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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 3-spanner 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 3-spanner 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 3-spanner 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 3-spanner 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).
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 t-spanner 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 t-spanner 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 t-spanner of essentially optimal size-stretch 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 t-spanner 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.
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 all-pairs 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 all-pairs 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 t-spanner of itself, the research as well as applications of spanners invariably deal with a t-spanner 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 t-spanner maintained at each stage by our algorithms matches, up to a poly-logarithmic factor, the worst case optimal size of a t-spanner. 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.
Fast c-k-r partitions of sparse graphs
- Chicago Journal of Theoretical Computer Science
"... We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1 ..."
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We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1
Improved Distributed Steiner Forest Construction Extended Abstract
"... We present new distributed algorithms for constructing a Steiner Forest in the congest model. Our deterministic algorithm finds, for any given constant ε> 0, a (2 + ε)-approximation in Õ(sk +√min {st, n}) rounds, where s is the shortest path diameter, t is the number of terminals, k is the numbe ..."
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We present new distributed algorithms for constructing a Steiner Forest in the congest model. Our deterministic algorithm finds, for any given constant ε> 0, a (2 + ε)-approximation in Õ(sk +√min {st, n}) rounds, where s is the shortest path diameter, t is the number of terminals, k is the number of terminal components in the input, and n is the number of nodes. Our randomized algorithm finds, with high probability, an O(logn)-approximation in time Õ(k + min {s,√n} + D), where D is the unweighted di-ameter of the network. We also prove a matching lower bound of Ω̃(k+min {s,√n}+D) on the running time of any distributed approximation algorithm for the Steiner Forest problem. Previous algorithms were randomized, and ob-tained either an O(logn)-approximation in Õ(sk) time, or an O(1/ε)-approximation in Õ((√n+ t)1+ε +D) time. 1.
Construction Locale de Sous-Graphes Couvrants Peu Denses
, 2008
"... Nous présentons un algorithme distribué déterministe qui calcule pour tout graphe simple non pondéré, un sous-graphe couvrant (spanner) avec O(kn 1+1/k) arêtes et un facteur d’étirement (2k−1,0), n étant le nombre de sommets du graphe et k un paramètre entier strictement positif. Si n est inconnu l’ ..."
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Nous présentons un algorithme distribué déterministe qui calcule pour tout graphe simple non pondéré, un sous-graphe couvrant (spanner) avec O(kn 1+1/k) arêtes et un facteur d’étirement (2k−1,0), n étant le nombre de sommets du graphe et k un paramètre entier strictement positif. Si n est inconnu l’algorithme termine en temps 3k − 2, sinon il termine en temps k. En se basant sur cet algorithme pour k = 2, nous construisons de façon déterministe un sous-graphe couvrant avec O(ε −2 n 3/2) arêtes et un facteur d’étirement (1+ε,2) en O(ε −1) temps, ε> 0 est un paramètre arbitrairement petit. Nous complétons nos algorithmes par des bornes inférieures. D’abord, nous montrons que k unités de temps sont nécessaires pour calculer un sous-graphe couvrant avec o(n 1+1/(k−1) ) arêtes et un facteur d’étirement (2k − 1,0) pour k ∈ {2,3,5}. Ensuite, nous montrons que pour tout k> 1, si un algorithme distribué construit un sous-graphe couvrant H ayant moins de n 1+1/k+ε arêtes en temps t � n ε, alors H a un facteur d’étirement au moins (1 + Ω(k/t),Ω(kn ε)). Nos bornes sont valables aussi bien pour des algorithmes déterministes que probabilistes, avec ou sans la connaissance de n.
Spanners and Sparsifiers in Dynamic Streams
"... Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing space-efficient algorithms for classica ..."
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Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing space-efficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM’12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketch-based algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use Õ(n1+1/k) bits of space and have stretch 2 k. While this stretch is larger than the conjectured optimal 2k − 1 for this amount of space, we show for an appropriate k that it implies the first 2-pass spectral sparsifier with n 1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n 1+c bits of space for a constant c> 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff. 1.
Parallel graph decomposition and diameter approximation in o(diameter) time and linear space,” arXiv 1407.3144
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Analyzing Massive Graphs in the Semi-streaming Model
"... Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semi-streaming model was proposed for processing massive graphs. In the semi-streaming model, we have a random accessible memory ..."
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Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semi-streaming model was proposed for processing massive graphs. In the semi-streaming model, we have a random accessible memory which is near-linear in the number of vertices. The input graph (or equivalently, edges in the graph) is presented as a sequential list of edges (insertion-only model) or edge insertions and deletions (dynamic model). The list is read-only but we may make multiple passes over the list. There has been a few results in the insertion-only model such as computing distance spanners and approximating the maximum matching. In this thesis, we present some algorithms and techniques for (i) solving more complex problems in the semi-streaming model, (for example, problems in the dynamic model) and (ii) having better solutions for the problems which have been studied (for example, the maximum matching problem). In course of both of these, we develop new techniques with broad applications and explore the rich trade-offs between the complexity of models (insertion-only streams vs. dynamic streams), the number of passes, space, accuracy, and running time. 1. We initiate the study of dynamic graph streams. We start with basic problems such as the connectivity problem and computing the minimum spanning tree. These problems are This dissertation is available at ScholarlyCommons:
