Leszek Gąsieniec

Boundary patrolling by mobile agents with distinct maximal speeds

A set of k mobile agents are placed on the boundary of a simply connected planar object represented by a cycle of unit length. Each agent has its own predefined maximal speed, and is capable of moving around this boundary without exceeding its maximal speed. The agents are required to protect the boundary from an intruder which attempts to penetrate to the interior of the object through a point of the boundary, unknown to the agents. The intruder needs some time interval of length τ to accomplish the intrusion. Will the intruder be able to penetrate into the object, or is there an algorithm allowing the agents to move perpetually along the boundary, so that no point of the boundary remains unprotected for a time period τ? Such a problem may be solved by designing an algorithm which defines the motion of agents so as to minimize the idle time I, i.e., the longest time interval during which any fixed boundary point remains unvisited by some agent, with the obvious goal of achieving I < τ. Depending on the type of the environment, this problem is known as either boundary patrolling or fence patrolling in the robotics literature. The most common heuristics adopted in the past include the cyclic strategy, where agents move in one direction around the cycle covering the environment, and the partition strategy, in which the environment is partitioned into sections patrolled separately by individual agents. This paper is, to our knowledge, the first study of the fundamental problem of boundary patrolling by agents with distinct maximal speeds. In this scenario, we give special attention to the performance of the cyclic strategy and the partition strategy. We propose general bounds and methods for analyzing these strategies, obtaining exact results for cases with 2, 3, and 4 agents. We show that there are cases when the cyclic strategy is optimal, cases when the partition strategy is optimal and, perhaps more surprisingly, novel, alternative methods have to be used to achieve optimality.

Almost optimal asynchronous rendezvous in infinite multidimensional grids

Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ > 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(dδ polylog d). This bound for the case of 2D-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(dδ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(dδ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O((d/r)δ polylog(d/r)), where r = min(r1, r2) and for r ≥ 1.

Oblivious gossiping in ad-hoc radio networks

We study oblivious deterministic and randomized algorithms for gossiping in unknown radio networks. In oblivious algorithms the fact (or probability in case of randomized algorithm) that a processor transmits or not at a given time-step depends solely on its identification number, the total number of processors and the number of the time-step. We distinguish oblivious deterministic algorithms which allow only one processor to transmit in each time-step and term them singleton algorithms. We also distinguish oblivious randomized algorithms where in each time-step all processors have equal probability of transmission, and call them uniform. The merit of oblivious algorithms, especially the singleton and uniform ones, is that they are simple and easy to implement. We observe that gossiping in unknown radio networks on <i>n</i> nodes can be completed in time (<i>n</i> - 1)(<i>n</i> - 2) + 4 by a singleton algorithm. On the other hand, we show that any singleton algorithm takes at least <i>n</i>² - <i>&Ogr;</i>(<i>n</i>^7/4+∈) steps, for any ∈ > 0, whereas any deterministic oblivious algorithm requires at least <i>n</i>²/2 - <i>&Ogr;</i>(<i>n</i>) steps to complete the gossiping. We prove also that there is an oblivious deterministic algorithm for gossiping working in time <i>n</i>² - <i>w</i>(<i>n</i>). Next we show that a uniform oblivious randomized algorithm completes gossiping with high probability in time <i>&Ogr;</i>(min{<i>m, Dd</i>} log² <i>n</i>), where <i>m</i> denotes the number of edges, <i>D</i> is the eccentricity and <i>d</i> the maximum, in-degree in the network. Note that this upper bound is poly-logarithmic in <i>n</i> if <i>D, d</i> = <i>&Ogr;</i>(<i>poly</i> log <i>n</i>). The best related deterministic gossiping algorithm, in terms of performance expressed with respect to <i>n</i>, <i>D</i>, <i>d</i>, has been previously given by Clementi et al. [13], it works in time <i>&Ogr;</i>(<i>Dd</i>² log³ <i>n</i>). We prove also that the upper bound attained by our uniform oblivious randomized algorithm is asymptotically optimal (up to a log-square factor) for a wide range of parameters <i>m</i>, <i>D</i> and <i>d</i> in the class of uniform oblivious randomized algorithms. Finally we observe that in case of symmetric networks the aforementioned oblivious randomized algorithm completes gossiping with high probability in time <i>&Ogr;</i>(<i>n</i> log² <i>n</i>) and that a known deterministic constructive broadcasting algorithm can be adopted to perform oblivious gossiping in time <i>&Ogr;</i>(<i>n</i>^3/2).

On adaptive deterministic gossiping in ad hoc radio networks

We study deterministic algorithms for gossiping problem in <i>ad hoc</i> radio networks. The <i>gossiping problem</i> is a communication task in which each node of the network possesses a unique single message that is to be communicated to all other nodes in the network. The efficiency of a communication algorithm in radio networks is very often expressed in terms of: <i>max-eccentricity D</i>, <i>max-indegree</i> Δ, and <i>size</i> (number of nodes) <i>n</i> of underlying graph of connections. The max-eccentricity <i>D</i> of a network is the maximum of the lengths of shortest directed paths from a node <i>u</i> to a node <i>v,</i> taken over all ordered pairs (<i>u, v</i>) of nodes in the network. The max-indegree Δ of a network is the maximum of indegrees of its nodes.We propose a new method that leads to several improvements in deterministic gossiping. It combines communication techniques designed for both known as well as unknown <i>ad hoc</i> radio networks. First we show how to subsume the <i>O</i>(<i>Dn</i>)-time bound yield by the Round-Robin procedure proposing a new <i>Õ</i>(√<i>Dn</i>)-time gossiping algorithm. Our algorithm is more efficient than the known Õ(<i>n</i>^3/2)-time gossiping algorithms [3, 6], whenever <i>D</i> = <i>O</i>(<i>n^α</i>) and α < 1. For large values of max-eccentricity <i>D,</i> we give another gossiping algorithm that works in time <i>O</i>(<i>D</i>Δ^3/2 log³ <i>n</i>) which subsumes the <i>O</i>(<i>D</i>&Delta² log³ <i>n</i>) upper bound presented in [4].

On the wake-up problem in radio networks

Radio networks model wireless communication when processing units communicate using one wave frequency. This is captured by the property that multiple messages arriving simultaneously to a node interfere with one another and none of them can be read reliably. We present improved solutions to the problem of waking up such a network. This requires activating all nodes in a scenario when some nodes start to be active spontaneously, while every sleeping node needs to be awaken by receiving successfully a message from a neighbor. Our contributions concern the existence and efficient construction of universal radio synchronizers, which are combinatorial structures introduced in [6] as building blocks of efficient wake-up algorithms. First we show by counting that there are (n,g)-universal synchronizers for

Optimal memory rendezvous of anonymous mobile agents in a unidirectional ring

g(k)={\mathcal O}(k \ {\rm log}\ k \ {\rm log}\ n)

Gathering few fat mobile robots in the plane

. Next we show an explicit construction of (n,g)-universal-synchronizers for

Tree exploration with logarithmic memory

g(k) = {\mathcal O}(k^{2}{\rm polylog}\ n)

Adaptive broadcasting with faulty nodes

. By way of applications, we obtain an existential wake-up algorithm which works in time

Tell Me Where I Am So I Can Meet You Sooner

{\mathcal O}(n {\rm log}^{2}n)