Algorithms For Spanners Research Articles

Minimum weight Euclidean [formula omitted]-spanners

Given a set S of n points in the plane and a parameter ɛ>0, a Euclidean (1+ɛ)-spanner is a geometric graph G=(S,E) that contains, for all p,q∈S, a pq-path of weight at most (1+ɛ)‖pq‖. We show that the minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit square [0,1]2 is O(ɛ−3/2n), and this bound is the best possible. The upper bound is based on a new spanner algorithm in the plane. It improves upon the baseline O(ɛ−2n), obtained by combining a tight bound for the weight of a Euclidean minimum spanning tree (MST) on n points in [0,1]2, and a tight bound for the lightness of Euclidean (1+ɛ)-spanners, which is the ratio of the spanner weight to the weight of the MST. Our result generalizes to Euclidean d-space for every constant dimension d∈N: The minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit cube [0,1]d is Od(ɛ(1−d2)/dn(d−1)/d), and this bound is the best possible.For the n×n section of the integer lattice in the plane, we show that the minimum weight of a Euclidean (1+ɛ)-spanner is between Ω(ɛ−3/4⋅n2) and O(ɛ−1log(ɛ−1)⋅n2). These bounds become Ω(ɛ−3/4⋅n) and O(ɛ−1log(ɛ−1)⋅n) when scaled to a grid of n points in the unit square. In particular, this shows that the integer grid is not an extremal configuration for minimum weight Euclidean (1+ɛ)-spanners.

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 [10] improved the state-of-the-art for light spanners by constructing a (2k−1)(1+ε)-spanner with O(n1+1k) edges and Oε(n1k) lightness. Soon after, Filtser and Solomon [18] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn1+1k) (which is faster than [10]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Ωε(kn1k), 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 Oε(n2+1k+ε′) time spanner construction which achieves the state-of-the-art bounds. Our second result is an Oε(m+nlog⁡n) time construction of a spanner with (2k−1)(1+ε) stretch, O(log⁡k⋅n1+1k) edges and Oε(log⁡k⋅n1k) 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 ε>0, we provide an O(m+n1+ε) 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=ω(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.

A Degree 3 Plane $5.19$-Spanner for Points in Convex Position

Let $S$ be a set of $n$ points in the plane that is in convex position. In this paper, using the well-known path-greedy spanner algorithm, we present an algorithm that constructs a plane $\frac{3+4\pi}{3}$-spanner $G$ of degree 3 on the point set $S$. Recently, Biniaz et al. ({\it Towards plane spanners of degree 3, Journal of Computational Geometry, 8 (1), 2017}) have proposed an algorithm that constructs a degree 3 plane $\frac{3+4\pi}{3}$-spanner $G'$ for $S$. We show that there is no upper bound with a constant factor on the total weight of $G'$, but the total weight of $G$ is asymptotically equal to the total weight of the minimum spanning tree of $S$.

On Constructing t‐Spanner in IoT under SINR

Following the recent advances in the Internet of Things (IoT), it is drawing lots of attention to design distributed algorithms for various network optimization problems under the SINR (Signal‐to‐Interference‐and‐Noise‐Ratio) interference model, such as spanner construction. Since a spanner can maintain a linear number of links while still preserving efficient routes for any pair of nodes in wireless networks, it is important to design distributed algorithms for spanners. Given a constant t > 1 as the required stretch factor, the problem of our concern is to design an efficient distributed algorithm to construct a t‐spanner of the communication graph under SINR such that the delay for the task completion is minimized, where the delay is the time interval between the time slot that the first node commences its operation to the time slot that all the nodes finish their task of constructing the t‐spanner. Our main contributions include four aspects. First, we propose a proximity range and proximity independent set (PISet) to increase the number of nodes transmitting successfully at the same time in order to reduce the delay. Second, we develop a distributed randomized algorithm SINR‐Spanner to construct a required t‐spanner with high probability. Third, the approximation ratio of SINR‐Spanner is proven to be a constant. Finally, extensive simulations are carried out to verify the effectiveness and efficiency of our proposed algorithm.

Traceability and Management Method of Supply Chain Information Based on Wireless Sensor Network

Wireless sensor network is an emerging technology with broad application prospects and will play an increasingly important role in many aspects of people’s production practices. This article is aimed at conducting research on supply chain information traceability and management methods from the perspective of wireless sensor networks. This paper uses simulation methods and comparative experiment methods. From the perspective of Spanner algorithm research, the wireless sensor network technology is researched, the explanation and simplification of Spanner algorithm in supply chain information traceability are designed, and the parameters are optimized while the algorithm is optimized. Performance analysis introduced the research on supply chain disturbance management and conducted the reliability analysis of the supply chain trust scale. The experimental results show that among the three dimensions of supply chain information sharing, the reliability coefficients of the two dimensions of information sharing content and information sharing quality are 0.803 and 0.825, respectively, which are both greater than 0.8, indicating that the reliability is very good. Cronbach’s coefficient in the information sharing level dimension is equal to 0.665, which is in an acceptable range, the CITC value of this dimension is all higher than 0.4, and the value of the deleted item is lower than 0.665, so all the items of the information sharing level will also be retained. The research of supply chain information traceability and management methods based on wireless sensor network is well completed.

Spanner Algorithm for Directed Graph Stream

Spanner Algorithm for Directed Graph Stream

Distribution-Sensitive Construction of the Greedy Spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take varOmega (n^2) time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an O(n log ^2 n log ^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an O((n + |E|) log ^2 n log log n) expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.

A fast online spanner for roadmap construction

This paper introduces a fast weighted streaming spanner algorithm (WSS) that trims edges from roadmaps generated by robot motion planning algorithms such as Probabilistic Roadmap (PRM) and variants (e.g. k-PRM*) as the edges are generated, but before collision detection; no route in the resulting graph is more than a constant factor larger than it would have been in the original roadmap. Experiments applying WSS to k-PRM* were conducted, and the results show our algorithm’s capability to filter graphs with up to 1.28 million vertices, discarding about three-quarters of the edges. Due to the fact that many collision detection steps can be avoided, the combination of WSS and k-PRM* is faster than k-PRM* alone. The paper further presents an online directed spanner algorithm that can be used for systems with non-holonomic constraints, with proof of correctness and experimental results.

Asymptotically Near-Optimal Planning With Probabilistic Roadmap Spanners

Asymptotically optimal motion planners guarantee that solutions approach optimal as more iterations are performed. A recently proposed roadmap-based method, i.e., the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{\tt PRM}^{*}$ </tex></formula> approach, provides this desirable property and minimizes the computational cost of generating the roadmap. Even for this method, however, the roadmap can be slow to construct and quickly grows too large for storage or fast online query resolution, especially for relatively high-dimensional instances. In graph theory, there are algorithms that produce sparse subgraphs, which are known as graph spanners, that guarantee near-optimal paths. This paper proposes different alternatives for interleaving graph spanners with the asymptotically optimal <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{\tt PRM}^{*}$</tex></formula> algorithm. The first alternative follows a sequential approach, where a graph spanner algorithm is applied to the output roadmap of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\hbox{\tt PRM}^{*}$</tex></formula> . The second one is an incremental method, where certain edges are not considered during the construction of the roadmap as they are not necessary for a roadmap spanner. The result in both cases is an asymptotically near-optimal motion planning solution. Theoretical analysis and experiments performed on typical, geometric motion planning instances show that large reductions in construction time, roadmap density, and online query resolution time can be achieved with a small sacrifice of path quality through roadmap spanners.