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Book ChapterDOI

A simple linear time algorithm for computing a (2k - 1)-spanner of o(n 1+1/k ) size in weighted graphs

30 Jun 2003-Vol. 2719, pp 384-396
TL;DR: This paper presents an extremely simple linear time randomized algorithm that constructs (2k - 1)-spanner of size matching the conjectured lower bound, and thus can be adapted suitably to obtain efficient distributed and parallel algorithms.
Abstract: Let G(V, E) be an undirected weighted graph with |V| = n, and |E| = m A t-spanner of the graph G(V, E) is a sub-graph G(V, ES) such that the distance between any pair of vertices in the spanner is at most t times the distance between the two in the given graph A 1963 girth conjecture of Erdos implies that Ω(n1+1/k) edges are required in the worst case for any (2k - 1)-spanner, which has been proved for k = 1, 2, 3, 5 There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn1/k) expected running time In this paper, we present an extremely simple linear time randomized algorithm that constructs (2k - 1)-spanner of size matching the conjectured lower bound Our algorithm requires local information for computing a spanner, and thus can be adapted suitably to obtain efficient distributed and parallel algorithms
Citations
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Journal ArticleDOI
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name "oracle", and it provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Abstract: Let G = (V,E) be an undirected weighted graph with vVv = n and vEv = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn1/k) expected time, constructing a data structure of size O(kn1p1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k−1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdos, implies that Ω(n1p1/k) space is needed in the worst case for any real stretch strictly smaller than 2kp1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n1p1/k) space had a query time of Ω(n1/k).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.

618 citations

Proceedings ArticleDOI
06 Jul 2001
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name ``oracle', which provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Abstract: Let G=(V,E) be an undirected weighted graph with |V|=n and |E|=m. Let k\ge 1 be an integer. We show that G=(V,E) can be preprocessed in O(kmn^{1/k}) expected time, constructing a data structure of size O(kn^{1+1/k}), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k-1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k-1. We show that a 1963 girth conjecture of Erd{\H{o}}s, implies that ω(n^{1+1/k}) space is needed in the worst case for any real stretch strictly smaller than 2k+1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name oracle. Previously, data structures that used only O(n^{1+1/k}) space had a query time of ω(n^{1/k}) and a slightly larger, non-optimal, stretch. Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.}

563 citations

Book ChapterDOI
11 Jul 2005
TL;DR: The first deterministic linear time algorithm for constructing optimal spanners of weighted graphs is obtained by derandomizing the O(km) expected time algorithm of Baswana and Sen for constructing (2k–1)-spanners of size O(kn) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.
Abstract: Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn$^{\rm 1/{\it k}}$) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn$^{\rm 1+1/{\it k}}$) capable of returning in O(k) time an approximation $\hat{\delta}(u,v)$ of the distance δ(u,v) from u to v in G that satisfies $\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)$, for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn$^{\rm 1+1/{\it k}}$log n). We present here a deterministic O(kmn$^{\rm 1/{\it k}}$) time algorithm for constructing oracles of size O(kn$^{\rm 1+1/{\it k}}$). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn$^{\rm 1+1/{\it k}}$) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.

202 citations


Cites methods from "A simple linear time algorithm for ..."

  • ...Due to lack of space we cannot describe here the deterministic version of the linear time spanner construction algorithm of Baswana and Sen [ 5 ]....

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  • ...The techniques we use to obtain the new deterministic algorithm can also be used to derandomize the expected linear time algorithm of Baswana and Sen [ 5 ] for constructing (2k − 1)-spanners of size O(kn 1+1/k ), retaining the linear running time and the O(kn 1+1/k ) size of the spanners....

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Journal ArticleDOI
TL;DR: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest- Paths problem.
Abstract: We obtain the following results related to dynamic versions of the shortest-paths problem: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of $\tilde {O}(m\sqrt{n})$ (we use $\tilde {O}$ to hide small poly-logarithmic factors) and a worst case query time is O(n3/4). A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.

200 citations

Proceedings ArticleDOI
22 Jan 2006
TL;DR: These are the first such results with additive error terms that are sublinear in the distance being approximated in the undirected and unweighted graph.
Abstract: Let k ≥ 2 be an integer. We show that any undirected and unweighted graph G = (V, E) on n vertices has a subgraph G' = (V, E') with O(kn1+1/k) edges such that for any two vertices u, v ∈ V, if δG(u, v) = d, then δG'(u, v) = d+O(d1-1/k-1). Furthermore, we show that such subgraphs can be constructed in O(mn1/k) time, where m and n are the number of edges and vertices in the original graph. We also show that it is possible to construct a weighted graph G* = (V, E*) with O(kn1+1/(2k-1)) edges such that for every u, v ∈ V, if δG(u, v) = d, then δ ≤ δG*(u, v) = d + O(d1-1/k-1). These are the first such results with additive error terms of the form o(d), i.e., additive error terms that are sublinear in the distance being approximated.

175 citations

References
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Journal ArticleDOI
TL;DR: A new simulation technique, referred to as a synchronizer, which is a new, simple methodology for designing efficient distributed algorithms in asynchronous networks, is proposed and is proved to be within a constant factor of the lower bound.
Abstract: The problem of simulating a synchronous network by an asynchronous network is investigated. A new simulation technique, referred to as a synchronizer, which is a new, simple methodology for designing efficient distributed algorithms in asynchronous networks, is proposed. The synchronizer exhibits a trade-off between its communication and time complexities, which is proved to be within a constant factor of the lower bound.

762 citations


"A simple linear time algorithm for ..." refers background in this paper

  • ...the design of synchronizers [2] and design of succinct...

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  • ...For other numerous applications, please refer to the papers [1,2,9,10]....

    [...]

Journal ArticleDOI
TL;DR: This paper gives a simple algorithm for constructing sparse spanners for arbitrary weighted graphs and applies this algorithm to obtain specific results for planar graphs and Euclidean graphs.
Abstract: Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v ?V, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

654 citations


"A simple linear time algorithm for ..." refers background or methods in this paper

  • ...Previously a number of papers [1,6,11] had addressed the problem of computing sparse spanners of graphs efficiently....

    [...]

  • ...For other numerous applications, please refer to the papers [1,2,9,10]....

    [...]

  • ...[1]....

    [...]

  • ...All the existing algorithms require computation of shortest distance information between many pairs of vertices [1], or computing shortest path trees from a set of (n1/k) vertices [11]....

    [...]

Proceedings ArticleDOI
06 Jul 2001
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name ``oracle', which provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Abstract: Let G=(V,E) be an undirected weighted graph with |V|=n and |E|=m. Let k\ge 1 be an integer. We show that G=(V,E) can be preprocessed in O(kmn^{1/k}) expected time, constructing a data structure of size O(kn^{1+1/k}), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k-1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k-1. We show that a 1963 girth conjecture of Erd{\H{o}}s, implies that ω(n^{1+1/k}) space is needed in the worst case for any real stretch strictly smaller than 2k+1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name oracle. Previously, data structures that used only O(n^{1+1/k}) space had a query time of ω(n^{1/k}) and a slightly larger, non-optimal, stretch. Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.}

563 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that if a graph G n has n vertices and at least 100kn 1+ 1 k edges, then G contains a cycle C 2 l of length 2 l for every integer l ∈ [k, kn 1 k ].

437 citations

Journal ArticleDOI
TL;DR: It is proved that any routing scheme for general networks that achieves a stretch factor k ≥ 1 must use a total of &OHgr; bits of routing information in the networks, which is a trade-off between the efficiency of a routing scheme and its space requirements.
Abstract: Two conflicting goals play a crucial role in the design of routing schemes for communication networks. A routing scheme should use paths that are as short as possible for routing messages in the network, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stretch factor-the maximum ratio between the length of a route computed by the scheme and that of a shortest path connecting the same pair of vertices.Most previous work has concentrated on finding good routing schemes (with a small fixed stretch factor) for special classes of network topologies. In this paper the problem for general networks is studied, and the entire range of possible stretch factors is examined. The results exhibit a trade-off between the efficiency of a routing scheme and its space requirements. Almost tight upper and lower bounds for this trade-off are presented. Specifically, it is proved that any routing scheme for general n-vertex networks that achieves a stretch factor k ≥ 1 must use a total of O(n1+1/(2k+4)) bits of routing information in the networks. This lower bound is complemented by a family K(k) of hierarchical routing schemes (for every k ≥ l) for unit-cost general networks, which guarantee a stretch factor of O(k), require storing a total of O(k3n1+(1/h)logn)- bits of routing information in the network, name the vertices with O(log2n)-bit names and use O(logn)-bit headers.

402 citations


Additional excerpts

  • ...For other numerous applications, please refer to the papers [1,2,9, 10 ]....

    [...]