The MST of symmetric disk graphs is light

AbstractSymmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in Rd (representing n transceivers) and a transmission range assignment r:S→R, the symmetric disk graph of S (denoted SDG(S)) is the undirected graph over S whose set of edges is E={(u,v)|r(u)⩾|uv| and r(v)⩾|uv|}, where |uv| denotes the Euclidean distance between points u and v. We prove that the weight of the MST of any connected symmetric disk graph over a set S of n points in the plane, is only O(logn) times the weight of the MST of the complete Euclidean graph over S. We then show that this bound is tight, even for points on a line.Next, we prove that if the number of different ranges assigned to the points of S is only k, k≪n, then the weight of the MST of SDG(S) is at most 2k times the weight of the MST of the complete Euclidean graph. Moreover, in this case, the MST of SDG(S) can be computed efficiently in time O(knlogn).We also present two applications of our main theorem, including an alternative proof of the Gap Theorem, and a result concerning range assignment in wireless networks.Finally, we show that in the non-symmetric model (where E={(u,v)|r(u)⩾|uv|}), the weight of a minimum spanning subgraph might be as big as Ω(n) times the weight of the MST of the complete Euclidean graph

Bounded-Angle Minimum Spanning Trees

Motivated by the connectivity problem in wireless networks with directional antennas, we study bounded-angle spanning trees. Let P be a set of points in the plane and let ? be an angle. An ?-ST of P is a spanning tree of the complete Euclidean graph on P with the property that all edges incident to each point p ? P lie in a wedge of angle ? centered at p. We study the following closely related problems for ? = 120 degrees (however, our approximation ratios hold for any ? ? 120 degrees). 1) The ?-minimum spanning tree problem asks for an ?-ST of minimum sum of edge lengths. Among many interesting results, Aschner and Katz (ICALP 2014) proved the NP-hardness of this problem and presented a 6-approximation algorithm. Their algorithm finds an ?-ST of length at most 6 times the length of the minimum spanning tree (MST). By adopting a somewhat similar approach and using different proof techniques we improve this ratio to 16/3. 2) To examine what is possible with non-uniform wedge angles, we define an ??-ST to be a spanning tree with the property that incident edges to all points lie in wedges of average angle ?. We present an algorithm to find an ??-ST whose largest edge-length and sum of edge lengths are at most 2 and 1.5 times (respectively) those of the MST. These ratios are better than any achievable when all wedges have angle ?. Our algorithm runs in linear time after computing the MST

A New Large N Expansion for General Matrix-Tensor Models

We define a new large $N$ limit for general $\text{O}(N)^{R}$ or $\text{U}(N)^{R}$ invariant tensor models, based on an enhanced large $N$ scaling of the coupling constants. The resulting large $N$ expansion is organized in terms of a half-integer associated with Feynman graphs that we call the index. This index has a natural interpretation in terms of the many matrix models embedded in the tensor model. Our new scaling can be shown to be optimal for a wide class of non-melonic interactions, which includes all the maximally single-trace terms. Our construction allows to define a new large $D$ expansion of the sum over diagrams of fixed genus in matrix models with an additional $\text{O}(D)^{r}$ global symmetry. When the interaction is the complete vertex of order $R+1$, we identify in detail the leading order graphs for $R$ a prime number. This slightly surprising condition is equivalent to the complete interaction being maximally single-trace.Comment: 57 pages, 20 figures (additional discussion in Sec. 4.1.1. and additional figure (Fig. 5)

A Novel Family of Geometric Planar Graphs for Wireless Ad Hoc Networks

International audienceWe propose a radically new family of geometric graphs, i.e., Hypocomb, Reduced Hypocomb and Local Hypocomb. The first two are extracted from a complete graph; the last is extracted from a Unit Disk Graph (UDG). We analytically study their properties including connectivity, planarity and degree bound. All these graphs are connected (provided the original graph is connected) planar. Hypocomb has unbounded degree while Reduced Hypocomb and Local Hypocomb have maximum degree 6 and 8, respectively. To our knowledge, Local Hypocomb is the first strictly-localized, degree-bounded planar graph computed using merely 1-hop neighbor position information. We present a construction algorithm for these graphs and analyze its time complexity. Hypocomb family graphs are promising for wireless ad hoc networking. We report our numerical results on their average degree and their impact on FACE routing. We discuss their potential applications and some open problems

Optical Torque from Enhanced Scattering by Multipolar Plasmonic Resonance

We present a theoretical study of the optical angular momentum transfer from a circularly polarized plane wave to thin metal nanoparticles of different rotational symmetries. While absorption has been regarded as the predominant mechanism of torque generation on the nanoscale, we demonstrate numerically how the contribution from scattering can be enhanced by using multipolar plasmon resonance. The multipolar modes in non-circular particles can convert the angular momentum carried by the scattered field, thereby producing scattering-dominant optical torque, while a circularly symmetric particle cannot. Our results show that the optical torque induced by resonant scattering can contribute to 80% of the total optical torque in gold particles. This scattering-dominant torque generation is extremely mode-specific, and deserves to be distinguished from the absorption-dominant mechanism. Our findings might have applications in optical manipulation on the nanoscale as well as new designs in plasmonics and metamaterials.Comment: main article 20 pages, 4 figures; supplementary material 6 pages, 2 figure