O ( √ n log n)approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonnegative edge lengths d: E → R ≥0 and a stretch k ≥ 1, a subgraph H = (V, EH) is a kspanner of G if for every edge (s, t) ∈ E, the graph H contains a path from s to t of length at most k · d(s, t). The previous best approximation ratio was Õ(n 2/3), due to Dinitz and Krauthgamer (STOC ’11). We also improve the approximation ratio for the important special case of directed 3spanners with unit edge lengths from Õ ( √ n) to O(n 1/3 log n). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer’s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly
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 ..."
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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.
Deterministic Distributed Construction of Linear Stretch Spanners in Polylogarithmic Time
, 2007
"... The paper presents a deterministic distributed algorithm that given an n node unweighted graph constructs an O(n 3/2) edge 3spanner for it in O(log n) time. This algorithm is then extended into a deterministic algorithm for computing an O(kn 1+1/k) edge O(k)spanner in O(log k−1 n) time for every i ..."
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The paper presents a deterministic distributed algorithm that given an n node unweighted graph constructs an O(n 3/2) edge 3spanner for it in O(log n) time. This algorithm is then extended into a deterministic algorithm for computing an O(kn 1+1/k) edge O(k)spanner in O(log k−1 n) time for every integer parameter k � 1. This establishes that the problem of the deterministic construction of a low stretch spanner with few edges can be solved in the distributed setting in polylogarithmic time. The paper also investigates the distributed construction of sparse spanners with almost pure additive stretch (1 + ɛ, β), i.e., such that the distance in the spanner is at most 1 + ɛ times the original distance plus β. It is shown, for every ɛ> 0, that in O(log n/ɛ) time one can deterministically construct a spanner with O(n 3/2) edges that is both a 3spanner and a (1 + ɛ, 8 log n)spanner. Furthermore, it is shown that in n O(1/ √ log n) + O(1/ɛ) time one can deterministically construct a spanner with O(n 3/2) edges which is both a 3spanner and a (1+ɛ, 4)spanner. (This algorithm can be transformed into a Las Vegas randomized algorithm with guarantees on the stretch and time, running in O(log n +1/ɛ) expected time).
Toward a Distance Oracle for BillionNode Graphs
, 2013
"... The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadthfirst search) and offline precomputing pairwise shortest distances are ..."
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The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadthfirst search) and offline precomputing pairwise shortest distances are prohibitive in time or space complexity for billionnode graphs. In this paper, we study the feasibility of building distance oracles for billionnode graphs. A distance oracle provides approximate answers to shortest distance queries by using a precomputed data structure for the graph. Sketchbased distance oracles are good candidates because they assign each vertex a sketch of bounded size, which means they have linear space complexity. However, stateoftheart sketchbased distance oracles lack efficiency or accuracy when dealing with big graphs. In this paper, we address the scalability and accuracy issues by focusing on optimizing the three key factors that affect the performance of distance oracles: landmark selection, distributed BFS, and answer generation. We conduct extensive experiments on both real networks and synthetic networks to show that we can build distance oracles of affordable cost and efficiently answer shortest distance queries even for billionnode graphs.
Faster streaming algorithms for graph spanners
 ArXiv:cs/0611023
, 2006
"... Given an undirected graph G = (V, E) on n vertices, m edges, and an integer t ≥ 1, a subgraph (V, ES), ES ⊆ E is called a tspanner if for any pair of vertices u, v ∈ V, the distance between them in the subgraph is at most t times the actual distance. We present streaming algorithms for computing a ..."
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Given an undirected graph G = (V, E) on n vertices, m edges, and an integer t ≥ 1, a subgraph (V, ES), ES ⊆ E is called a tspanner if for any pair of vertices u, v ∈ V, the distance between them in the subgraph is at most t times the actual distance. We present streaming algorithms for computing a tspanner of essentially optimal sizestretch trade offs for any undirected graph. Our first algorithm is for the classical streaming model and works for unweighted graphs only. The algorithm performs a single pass on the stream of edges and requires O(m) time to process the entire stream of edges. This drastically improves the previous best single pass streaming algorithm for computing a tspanner which requires θ(mn 2 t) time to process the stream and computes spanner with size slightly larger than the optimal. Our second algorithm is for StreamSort model introduced by Aggarwal et al. [2], which is the streaming model augmented with a sorting primitive. The StreamSort model has been shown to be a more powerful and still very realistic model than the streaming model for massive data sets applications. Our algorithm, which works of weighted graphs as well, performs O(t) passes using O(log n) bits of working memory only. Our both the algorithms require elementary data structures.
New approximation algorithms for minimum cycle bases of graphs
 In STACS 2007, 24th Annual Symposium on Theoretical Aspects of Computer Science, volume 4393 of LNCS
, 2007
"... We consider the problem of computing an approximate minimum cycle basis of an undirected nonnegative edgeweighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space ov ..."
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We consider the problem of computing an approximate minimum cycle basis of an undirected nonnegative edgeweighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over F2 generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. Although in most such applications any cycle basis can be used, a low weight cycle basis often translates to better performance and/or numerical stability. Despite the fact that the problem can be solved exactly in polynomial time, we design approximation algorithms since the performance of the exact algorithms may be too expensive for some practical applications. We present two new algorithms to compute an approximate minimum cycle basis. For any integer k ≥ 1, we give (2k−1)approximation algorithms with expected running time O(kmn 1+2/k + mn (1+1/k)(ω−1) ) and deterministic running time O(n 3+2/k), respectively. Here ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. Both algorithms are o(m ω) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(m ω) bound. We also present a 2approximation algorithm with expected running time O(m ω √ nlogn), a linear time 2approximation algorithm for planar graphs and an O(n 3) time 2.42approximation algorithm for the complete Euclidean graph in the plane. 1
Fully Dynamic Randomized Algorithms for Graph Spanners
, 2008
"... Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. ..."
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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a tspanner of itself, the research as well as applications of spanners invariably deal with a tspanner which has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a tspanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a tspanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
Sparse faulttolerant BFS trees
 In ESA
, 2013
"... 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 considers breadthfirst search (BFS) spanning trees, and addresses the problem of designing a sparse faulttolerant BFS tree, or FTBFS tree for ..."
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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 considers breadthfirst search (BFS) spanning trees, and addresses the problem of designing a sparse faulttolerant BFS tree, or FTBFS tree for short, namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T ′ of T still contains a BFS spanning tree for (the surviving part of) G. For a source node s, a target node t and an edge e ∈ G, the shortest s − t path Ps,t,e that does not go through e is known as a replacement path. Thus, our FTBFS tree contains the collection of all replacement paths Ps,t,e for every t ∈ V (G) and every failed edge e ∈ E(G). Our main results are as follows. We present an algorithm that for every nvertex graph G and source node s constructs a (single edge failure) FTBFS tree rooted at s with O(n · min{Depth(s),√n}) edges, where Depth(s) is the depth of the BFS tree rooted at s. This result is complemented by a matching lower bound, showing that there exist nvertex graphs with a source node s for which any edge (or vertex) FTBFS tree rooted at s has Ω(n3/2) edges. We then consider faulttolerant multisource BFS trees, or FTMBFS trees for short, aiming to provide (following a failure) a BFS tree rooted at each source s ∈ S for some subset of sources S ⊆ V. Again, tight bounds are provided, showing that there exists a polytime algorithm that for every nvertex graph and source set S ⊆ V of size σ constructs a (single failure) FTMBFS tree T ∗(S) from each source si ∈ S, with O( σ · n3/2) edges, and on the other hand there exist nvertex graphs with source sets S ⊆ V of cardinality σ, on which any FTMBFS tree from S has Ω( σ · n3/2) edges.