Linear search with terrain-dependent speeds

We revisit the linear search problem where a robot, initially placed at the origin on an infinite line, tries to locate a stationary tar-get placed at an unknown position on the line. Unlike previous studies, in which the robot travels along the line at a constant speed, we con-sider settings where the robot’s speed can depend on the direction of travel along the line, or on the profile of the terrain, e.g. when the line is inclined, and the robot can accelerate. Our objective is to design search algorithms that achieve good competitive ratios for the time spent by the robot to complete its search versus the time spent by an omniscient robot that knows the location of the target. We consider several new robot mobility models in which the speed of the robot depends on the terrain. These include (1) different con-stant speeds for different directions, (2) speed with constant acceleration and/or variability depending on whether a certain segment has already been searched, (3) speed dependent on the incline of the terrain. We pro-vide both upper and lower bounds on the competitive ratios of search algorithms for these models, and in many cases, we derive optimal algo-rithms for the search time

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 (

Search on a line with faulty robots

We consider the problem of searching on a line using n mobile robots, of which at most f are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a target placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target. If n ≥ 2f + 2, there is a simple algorithm with competitive ratio 1. For f < n < 2f + 2 we develop a new class of algorithms, called proportional schedule algorithms. For any given (n, f), we give a proportional schedule algorithm A(n, f), whose competitive ratio is (equition presented) Setting a = n/f as a constant, the asymptotic competitive ratio is (4/a)2/a (4/a - 2)1-2/a + 1. Our search algorithm is easily seen to be optimal for the case n = f + 1. We also show that as n tends to ∞ the competitive ratio of our algorithm for the case n = 2f + 1 approaches 3 and this is optimal. More precisely, we show that asymptotically, the competitive ratio of our propor-tional schedule algorithm A(2f + 1, f) is at most 3+ 4 lnn/n , while any search algorithm has a lower bound 3 + 2 lnn/ n on its competitive ratio

Search on a line with faulty robots

We consider the problem of searching on a line using n mobile robots, of which at most f are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a target placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target. If (Formula presented.), there is a simple algorithm with a competitive ratio of 1. For (Formula presented.) we develop a new class of algorithms, called proportional schedule algorithms. For any given (n, f), we give a proportional schedule algorithm A(n, f), whose competitive ratio is (Formula presented.)Setting (Formula presented.) as a constant, the asymptotic competitive ratio is (Formula presented.). Our search algorithm is easily seen to be optimal for the case (Formula presented.). We also show that as n tends to (Formula presented.) the competitive ratio of our algorithm for the case (Formula presented.) approaches 3 and this is optimal. More precisely, we show that asymptotically (after excluding small order terms), the competitive ratio of our proportional schedule algorithm (Formula presented.) is at most (Formula presented.), while any search algorithm has a lower bound (Formula presented.) on its competitive ratio

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

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

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

Distributed algorithms for barrier coverage using relocatable sensors

We study the barrier coverage problem using relocatable sensor nodes. We assume each sensor can sense an intruder or event inside its sensing range. Sensors are initially located at arbitrary positions on the barrier and can move along the barrier. The goal is to find final positions for sensors so that the entire barrier is covered. In recent years, the problem has b

Morelia test: Improving the efficiency of the Gabriel test and face routing in ad-hoc networks

An important technique for discovering routes between two nodes in an ad-hoc network involves applying the face routing algorithm on a planar spanner of the network. Face routing guar