On sparse spanners of weighted graphs

Althöfer, Ingo; Das, Gautam; Dobkin, David P.; Joseph, Deborah; Soares, Jacyra Ramos

doi:10.1007/bf02189308

Cited by 481 publications

(487 citation statements)

References 18 publications

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“…It was shown [1,2] that these techniques produce t-spanners with n 1+O( 1 t−1 ) edges on general graphs of n nodes.…”

Section: T-spanner Construction Algorithmsmentioning

confidence: 99%

t-Spanners for metric space searching

Navarro¹,

Paredes²,

Chávez³

2007

Data & Knowledge Engineering

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The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G ′ (V, E) as the representation of the metric database. A t-spanner is a subgraph G ′ (V, E) of a graph G(V, A), such that E ⊆ A and G ′ approximates the shortest path costs over G within a precision factor t.Our key idea is to regard the t-spanner as an approximation to the complete graph of distances among the objects, and to use it as a compact device to simulate the large matrix of distances required by successful search algorithms such as AESA. The t-spanner properties imply that we can use shortest paths over G ′ to estimate any distance with bounded error factor t.For this sake, several t-spanner construction, updating, and search algorithms are proposed and experimentally evaluated. We show that our technique is competitive against current approaches. For example, in a metric space of documents our search time is only 9% over AESA, yet we need just 4% of its space requirement. Similar results are obtained in other metric spaces.Finally, we conjecture that the essential metric space property to obtain good tspanner performance is the existence of clusters of elements, and enough empirical evidence is given to support this claim. This property holds in most real-world metric spaces, so we expect that t-spanners will display good behavior in most practical applications. Furthermore, we show that t-spanners have a great potential for improvements.

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“…It was shown [1,2] that these techniques produce t-spanners with n 1+O( 1 t−1 ) edges on general graphs of n nodes.…”

Section: T-spanner Construction Algorithmsmentioning

confidence: 99%

t-Spanners for metric space searching

Navarro¹,

Paredes²,

Chávez³

2007

Data & Knowledge Engineering

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“…A lower bound on the number of edges is n -1 because a spanning tree is a connected graph with the fewest edges. Spanners with O(n) edges and constant stretch factors have been designed in several papers, but in most of these spanners the c~nstant implicit in the O-notation depends on the dimension [1,2,3,4,7,9,10,14]. The degree-4 spanner constructed in [11] is an improvement, because it has at most 2. n edges.…”

Section: B For K < 3 Its Weight Is O(1) Wt(mst) and For K > 3 Its mentioning

confidence: 99%

“…This has been answered in the affirmative in 2-dimensions. In [1] it is shown that, given any 5 > 1, a spanner with a constant stretch factor (which depends on 5) can be constructed with at most n. 5 edges. However, the techniques employ planarity properties and do not extend to higher dimensions.…”

Section: B For K < 3 Its Weight Is O(1) Wt(mst) and For K > 3 Its mentioning

confidence: 99%

“…The minimum spanning tree of V, MST, is obviously the sparsest possible connected spanner both in number of edges and weight (although not in degree), but its "Supported in part by NSF grant ]2 stretch factor can be as bad as n -1 [1]. The sparseness of a spanner is frequently judged by comparing it to the number of edges and weight of the MST.…”

Section: Introductionmentioning

confidence: 99%

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Constructing degree-3 spanners with other sparseness properties

Das

Heffernan

1993

Algorithms and Computation

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Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any ~ and ~ in V, the length of the shortest path from u to v in the spanner is at most t times d(~, ~). We show that for any 5 > 1, there exists a polynomial-time constructihle tspanner (where ~ is a constant that depends only on 5 and k) with the following properties. Its maximum degree is 3, it has at most n 9 6 edges, and its total edge weight is comparable to the minimum spanning tree of V (for/~ < 3 its weight is O(1). wt(MgT), and for k > 3 its weight is O(log n). wt(MST)).

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“…Considerable work has been done on the spanner of a graph. In general, a t-spanner of G is a low-weight subgraph of G such that, for any two vertices, the distance within the spanner is at most t times the distance in G. Some results of finding spanners of a weighted graph can be found in [1]. Obviously, the spanners can be used to approximate the minimum routing cost subgraph with arbitrary requirements.…”

Section: Introductionmentioning

confidence: 99%

Light graphs with small routing cost

Chao²,

Tang

2002

Networks

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Let G = ({1,…, n}, E, w) be an undirected graph with nonnegative edge weights w and let aij be the nonnegative requirement between vertices i and j. For any spanning subgraph H of G, the weight of H is the total weight of its edges and the routing cost of H is Σi<j aijdH(i, j), where dH(i, j) is the distance between i and j in H. In this paper, we investigated two special cases of the problem of finding a spanning subgraph with small weight and small routing cost. For the case where all the distances in G are 1, we show that the problem is NP‐complete, and give a simple approximation algorithm for it. Furthermore, we define some sufficient conditions for the problem to be polynomial‐time solvable. For the case where all the requirements are 1, we develop an algorithm for finding a spanning tree with small weight and small routing cost. The algorithm provides trade‐offs among tree weights, routing costs, and time complexity. We also extend the results to some other related problems. © 2002 Wiley Periodicals, Inc.

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On sparse spanners of weighted graphs

Cited by 481 publications

References 18 publications

t-Spanners for metric space searching

t-Spanners for metric space searching

Constructing degree-3 spanners with other sparseness properties

Light graphs with small routing cost

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