Jurek Czyzowicz

Gathering few fat mobile robots in the plane

Autonomous identical robots represented by unit discs move deterministically in the plane. They do not have any common coordinate system, do not communicate, do not have memory of the past and are totally asynchronous. Gathering such robots means forming a configuration for which the union of all discs representing them is connected. We solve the gathering problem for at most four robots. This is the first algorithmic result on gathering robots represented by two-dimensional figures rather than points in the plain: we call such robots fat.

On minimizing the sum ofensor movements for barrier coverage of a line segment

A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned.

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.

Ulam's searching game with lies

Abstract We determine the minimal number of yes-no queries sufficient to find an unknown integer between 1 and 2 m if at most two of the answers may be erroneous.

Strategies for Hotlink Assignments

Consider a DAG (directed acyclic graph) G = (V, E) representing a collection V of web pages connected via links E. All web pages can be reached from a designated source page, represented by a source node s of G. Each web page carries a weight representative of the frequency with which it is visited. By adding hotlinks, at most one per page, we are interested in minimizing the expected number of steps needed to visit a selected set of web pages from the source page. For arbitrary DAGs we show that the problem is NP-complete. We also give algorithms for assigning hotlinks, as well as upper and lower bounds on the expected number of steps to reach the leaves from the source page s located at the root of a complete binary tree. Depending on the probability distribution (arbitrary, uniform, Zipf) the expected number of steps is at most c ċ n, where c is a constant less than 1. For the geometric distribution we show how to obtain a constant average number of steps.

Searching for a Black Hole in Synchronous Tree 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. We consider the task of locating a black hole in a (partially) synchronous tree network, assuming an upper bound on the time of any edge traversal by an agent. The minimum number of agents capable of identifying a black hole is two. For a given tree and given starting node we are interested in the fastest-possible black hole search by two agents. For arbitrary trees we give a 5/3-approximation algorithm for this problem. We give optimal black hole search algorithms for two ‘extreme’ classes of trees: the class of lines and the class of trees in which any internal node (including the root which is the starting node) has at least two children.

Immobilizing a polytope

We say that a polygon P is immobilized by a set of points I on its boundary if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P. Three immobilization points are always sufficient for a polygon with vertices in general positions, but four points are necessary for some polygons with parallel edges. An O(n log n) algorithm that finds a set of 3 points that immobilize a given polygon with vertices in general positions is suggested. The algorithm becomes linear for convex polygons. Some results are generalized for d-dimensional polytopes, where 2d points are always sufficient and sometimes necessary to immobilize. When the polytope has vertices in general position d+1 points are sufficient to immobilize.

Local algorithms for dominating and connected dominating sets of unit disk graphs with location aware nodes

Many protocols in ad-hoc networks use dominating and connected dominating sets, for example for broadcasting and routing. For large ad hoc networks the construction of such sets should be local in the sense that each node of the network should make decisions based only on the information obtained from nodes located a constant number of hops from it. In this paper we use the location awareness of the network, i.e. the knowledge of position of nodes in the plane to provide local, constant approximation, deterministic algorithms for the construction of dominating and connected dominating sets of a Unit Disk Graph (UDG). The size of the constructed set, in the case of the dominating set, is shown to be 5 times the optimal, while for the connected dominating set 7.453 + Ɛ the optimal, for any arbitrarily small Ɛ > 0. These are to our knowledge the first local algorithms whose time complexities and approximation bounds are independent of the size of the network.

Searching for a black hole in tree 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. We consider the task of locating a black hole in a (partially) synchronous tree network, assuming an upper bound on the time of any edge traversal by an agent. The minimum number of agents capable to identify a black hole is two. For a given tree and given starting node we are interested in the fastest possible black hole search by two agents. For arbitrary trees we give a 5/3-approximation algorithm for this problem. We give optimal black hole search algorithms for two “extreme” classes of trees: the class of lines and the class of trees in which any internal node (including the root which is the starting node) has at least 2 children.