Ralf Klasing

Gathering asynchronous oblivious mobile robots in a ring

We consider the problem of gathering identical, memoryless, mobile robots in one node of an anonymous unoriented ring. Robots start from different nodes of the ring. They operate in Look-Compute-Move cycles and have to end up in the same node. In one cycle, a robot takes a snapshot of the current configuration (Look), makes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case makes an instantaneous move to this neighbor (Move). Cycles are performed asynchronously for each robot. For an odd number of robots we prove that gathering is feasible if and only if the initial configuration is not periodic, and we provide a gathering algorithm for any such configuration. For an even number of robots we decide the feasibility of gathering except for one type of symmetric initial configurations, and provide gathering algorithms for initial configurations proved to be gatherable.

Dissemination of Information in Interconnection Networks (Broadcasting & Gossiping)

Considerable attention in recent theoretical computer science is devoted to parallel computing. Here, we would like to present a special part of this large topic, namely, the part devoted to an abstract study of the dissemination of information in interconnection networks. The importance of this research area lies in the fact that the ability of a network to effectively disseminate information is an important qualitative measure for the suitabilty of the network for parallel computing. This follows simply from the observation that the communication among processes working in parallel is one of the main parts of the whole parallel computation. So, the effectivity of information exchange among processors essentially influences the effectivity of the whole computation process.

Searching for black-hole faults in a network using multiple agents

We consider a fixed communication network where (software) agents can move freely from node to node along the edges. A black hole is a faulty or malicious node in the network such that if an agent enters this node, then it immediately “dies.” We are interested in designing an efficient communication algorithm for the agents to identify all black holes. We assume that we have k agents starting from the same node s and knowing the topology of the whole network. The agents move through the network in synchronous steps and can communicate only when they meet in a node. At the end of the exploration of the network, at least one agent must survive and must know the exact locations of the black holes. If the network has n nodes and b black holes, then any exploration algorithm needs Ω(n/k + Db) steps in the worst-case, where Db is the worst case diameter of the network with at most b nodes deleted. We give a general algorithm which completes exploration in O((n/k)logn/loglogn + bDb) steps for arbitrary networks, if b≤k/2. In the case when b≤k/2, and , we give a refined algorithm which completes exploration in asymptotically optimal O(n/k) steps.

Hardness and approximation results for Black Hole Search in arbitrary networks

A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node without leaving any trace. The Black Hole Search is the task of locating all black holes in a network by exploring it with mobile agents. We consider the problem of designing the fastest Black Hole Search, given the map of the network and the starting node. We study the version of this problem that assumes that there is at most one black hole in the network and there are two agents, which move in synchronized steps. We prove that this problem is NP-hard in arbitrary graphs (even in planar graphs), solving an open problem stated in [J. Czyzowicz, D. Kowalski, E. Markou, A. Pelc, Searching for a black hole in tree networks, in: Proc. 8th Int. Conf. on Principles of Distributed Systems, OPODIS 2004, 2004, pp. 34-35. Also: Springer LNCS, vol. 3544, pp. 67-80]. We also give a -approximation algorithm, showing the first non-trivial approximation ratio upper bound for this problem. Our algorithm follows a natural approach of exploring networks via spanning trees. We prove that this approach cannot lead to an approximation ratio bound better than 3/2.

Taking advantage of symmetries: Gathering of many asynchronous oblivious robots on a ring

One of the recently considered models of robot-based computing makes use of identical, memoryless mobile units placed in nodes of an anonymous graph. The robots operate in Look-Compute-Move cycles; in one cycle, a robot takes a snapshot of the current configuration (Look), takes a decision whether to stay idle or to move to one of the nodes adjacent to its current position (Compute), and in the latter case makes an instantaneous move to this neighbor (Move). Cycles are performed asynchronously for each robot. In such a restricted scenario, we study the influence of symmetries of the robot configuration on the feasibility of certain computational tasks. More precisely, we deal with the problem of gathering all robots at one node of the graph, and propose a solution based on a symmetry-preserving strategy. When the considered graph is an undirected ring and the number of robots is sufficiently large (more than 18), such an approach is proved to solve the problem for all starting situations, as long as gathering is feasible. In this way we also close the open problem of characterizing symmetric situations on the ring which admit a gathering [R. Klasing, E. Markou, A. Pelc: Gathering asynchronous oblivious mobile robots in a ring, Theoret. Comput. Sci. 390 (1) (2008) 27-39]. The proposed symmetry-preserving approach, which is complementary to symmetry-breaking techniques found in related work, appears to be new and may have further applications in robot-based computing.

Hardness results and approximation algorithms of k -tuple domination in graphs

. In a graphG, a vertex is said todominate itself and all of its neighbors. Adominating set of G = (V ,E) is a subsetD of V such that every vertex inV is dominated by at least one vertex in D. Domination and its variations have many applications, and have b extensively studied in the literature, see [4,8,9]. Among the variations of domination, the k-tuple domination problem was introduced in [7,8]. For fixed positive integerk, a k-tuple dominating set of G = (V ,E) is a subsetDk of V such that every vertex inV is dominated by at least k vertices ofD. The special case when k = 1 is the usual domination The case whenk = 2 is called double domination in [7] where exact values of the double dominati

Hardness and approximation of gathering in static radio networks

In this paper, we address the problem of gathering information in a central node of a radio network, where interference constraints are present. We take into account the fact that, when a node transmits, it produces interference in an area bigger than the area in which its message can actually be received. The network is modeled by a graph; a node is able to transmit one unit of information to the set of vertices at distance at most d/sub T/ in the graph, but when doing so it generates interference that does not allow nodes at distance up to d/sub I/ (d/sub I/ /spl ges/ d/sub T/) to listen to other transmissions. Time is synchronous and divided into time-steps in each of which a round (set of non-interfering radio transmissions) is performed. We give a general lower bound on the number of rounds required to gather on any graph, and present an algorithm working on any graph, with an approximation factor of 4. We also show that the problem of finding an optimal strategy for gathering (one that uses a minimum number of time-steps) does not admit a fully polynomial time approximation scheme if d/sub I/ > d/sub T/, unless P=NP, and in the case d/sub I/ = d/sub T/ the problem is NP-hard.

Improved approximation results for the minimum energy broadcasting problem

In this paper we present new results on the performance of the Minimum Spanning Tree heuristic for the Minimum-Energy Broadcast Routing (MEBR) problem. We first prove that, for any number of dimensions d ≥ 2, the approximation ratio of the heuristic does not increase when the power attenuation coefficient α, that is the exponent to which the coverage distance must be raised to give the emission power, grows. Moreover, we show that, as a limit for α going to infinity, the ratio tends to the lower bound of [3, 15] given by the d-dimensional kissing number, thus closing the existing gap between the upper and the lower bound. We then introduce a new analysis allowing to establish a 7.6-approximation ratio for the 2-dimensional case, thus signifcantly decreasing the previously known 12 upper bound [15] (actually corrected to 12.15 in [10]). Starting from the above results, such an approximation holds for any α ≥ 2. Finally, we extend our analysis to any number of dimensions d ≥ 2 and any α ≥ d, obtaining a general approximation ratio of 3d-1, independent of α. The improvements of the approximation ratios are specifically significant in comparison with the lower bounds given by the kissing numbers, as these grow at least exponentially with respect to d. Note that for α ‹ d the ratios cannot be bounded by any function of α and d [3].

Adaptive Broadcast Consumption (ABC), a New Heuristic and New Bounds for the Minimum Energy Broadcast Routing Problem

In this paper we present a new heuristic called Adaptive Broadcast Consumption (ABC for short) for the Minimum-Energy Broadcast Routing (MEBR) problem. We first investigate the problem trying to understand which are the main properties not taken into account by the classic and well-studied MST and BIP heuristics, then we propose a new algorithm proving that it computes the MEBR with an approximation ratio less than or equal to MST, for which we prove an approximation ratio of at most 12.15 instead of the well-known 12 [10]. Finally we present experimental results supporting our intuitive ideas, comparing ABC with other heuristics presented in the literature and showing its good performance on random instances even compared to the optimum.

Taking Advantage of Symmetries: Gathering of Asynchronous Oblivious Robots on a Ring

One of the recently considered models of robot-based computing makes use of identical, memoryless mobile units placed in nodes of an anonymous graph. The robots operate in Look-Compute-Move cycles; in one cycle, a robot takes a snapshot of the current configuration (Look), takes a decision whether to stay idle or to move to one of the nodes adjacent to its current position (Compute), and in the latter case makes an instantaneous move to this neighbor (Move). Cycles are performed asynchronously for each robot. In such a restricted scenario, we study the influence of symmetries of the robot configuration on the feasibility of certain computational tasks. More precisely, we deal with the problem of gathering all robots at one node of the graph, and propose a solution based on a symmetry-preserving strategy. When the considered graph is an undirected ring and the number of robots is sufficiently large (more than 18), such an approach is proved to solve the problem for all starting situations, as long as gathering is feasible. In this way we also close the open problem of characterizing symmetric situations on the ring which admit a gathering [R. Klasing, E. Markou, A. Pelc: Gathering asynchronous oblivious mobile robots in a ring, Theor. Comp. Sci. 390(1), 27-39, 2008]. The proposed symmetry-preserving approach, which is complementary to symmetry-breaking techniques found in related work, appears to be new and may have further applications in robot-based computing.