Deterministic Symmetry Breaking in Ring Networks

We study a distributed coordination mechanism for uniform agents located on a circle. The agents perform their actions in synchronised rounds. At the beginning of each round an agent chooses the direction of its movement from clockwise, anticlockwise, or idle, and moves at unit speed during this round. Agents are not allowed to overpass, i.e., when an agent collides with another it instantly starts moving with the same speed in the opposite direction (without exchanging any information with the other agent). However, at the end of each round each agent has access to limited information regarding its trajectory of movement during this round. We assume that $n$ mobile agents are initially located on a circle unit circumference at arbitrary but distinct positions unknown to other agents. The agents are equipped with unique identifiers from a fixed range. The {\em location discovery} task to be performed by each agent is to determine the initial position of every other agent. Our main result states that, if the only available information about movement in a round is limited to %information about distance between the initial and the final position, then there is a superlinear lower bound on time needed to solve the location discovery problem. Interestingly, this result corresponds to a combinatorial symmetry breaking problem, which might be of independent interest. If, on the other hand, an agent has access to the distance to its first collision with another agent in a round, we design an asymptotically efficient and close to optimal solution for the location discovery problem.Comment: Conference version accepted to ICDCS 201

Faster Geometric Algorithms via Dynamic Determinant Computation

The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total computation time is consumed by these computations. In this paper we study the sequences of determinants that appear in geometric algorithms. The computation of a single determinant is accelerated by using the information from the previous computations in that sequence. We propose two dynamic determinant algorithms with quadratic arithmetic complexity when employed in convex hull and volume computations, and with linear arithmetic complexity when used in point location problems. We implement the proposed algorithms and perform an extensive experimental analysis. On one hand, our analysis serves as a performance study of state-of-the-art determinant algorithms and implementations. On the other hand, we demonstrate the supremacy of our methods over state-of-the-art implementations of determinant and geometric algorithms. Our experimental results include a 20 and 78 times speed-up in volume and point location computations in dimension 6 and 11 respectively.Comment: 29 pages, 8 figures, 3 table