Push & Pull: autonomous deployment of mobile sensors for a complete coverage

Mobile sensor networks are important for several strategic applications devoted to monitoring critical areas. In such hostile scenarios, sensors cannot be deployed manually and are either sent from a safe location or dropped from an aircraft. Mobile devices permit a dynamic deployment reconfiguration that improves the coverage in terms of completeness and uniformity. In this paper we propose a distributed algorithm for the autonomous deployment of mobile sensors called Push&Pull. According to our proposal, movement decisions are made by each sensor on the basis of locally available information and do not require any prior knowledge of the operating conditions or any manual tuning of key parameters. We formally prove that, when a sufficient number of sensors are available, our approach guarantees a complete and uniform coverage. Furthermore, we demonstrate that the algorithm execution always terminates preventing movement oscillations. Numerous simulations show that our algorithm reaches a complete coverage within reasonable time with moderate energy consumption, even when the target area has irregular shapes. Performance comparisons between Push&Pull and one of the most acknowledged algorithms show how the former one can efficiently reach a more uniform and complete coverage under a wide range of working scenarios.Comment: Technical Report. This paper has been published on Wireless Networks, Springer. Animations and the complete code of the proposed algorithm are available for download at the address: http://www.dsi.uniroma1.it/~novella/mobile_sensors

Minimum-energy broadcast in random-grid ad-hoc networks: approximation and distributed algorithms

The Min Energy broadcast problem consists in assigning transmission ranges to the nodes of an ad-hoc network in order to guarantee a directed spanning tree from a given source node and, at the same time, to minimize the energy consumption (i.e. the energy cost) yielded by the range assignment. Min energy broadcast is known to be NP-hard. We consider random-grid networks where nodes are chosen independently at random from the $n$ points of a $\sqrt n \times \sqrt n$ square grid in the plane. The probability of the existence of a node at a given point of the grid does depend on that point, that is, the probability distribution can be non-uniform. By using information-theoretic arguments, we prove a lower bound $(1-\epsilon) \frac n{\pi}$ on the energy cost of any feasible solution for this problem. Then, we provide an efficient solution of energy cost not larger than $1.1204 \frac n{\pi}$. Finally, we present a fully-distributed protocol that constructs a broadcast range assignment of energy cost not larger than $8n$,thus still yielding constant approximation. The energy load is well balanced and, at the same time, the work complexity (i.e. the energy due to all message transmissions of the protocol) is asymptotically optimal. The completion time of the protocol is only an $O(\log n)$ factor slower than the optimum. The approximation quality of our distributed solution is also experimentally evaluated. All bounds hold with probability at least $1-1/n^{\Theta(1)}$.Comment: 13 pages, 3 figures, 1 tabl

On strongly chordal graphs that are not leaf powers

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv is an edge if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with determining the complexity of their recognition. This is in part due to the fact that few graphs are known to not be leaf powers, as such graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf powers could be characterized by strong chordality and a finite set of forbidden subgraphs. In this paper, we provide a negative answer to this question, by exhibiting an infinite family \G of (minimal) strongly chordal graphs that are not leaf powers. During the process, we establish a connection between leaf powers, alternating cycles and quartet compatibility. We also show that deciding if a chordal graph is \G-free is NP-complete, which may provide insight on the complexity of the leaf power recognition problem

Upward Three-Dimensional Grid Drawings of Graphs

A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with $(n^3)$ volume, which is tight for the complete dag. The previous best upper bound was $O(n^4)$. Our main result is that every $c$-colourable directed acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with $O(n^2)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $O(n^3)$ for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar

Maximizing the Number of Broadcast Operations in Static Random Geometric Ad-Hoc Networks

We consider static ad-hoc wireless networks where nodes have the same initial battery charge and they may dynamically change their transmission range at every time slot. When a node v transmits with range r(v), its battery charge is decreased by β×r(v)2 where β> 0 is a fixed constant. The goal is to provide a range assignment schedule that maximizes the number of broadcast operations from a given source (this number is denoted as the length of the schedule). This maximization problem, denoted as , is known to be -hard and the best algorithm yields worst-case approximation ratio Θ(logn), where n is the number of nodes of the network [5]. We consider random geometric instances formed by selecting n points independently and uniformly at random from a square of side length n‾‾√ in the Euclidean plane. We first present an efficient algorithm that constructs a range assignment schedule having length, with high probability, not smaller than 1/12 of the optimum. We then design an efficient distributed version of the above algorithm where nodes initially know n and their own position only. The resulting schedule guarantees the same approximation ratio achieved by the centralized version thus obtaining the first distributed algorithm having provably-good performance for this problem

Maximizing the number of broadcast operations in random geometric ad-hoc wireless networks

We consider static ad hoc wireless networks whose nodes, equipped with the same initial battery charge, may dynamically change their transmission range. When a node v transmits with range r(v), its battery charge is decreased by \beta r(v)^2, where \beta >0 is a fixed constant. The goal is to provide a range assignment schedule that maximizes the number of broadcast operations from a given source (this number is denoted by the length of the schedule). This maximization problem, denoted by Max LifeTime, is known to be NP-hard and the best algorithm yields worst-case approximation ratio \Theta (\log n), where n is the number of nodes of the network. We consider random geometric instances formed by selecting n points independently and uniformly at random from a square of side length \sqrt{n} in the euclidean plane. We present an efficient algorithm that constructs a range assignment schedule having length not smaller than 1/12 of the optimum with high probability. Then we design an efficient distributed version of the above algorithm, where nodes initially know n and their own position only. The resulting schedule guarantees the same approximation ratio achieved by the centralized version, thus, obtaining the first distributed algorithm having provably good performance for this problem

Exploring pairwise compatibility graphs

A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge weight function w on T, and two non-negative real numbers d(mm) <= d(max), such that each leaf l(u) of T corresponds to a vertex u is an element of V and there is an edge (u, V) is an element of E if and only if d(min) <= d(T,w)(l(u), l(v)) <= d(max) where d(T,w)(l(u), l(v)) is the sum of the weights of the edges on the unique path from l(u), to l(v) in T. In this paper we analyze the class of PCGs in relation to two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves concern only d(max) (LPG) or only d(min) (mLPG). In particular, we show that the union of LPG and mLPG classes does not coincide with the whole class of PCGs, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, we study the closure properties of the classes PCG, mLPG and LPG, under some common graph operations. In particular, we consider the following operations: adding an isolated or universal vertex, adding a pendant vertex, adding a false or a true twin, taking the complement of a graph and taking the disjoint union of two graphs. (C) 2012 Elsevier B.V. All rights reserved