Edge-disjoint spanners in Cartesian products of graphs

AbstractA spanning subgraph S=(V,E′) of a connected graph G=(V,E) is an (x+c)-spanner if for any pair of vertices u and v, dS(u,v)⩽dG(u,v)+c where dG and dS are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs, focusing on graphs formed as Cartesian products. Our approach is to construct sets of edge-disjoint spanners in a product based on sets of edge-disjoint spanners and colorings of the component graphs. We present several results on general products and then narrow our focus to hypercubes

Weak coverage of a rectangular barrier

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak bar-rier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (

Strongly connected orientations of plane graphs

We study the problem of orienting a subset of edges of a given plane graph such that the resulting sub-digraph is strongly connected and spans all vertices of the graph. We are interested in orientations with minimum number of arcs which at the same time produce a digraph with smallest possible stretch factor. Such orientations have applications into the problem of establishing strongly connected sensor network when sensors are equipped with directional antennae. We present three constructions for such orientations. Let G = (V, E) be a 2-edge connected plane graph and let Φ(G) be the degree of the largest face in G. Our constructions are based on a face coloring of G, say with λ colors. The first construction gives a strongly connected orientation with at most (2 - 4λ-6/λ(λ-1) |E| arcs and the stretch factor at most Φ(G) - 1. The second construction gives a strongly connected orientation with |E| arcs and the stretch factor at most (Φ(G)-1) ⌈λ+1/2⌉. The third construction can be applied to plane graphs which are 3-edge connected. It uses a particular 6-face coloring and for any integer k ≥ 1 it produces a strongly connected orientation with at most (1 - k/10(k+1))|E| arcs and the stretch factor at most Φ 2(G)(Φ(G)-1) 2k+4. Since the stretch factor solely depends only on Φ(G), λ, and k, if these three parameters are bounded, our constructions result in orientations with bounded stretch factor. Crown Copyrigh

Maximum interference of random sensors on a line

Consider n sensors whose positions are represented by n uniform, independent and identically distributed random variables assuming values in the open unit interval (0,1). A natural way to guarantee connectivity in the resulting sensor network is to assign to each sensor as range the maximum of the two possible distances to its two neighbors. The interference at a given sensor is defined as the number of sensors that have this sensor within their range. In this paper we prove that the expected maximum interference is Ω(ln ln n), and that for any ∈ > 0, it is O((ln n)1/2+∈)

Asymptotic expected number of base pairs in optimal secondary structure for random RNA using the Nussinov-Jacobson energy model

Motivated by computer experiments, we study asymptotics of the expected maximum number of base pairs in secondary structures for random RNA sequences of length n. After proving a general limit result, we provide estimates of the limit for the binary alphabet { G, C } with thresholds k ≥ 0. We prove a general theorem entailing the existence of an asymptotic limit for the mean and standard deviation of free energy per nucleotide, as computed by mfold, for random RNA of any fixed compositional frequency; higher order moment limits are additionally shown to exist

Local construction of planar spanners in unit disk graphs with irregular transmission ranges

We give an algorithm for constructing a connected spanning subgraphs(panner) of a wireless network modelled as a unit disk graph with nodes of irregular transmission ranges, whereby for some parameter 0 < r ≤ 1 the transmission range of a node includes the entire disk around the node of radius at least r and it does not include any node at distance more than one. The construction of a spanner is distributed and local in the sense that nodes use only information at their vicinity, moreover for a given integer k ≥ 2 each node needs only consider all the nodes at distance at most k hop

Strong connectivity in sensor networks with given number of directional antennae of bounded angle

Given a set S of n sensors in the plane we consider the problem of establishing an ad hoc network from these sensors using directional antennae. We prove that for each given integer 1 ≤ k ≤ 5 there is a strongly connected spanner on the set of points so that each sensor uses at most k such directional antennae whose range differs from the optimal range by a multiplicative factor of at most 2·sin (π/k+1). Moreover, given a minimum spanning tree on the set of points the spanner can be constructed in additional O(n) time. In addition, we prove NP completeness results for k = 2 antennae

Half-space proximal: A new local test for extracting a bounded dilation spanner of a unit disk graph

We give a new local test, called a Half-Space Proximal or HSP test, for extracting a sparse directed or undirected subgraph of a given unit disk graph. The HSP neighbors of each vertex are unique, given a fixed underlying unit disk graph. The HSP test is a fully distributed, computationally simple algorithm that is applied independently to each vertex of a unit disk graph. The directed spanner obtained by this test is shown to be strongly connected, has out-degree at most six, its dilation is at most 2π + 1, contains the minimum weight spanning tree as its subgraph and, unlike the Yao graph, it is rotation invariant. Since no coordinate assumption is needed to determine the HSP nodes, the test can be applied in any metric space

Weak Coverage of a Rectangular Barrier

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. Each sensor can monitor a circular area of specific diameter around its position, called the sensor diameter. Sensors are required to move to final locations so that they can there detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by any sensor (MinMax). We give an O(n3 / 2) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions; in cont