220 research outputs found
Constructing Light Spanners Deterministically in Near-Linear Time
Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn^(1+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(1/k)), even when randomization is used.
The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an O_epsilon(n^(2+1/k+epsilon\u27)) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(1/k)) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log n, for every constant epsilon>0, we provide an O(m+n^(1+epsilon)) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = omega(1).
To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest
Experimental study of geometric t-spanners : a running time comparison
The construction of t-spanners of a given point set has received a lot of attention, especially from a theoretical perspective. We experimentally study the performance of the most common construction algorithms for points in the Euclidean plane. In a previous paper [10] we considered the properties of the produced graphs from five common algorithms. We consider several additional algorithms and focus on the running times. This is the first time an extensive comparison has been made between the running times of construction algorithms of t-spanners
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
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