Tobias Langner

A tight runtime bound for synchronous gathering of autonomous robots with limited visibility

The problem of gathering n autonomous robots in the Euclidean plane at one (not predefined) point is well-studied under various restrictions on the capabilities of the robots and in several time models. However, only very few runtime bounds are known. We consider the scenario of local algorithms in which the robots can only observe their environment within a fixed viewing range and have to base their decision where to move in the next step solely on the relative positions of the robots within their viewing range. Such local algorithms have to guarantee that the (initially connected) unit disk graph defined by the viewing range of the robots stays connected at all times. In this paper, we focus on the synchronous setting in which all robots are activated concurrently. Ando et al. [2] presented an algorithm where a robot essentially moves to the center of the smallest enclosing circle of the robots in its viewing range and showed that this strategy performs gathering of the robots in finite time. However, no bounds on the number of rounds needed by the algorithm are known. We present a lower bound of ©(n2) for the number of rounds as well as a matching upper bound of O(n2) and thereby obtain a tight runtime analysis of the algorithm of Θ(n).

Solving the ANTS Problem with Asynchronous Finite State Machines

Consider the Ants Nearby Treasure Search (ANTS) problem introduced by Feinerman, Korman, Lotker, and Sereni (PODC 2012), where n mobile agents, initially placed in a single cell of an infinite grid, collaboratively search for an adversarially hidden treasure. In this paper, the model of Feinerman et al. is adapted such that each agent is controlled by an asynchronous (randomized) finite state machine: they possess a constant-size memory and can locally communicate with each other through constant-size messages. Despite the restriction to constant-size memory, we show that their collaborative performance remains the same by presenting a distributed algorithm that matches a lower bound established by Feinerman et al. on the run-time of any ANTS algorithm.

How Many Ants Does It Take to Find the Food

Consider the Ants Nearby Treasure Search (ANTS) problem, where n mobile agents, initially placed at the origin of an infinite grid, collaboratively search for an adversarially hidden treasure. The agents are controlled by deterministic/randomized finite or pushdown automata and are able to communicate with each other through constant-size messages. We show that the minimum number of agents required to solve the ANTS problem crucially depends on the computational capabilities of the agents as well as the timing parameters of the execution environment. We give lower and upper bounds for different scenarios.

Fault-Tolerant ANTS

In this paper, we study a variant of the Ants Nearby Treasure Search problem, where n mobile agents, controlled by finite automata, search collaboratively for a treasure hidden by an adversary. In our version of the model, the agents may fail at any time during the execution. We provide a distributed protocol that enables the agents to detect failures and recover from them, thereby providing robustness to the protocol. More precisely, we provide a protocol that allows the agents to locate the treasure in time \(\mathcal{O}(D + D^2/n + Df)\) where D is the distance to the treasure and \(f \in \mathcal{O}(n)\) is the maximum number of failures.

The Price of Matching with Metric Preferences

We consider a version of the Gale-Shapley stable matching setting, where each pair of nodes is associated with a (symmetric) matching cost and the preferences are determined with respect to these costs. This stable matching version is analyzed through the Price of Anarchy (PoA) and Price of Stability (PoS) lens under the objective of minimizing the total cost of matched nodes (for both the marriage and roommates variants). A simple example demonstrates that in the general case, the PoA and PoS are unbounded, hence we restrict our attention to metric costs. We use the notion of α-stability, where a pair of unmatched nodes defect only if both improve their costs by a factor greater than α ≥ 1. Our main result is an asymptotically tight trade-off, showing that with respect to α-stable matchings, the Price of Stability is \(\Theta \big( n^{\log ( 1 + \frac{1}{2 \alpha} )} \big)\). The proof is constructive: we present a simple algorithm that outputs an α-stable matching satisfying this bound.

How many ants does it take to find the food

Consider the Ants Nearby Treasure Search (ANTS) problem, where n mobile agents, initially placed at the origin of an infinite grid, collaboratively search for an adversarially hidden treasure. The agents are controlled by deterministic/randomized finite or pushdown automata and are able to communicate with each other through constant-size messages. We show that the minimum number of agents required to solve the ANTS problem crucially depends on the computational capabilities of the agents as well as the timing parameters of the execution environment. We give lower and upper bounds for different scenarios.

Versioning Tree Structures by Path-Merging

We propose path-merging as a refinement of techniques used to make linked data structures partially persistent. Path-merging supports bursts of operations between any two adjacent versions in contrast to only one operation in the original variant. The superiority of the method is shown both theoretically and experimentally. Details of the technique are explained for the case of binary search trees. Path-merging is particularly useful for the implementation of scan-line algorithms where many update operations on the sweep status structure have to be performed at the same event points. Examples are algorithms for planar point location, for answering intersection queries for sets of horizontal line segments, and for detecting conflicts in sets of 1-dim IP packet filters. Subject Classifications: E.1 [ Data ]: Data Structures --- trees; E.2 [ Data ]: Data Storage Representations --- linked representations; F.2.2 [ Analysis of Algorithms and Problem Complexity ] Nonnumerical Algorithms and Problems --- Geometrical problems and computations.