Near-Optimal Light Spanners

@article{Chechik2016NearOptimalLS,
  title={Near-Optimal Light Spanners},
  author={Shiri Chechik and Christian Wulff-Nilsen},
  journal={ACM Transactions on Algorithms (TALG)},
  year={2016},
  volume={14},
  pages={1 - 15}
}
A spanner H of a weighted undirected graph G is a “sparse” subgraph that approximately preserves distances between every pair of vertices in G. We refer to H as a δ-spanner of G for some parameter δ ≥ 1 if the distance in H between every vertex pair is at most a factor δ bigger than in G. In this case, we say that H has stretch δ. Two main measures of the sparseness of a spanner are the size (number of edges) and the total weight (the sum of weights of the edges in the spanner). It is well… 

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References

SHOWING 1-10 OF 39 REFERENCES
Light Spanners
TLDR
It is shown that for any parameters k ≥ 1 and ε > 0, any weighted graph G on n vertices admits a (2k − 1) · (1 + ε)-stretch spanner of weight at most w(MST (G) · Oε(kn/ log k), where w(G)) is the weight of a minimum spanning tree of G.
Lower Bounds for Additive Spanners, Emulators, and More
  • David P. Woodruff
  • Mathematics, Computer Science
    2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
  • 2006
TLDR
The study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin by considering their approximate variants and their relaxation to emulators and proves the first lower bounds for such graphs.
Additive Spanners in Nearly Quadratic Time
TLDR
This work considers the problem of efficiently finding an additive C-spanner of an undirected unweighted graph G so that for all pairs of vertices u,v, δ H (u,v) ≤ δ G (u-v) + C, where δ denotes shortest path distance.
Approximate distance oracles with improved preprocessing time
TLDR
This work shows that for some universal constant c, a (2k − 1)-approximate distance oracle for G of size O(kn1+1/k) can be constructed in [EQUATION] time and can answer queries in O(k) time and gives an oracle which is faster for smaller k.
Approximate distance oracles
TLDR
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.
All-Pairs Almost Shortest Paths
TLDR
A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication, and describes an APASP2 algorithm, which is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm.
Optimal euclidean spanners: really short, thin and lanky
TLDR
This paper resolves the long-standing conjecture of Arya et al. that the weight bound can be improved by a logarithmic factor, without increasing the degree and the diameter of the spanner, and within the same running time, and demonstrates that the conjecture holds true not only in constant-dimensional Euclidean spaces, but also in doubling metrics.
On sparse spanners of weighted graphs
TLDR
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.
The Greedy Spanner is Existentially Optimal
TLDR
It is concluded that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area.
A Light Metric Spanner
  • Lee-Ad Gottlieb
  • Mathematics
    2015 IEEE 56th Annual Symposium on Foundations of Computer Science
  • 2015
TLDR
This paper shows that doubling spaces admit (1 + ε)-stretch spanners with lightness WD = (ddim /ε)<sup>O(ddim)</sup>.
...
1
2
3
4
...