Which triangulations approximate the complete graph?

Das, Gautam; Joseph, Deborah

doi:10.1007/3-540-51859-2_15

Cited by 84 publications

(85 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Section 3 contains a proof of Theorem 1. This proof is obtained by showing that the Delaunay graph satisfies the "diamond property" and a variant of the "good polygon property" of Das and Joseph [6]. The proof of the latter property is obtained by generalizing the analysis of Dobkin et al [8] for the lengths of so-called one-sided paths.…”

Section: Introductionmentioning

confidence: 94%

On the Stretch Factor of Convex Delaunay Graphs

Bose

Carmi

Collette

et al. 2008

Algorithms and Computation

View full text Add to dashboard Cite

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DG C (S) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex distance function defined by C. We prove that DG C (S) is a t-spanner for S, for some constant t that depends only on the shape of the set C. Thus, for any two points p and q in S, the graph DG C (S) contains a path between p and q whose Euclidean length is at most t times the Euclidean distance between p and q.

show abstract

Section: Introductionmentioning

confidence: 94%

On the Stretch Factor of Convex Delaunay Graphs

Bose

Carmi

Collette

et al. 2008

Algorithms and Computation

View full text Add to dashboard Cite

show abstract

“…Let wt(G) be the total edge weight of a graph G. In [5,8] it is shown that a 2-dimensional spanner exists with weight at most O(1) 9 wt(MST), which is an asymptotically optimal result. Both papers exploit planarity and the techniques 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%

Constructing degree-3 spanners with other sparseness properties

Das

Heffernan

1993

Algorithms and Computation

Self Cite

View full text Add to dashboard Cite

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)).

show abstract

“…Graph spanners arise in many applications, including communication networks, computational biology, computational geometry, robotics and distributed computing [1,2,3,5,6,7,9,10,11,12,13,14,15]. Intuitively, a spanner of a graph is a subgraph that preserves approximate distances between all pairs of vertices.…”

Section: Introductionmentioning

confidence: 99%

Small Stretch Spanners on Dynamic Graphs

Ausiello¹,

Franciosa²,

Italiano³

2006

JGAA

View full text Add to dashboard Cite

Abstract. We present fully dynamic algorithms for maintaining 3-and 5-spanners of undirected graphs. For unweighted graphs we maintain a 3-or 5-spanner under insertions and deletions of edges in O(n) amortized time per operation over a sequence of Ω(n) updates. The maintained 3-spanner (resp., 5-spanner) has O(n 3/2 ) edges (resp., O(n 4/3 ) edges), which is known to be optimal. On weighted graphs with d different edge cost values, we maintain a 3-or 5-spanner in O(n) amortized time per operation over a sequence of Ω(d · n) updates. The maintained 3-spanner (resp., 5-spanner) has O(d·n 3/2 ) edges (resp., O(d·n 4/3 ) edges). The same approach can be extended to graphs with real-valued edge costs in the range [1, C]. All our algorithms are deterministic and are substantially faster than recomputing a spanner from scratch after each update.

show abstract

Which triangulations approximate the complete graph?

Cited by 84 publications

References 5 publications

On the Stretch Factor of Convex Delaunay Graphs

On the Stretch Factor of Convex Delaunay Graphs

Constructing degree-3 spanners with other sparseness properties

Small Stretch Spanners on Dynamic Graphs

Contact Info

Product

Resources

About