When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots

A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots move perpetually along the domain, represented by all points belonging to the graph edges, without exceeding their maximum speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e., they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolman? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by Ifk(G) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing Ifk(G). We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness Ifk(G)=(f+1)|E|/k, where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs—a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs

Computing Boolean Functions on Anonymous Networks

AbstractWe study the bit-complexity of computing Boolean functions on anonymous networks. Let N be the number of nodes, δ the diameter, and d the maximal node degree of the network. For arbitrary, anonymous networks we give a general algorithm of polynomial bit complexity O(N3 · δ · d2 · log N) for computing any Boolean function which is computable on the network. This improves upon the previous best known algorithm, which was of exponential bit complexity O(dN2). For symmetric functions on arbitrary networks we give an algorithm with bit complexity O(N3· δ · d2 · log2N). This same algorithm is shown to have even lower bit complexity for a number of specific networks, for example tori, hypercubes, and random regular graphs. We also consider the class of distance regular unlabeled networks and show that on such networks symmetric functions can be computed efficiently in O(N · δ · d · log N) bits

Randomized rendez-vous with limited memory

We present a tradeoff between the expected time for two identical agents to rendez-vous on a synchronous, anonymous, oriented ring and the memory requirements of the agents. In particular, we show that there exists a 2t state agent, which can achieve rendez-vous on an n node ring in expected time O( n 2/2 t ∈+∈2 t ) and that any t/2 state agent requires expected time Ω( n 2/2 t ). As a corollary we observe that Θ(loglogn) bits of memory are necessary and sufficient to achieve rendez-vous in linear time

Asymptotic structural properties of quasi-random saturated structures of RNA

Background: RNA folding depends on the distribution of kinetic traps in the landscape of all secondary structures. Kinetic traps in the Nussinov energy model are precisely those secondary structures that are saturated, meaning that no base pair can be added without introducing either a pseudoknot or base triple. In previous work, we investigated asymptotic combinatorics of both random saturated structures and of quasi-random saturated structures, where the latter are constructed by a natural stochastic process.Results: We prove that for quasi-random saturated structures with the uniform distribution, the asymptotic expected number of external loops is O(logn) and the asymptotic expected maximum stem length is O(logn), while under the Zipf distribution, the asymptotic expected number of external loops is O(log2n) and the asymptotic expected maximum stem length is O(logn/log logn).Conclusions: Quasi-random saturated structures are generated by a stochastic greedy method, which is simple to implement. Structural features of random saturated structures appear to resemble those of quasi-random saturated structures, and the latter appear to constitute a class for which both the generation of sampled structures as well as a combinatorial investigation of structural features may be simpler to undertake

Range Quantile Queries: Another Virtue of Wavelet Trees

We show how to use a balanced wavelet tree as a data structure that stores a list of numbers and supports efficient {\em range quantile queries}. A range quantile query takes a rank and the endpoints of a sublist and returns the number with that rank in that sublist. For example, if the rank is half the sublist's length, then the query returns the sublist's median. We also show how these queries can be used to support space-efficient {\em coloured range reporting} and {\em document listing}.Comment: Added note about generalization to any constant number of dimensions

Routing on delay tolerant sensor networks

Delay (or disruption) tolerant sensor networks may be modeled as Markovian evolving graphs [1]. We present experimental evidence showing that considering multiple (possibly not shortest) paths instead of one fixed (greedy) path can decrease the expected time to deliver a packet on such a network by as much as 65 per cent depending on the probability that an edge exists in a given time interval. We provide theoretical justification for this result by studying a special case of the Markovian evolving grid graph. We analyze a natural algorithm for routing on such networks and show that it is possible to improve the expected time of delivery by up to a factor of two depending upon the probability of an edge being up during a time step and the relative positions of the source and destination. Furthermore we show that this is optimal, i.e., no other algorithm can achieve a better expected running time. As an aside, our results give high probability bounds for Knuth's toilet paper problem [11]

Station layouts in the presence of location constraints

In wireless communication, the signal of a typical broadcast station is transmited from a broadcast center p and reaches objects at a distance, say, R from it. In addition there is a radius r, r < R, such that the signal originating from the center of the station is so strong that human habitation within distance r from the center p should be avoided. Thus every station determines a region which is an “annulus of permissible habitation". We consider the following station layout (SL) problem: Cover a given (say, rectangular) planar region which includes a collection of orthogonal buildings with a minimum number of stations so that every point in the region is within the reach of a station, while at the same time no building is within the dangerous range of a station. We give algorithms for computing such station layouts in both the one-and two-dimensional cases

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