Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$

A clustered back-bone for routing in ad-hoc networks

In the recent years, a lot of research work has been undertaken in the area of ad-hoc networks due to the increasing potential of putting them to commercial use in various types of mobile computing devices. Topology control in ad-hoc networks is a widely researched topic; with a number of algorithms being proposed for the construction of a power-efficient topology that optimizes the battery usage of the mobile nodes. This research proposes a novel technique of partitioning the ad-hoc network into virtually-disjoint clusters. The ultimate aim of forming a routing graph over which power-efficient routing can be implemented in a simple and effective manner is realized by partitioning the network into disjoint clusters and thereafter joining them through gateways to form a connected, planar back-bone which is also a t-spanner of the original Unit Disk Graph (UDG). Some of the previously proposed algorithms require the nodes to construct local variations of the Delaunay Triangulation and undertake several complicated steps for ensuring the planarity of the back-bone graph. The construction of the Delaunay Triangulation is very complex and time-consuming. This work achieves the objective of constructing a routing graph which is a planar spanner, without requiring the expensive construction of the Delaunay Triangulation, thus saving the node power, an important resource in the ad-hoc network. Moreover, the algorithm guarantees that the total number of messages required to be sent by each node is O(n). This makes the topology easily reconfigurable in case of node motion

Minimum Enclosing Circle of a Set of Fixed Points and a Mobile Point

Given a set S of n static points and a mobile point p in ℝ2, we study the variations of the smallest circle that encloses S ∪ {p} when p moves along a straight line ℓ. In this work, a complete characterization of the locus of the center of the minimum enclosing circle (MEC) of S ∪ {p}, for p ∈ ℓ, is presented. The locus is a continuous and piecewise differentiable linear function, and each of its differentiable pieces lies either on the edges of the farthest-point Voronoi diagram of S, or on a line segment parallel to the line ℓ. Moreover, the locus has differentiable pieces, which can be computed in linear time, given the farthest-point Voronoi diagram of S

Spreadable Connected Autonomic Networks (SCAN)

A Spreadable Connected Autonomic Network (SCAN) is a mobile network that automatically maintains its own connectivity as nodes move. We envision SCANs to enable a diverse set of applications such as self-spreading mesh networks and robotic search and rescue systems. This paper describes our experiences developing a prototype robotic SCAN built from commercial, off-the-shelf hardware, to support such applications. A major contribution of our work is the development of a protocol, called SCAN1, which maintains network connectivity by enabling individual nodes to determine when they must constrain their mobility in order to avoid disconnecting the network. SCAN1 achieves its goal through an entirely distributed process in which individual nodes utilize only local (2-hop) knowledge of the network's topology to periodically make a simple decision: move, or freeze in place. Along with experimental results from our hardware testbed, we model SCAN1's performance, providing both supporting analysis and simulation for the efficacy of SCAN1 as a solution to enable SCANs. While our evaluation of SCAN1 in this paper is limited to systems whose capabilities match those of our testbed, SCAN1 can be utilized in conjunction with a wide-range of potential applications and environments, as either a primary or backup connectivity maintenance mechanism

Kinetic and dynamic data structures for convex hulls and upper envelopes

AbstractLet S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O(n2βs+2(n)logn) critical events, each in O(log2n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, βs(q)=λs(q)/q, and λs(q) is the maximum length of Davenport–Schinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [J. Basch, L.J. Guibas, J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1999) 1–28], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic

Spreadable Connected Autonomic Networks (SCAN)

A Spreadable Connected Autonomic Network (SCAN) is a mobile network that automatically maintains its own connectivity as nodes move. We envision SCANs to enable a diverse set of applications such as self-spreading mesh networks and robotic search and rescue systems. This paper describes our experiences developing a prototype robotic SCAN built from commercial, off-the-shelf hardware, to support such applications. A major contribution of our work is the development of a protocol, called SCAN1, which maintains network connectivity by enabling individual nodes to determine when they must constrain their mobility in order to avoid disconnecting the network. SCAN1 achieves its goal through an entirely distributed process in which individual nodes utilize only local (2-hop) knowledge of the network's topology to periodically make a simple decision: move, or freeze in place. Along with experimental results from our hardware testbed, we model SCAN1's performance, providing both supporting analysis and simulation for the efficacy of SCAN1 as a solution to enable SCANs. While our evaluation of SCAN1 in this paper is limited to systems whose capabilities match those of our testbed, SCAN1 can be utilized in conjunction with a wide-range of potential applications and environments, as either a primary or backup connectivity maintenance mechanism

Pienen dominoivan joukon etsiminen tasoverkossa hajautetusti

Dominoiva joukko D on jonkin verkon solmujen osajoukko siten, että verkon jokainen solmu joko kuuluu D:hen tai on siihen kuuluvan solmun naapuri. Verkon pienimmän dominoivan joukon löytäminen verkossa on NP-kova ongelma. Mikäli algoritmi etsii pienintä joukkoa, jonka koko on x, mutta sen voidaan taata löytävän vain joukon, jonka koko on enintään ax jollakin vakiolla a, kutsutaan tulosjoukkoa oikean ratkaisun a-approksimaatioksi. Rajoitetussa verkossa voidaan löytää pienimmän dominoivan joukon approksimaatioita nopeasti. Hajautetussa laskennassa verkon solmut ovat rinnakkaisesti toimivia prosessoreita, jotka suorittavat kaikki samaa algoritmia. Vakioaikaisuudella tarkoitetaan hajautetun laskennan kontekstissa sitä, että prosessorit saavat vaihtaa naapuriprosessoriensa kanssa vain vakiomäärän viestejä. Algoritmi palauttaa valmistuttuaan jokaiselle solmulle tulosteen, esimerkiksi tiedon siitä, kuuluuko solmu dominoivaan joukkoon. Tässä tutkielmassa tarkastellaan pienimmän dominoivan joukon vakioaikaisia approksimaatioita tasoverkossa hajautetun laskennan näkökulmasta. Ensiksi tutkielmassa esitellään todistus, että ei ole olemassa hajautettua algoritmia, joka löytäisi (5 − epsilon)-approksimaation pienimmästä dominoivasta joukosta, kun epsilon > 0. Tämän jälkeen todistetaan, että myös tiukempi (7 − epsilon)-approksimaatioalaraja pätee. Lopuksi esitellään hajautettu algoritmi joka löytää ainakin 52-approksimaation pienimmästä dominoivasta joukosta tasoverkossa. Algoritmi käyttää uniikkeja tunnisteita eikä prosessorien välisten viestien kokoja ole rajoitettu. Algoritmin analyysissä yhdistetään Lenzenin et al. (2010) esittämä alkuperäinen analyysi ja Wawrzyniakin (2013) esittämä analyysin toisen osan parannus

Deployment and coverage maintenance in mobile sensor networks

Deployment of mobile nodes in a region of interest is a critical issue in building a mobile sensor network because it affects cost and detection capabilities of the system. The deployment of mobile sensors in essence is the movement of sensors from an initial position to a final optimal location. Considerable attention has recently been given to this deployment issue. Many of the distributed deployment schemes use the potential field method. In most cases, the negative gradient of the potential function becomes the feedback control input to a node. This assumes that the potential function is differentiable over the entire region. This assumption is valid primarily when the topology of the network is fixed. In this research, we analyze the stability of a network that uses piecewise smooth potential functions. A gravitation-like force is proposed to deploy a group of agents and to form a certain configuration. We use a nonsmooth version of the Lyapunov stability theory and LaSalle’s invariance principle to show asymptotic stability of the network which is governed by discontinuous dynamics. We propose a hierarchical structure using potential fields for mobile sensor network deployment. A group of mobile nodes first form a cluster using a potential field method and then cluster heads are used to establish a hexagonal structure that employs a higher level potential field. We consider specifically the problem of deploying a mobile sensor network so that a certain area coverage is realized and maintained. And we propose an algorithm for main taining the desired coverage that assumes the availability of a stochastic sensor model. The model reflects the decline of the sensor accuracy as the distance increases from the sensor. It is further assumed that each node’s sensor has a different sensing range to represent sensor performance deterioration due to power decay. The network deployment scheme combines artificial forces with individual sensor ranges. The validity and the effectiveness of the proposed algorithm are compared to the conventional methods in simulations. Simulation results confirm the effectiveness of the proposed algorithms with respect to a defined performance metric

An Update Algorithm for Restricted Random Walk Clusters

This book presents the dynamic extension of the Restricted Random Walk Cluster Algorithm by Schöll and Schöll-Paschinger. The dynamic variant allows to quickly integrate changes in the underlying object set or the similarity matrix into the clusters; the results are indistinguishable from the renewed execution of the original algorithm on the updated data set