Why Extension-Based Proofs Fail

We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve k-set agreement among n > k > 1 processes in a wait-free manner in certain asynchronous models. However, it was unknown whether proofs based on simpler techniques were possible. We show that this impossibility result cannot be obtained for one of these models by an extension-based proof and, hence, extension-based proofs are limited in power.Comment: This version of the paper is for the NIS model. Previous versions of the paper are for the NIIS mode

Searching Polyhedra by Rotating Half-Planes

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.Comment: 45 pages, 26 figure

Gathering in Dynamic Rings

The gathering problem requires a set of mobile agents, arbitrarily positioned at different nodes of a network to group within finite time at the same location, not fixed in advanced. The extensive existing literature on this problem shares the same fundamental assumption: the topological structure does not change during the rendezvous or the gathering; this is true also for those investigations that consider faulty nodes. In other words, they only consider static graphs. In this paper we start the investigation of gathering in dynamic graphs, that is networks where the topology changes continuously and at unpredictable locations. We study the feasibility of gathering mobile agents, identical and without explicit communication capabilities, in a dynamic ring of anonymous nodes; the class of dynamics we consider is the classic 1-interval-connectivity. We focus on the impact that factors such as chirality (i.e., a common sense of orientation) and cross detection (i.e., the ability to detect, when traversing an edge, whether some agent is traversing it in the other direction), have on the solvability of the problem. We provide a complete characterization of the classes of initial configurations from which the gathering problem is solvable in presence and in absence of cross detection and of chirality. The feasibility results of the characterization are all constructive: we provide distributed algorithms that allow the agents to gather. In particular, the protocols for gathering with cross detection are time optimal. We also show that cross detection is a powerful computational element. We prove that, without chirality, knowledge of the ring size is strictly more powerful than knowledge of the number of agents; on the other hand, with chirality, knowledge of n can be substituted by knowledge of k, yielding the same classes of feasible initial configurations

Continuous Tasks and the Asynchronous Computability Theorem

The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized distributed tasks that are wait-free solvable and uncovered deep connections with combinatorial topology. We provide an alternative characterization of those tasks by means of the novel concept of continuous tasks, which have an input/output specification that is a continuous function between the geometric realizations of the input and output complex: We state and prove a precise characterization theorem (CACT) for wait-free solvable tasks in terms of continuous tasks. Its proof utilizes a novel chromatic version of a foundational result in algebraic topology, the simplicial approximation theorem, which is also proved in this paper. Apart from the alternative proof of the ACT implied by our CACT, we also demonstrate that continuous tasks have an expressive power that goes beyond classic task specifications, and hence open up a promising venue for future research: For the well-known approximate agreement task, we show that one can easily encode the desired proportion of the occurrence of specific outputs, namely, exact agreement, in the continuous task specification

Termination Detection of Local Computations

Contrary to the sequential world, the processes involved in a distributed system do not necessarily know when a computation is globally finished. This paper investigates the problem of the detection of the termination of local computations. We define four types of termination detection: no detection, detection of the local termination, detection by a distributed observer, detection of the global termination. We give a complete characterisation (except in the local termination detection case where a partial one is given) for each of this termination detection and show that they define a strict hierarchy. These results emphasise the difference between computability of a distributed task and termination detection. Furthermore, these characterisations encompass all standard criteria that are usually formulated : topological restriction (tree, rings, or triangu- lated networks ...), topological knowledge (size, diameter ...), and local knowledge to distinguish nodes (identities, sense of direction). These results are now presented as corollaries of generalising theorems. As a very special and important case, the techniques are also applied to the election problem. Though given in the model of local computations, these results can give qualitative insight for similar results in other standard models. The necessary conditions involve graphs covering and quasi-covering; the sufficient conditions (constructive local computations) are based upon an enumeration algorithm of Mazurkiewicz and a stable properties detection algorithm of Szymanski, Shi and Prywes

Komplex hálózatok szerkezete és dinamikája = Structure and dynamics of complex networks

A komplex rendszerek tanulmányozásának jelenleg legsikeresebb eszköze a hálózati megközelítés. Az elméleti leírás kereteit tágítottuk azzal, hogy fogalmakat általánosítottunk a súlyozott hálózatok esetére, részletesen elemeztük a modulok meghatározásához használt algoritmusokat, új módszert dolgoztunk ki, valamint elemeztük az eljárások korlátait. A tőzsdei adatok példáján a korrelációs mátrix hatékony zajmentesítési lehetőségeit taulmányoztuk. Kommunikációs adatok elemzésével először sikerült a szociális hálózatra vonatkozó Granovetter-hipotézist, (""a gyenge kötések ereje"") társadalmi méretekben igazolni, és ennek alapján működő modellt konstruálni. A hálózatokon zajló dinamikai jelenségek közül a terjedés az egyik legfontosabb. Vizsgáltuk, hogyan hat a topológia és az élsúlyok kapcsolata az ilyen jelenségekre és mi a katasztrofális kaszkádok mechanizmusa. Bebizonyítottuk, hogy az emberi viselkedés rendkívül inhomogén jellege lényegesen befolyásolja az információterjedés sebességét. Vizsgálatainkból azt a következtetést lehet levonni, hogy annak ellenére, hogy nagyon különböző hálózatok meglepően hasonló sajátosságokat mutathatnak, működési szempontból igen eltérő optimalizációs elveknek felelnek meg. Végül megmutattuk, hogy a komplex hálózatokon, de általában a komplex rendszerekben lezajló dinamika általánosan mutatja a fluktuációs skálázást, elemeztük ennek lehetséges okait, valamint az egyszerű skálázáson túlmutató jelenségeket. | The network approach is presently the most efficient tool to study complex systems. We broadened the framework of theoretical description by generalizing concepts to the case of weighted networks, analyzing in detail community detection algorithms, constructing a new detection method and analyzed the limitations of the procedures. On the example of stock market data we studied the possibilities of denoising efficiently the correlation matrix. Using communication data we proved for the first time on a societal scale the Granovetter hypothesis (""The strength of weak ties"") on the social network. One of the most important dynamic phenomena on networks is that of spreading. We investigated how the topology and its relation to the link weights affect such phenomena and what is the mechanism of catastrophic cascades. We proved that the inhomogeneous, bursty character of human behavior substantially influences the speed of spreading of information. We can conclude from our investigations that in spite of the fact that very different networks may show surprisingly similar properties, they obey very different optimization principles from the point of view of their functioning. Finally, we showed that dynamics in complex networks but in complex systems in general shows fluctuation scaling, we analyzed the possible origins and the phenomena, which go beyond simple scaling