Locally finding a solution to symmetry-breaking tasks such as vertex-coloring, edge-coloring, maximal matching, maximal independent set, etc., is a long-standing challenge in distributed network computing. More recently, it has also become a challenge in the framework of centralized local computation. We introduce conflict coloring as a general symmetry-breaking task that includes all the aforementioned tasks as specific instantiations -conflict coloring includes all locally checkable labeling tasks from [Naor & Stockmeyer, STOC 1993]. Conflict coloring is characterized by two parameters l and d, where the former measures the amount of freedom given to the nodes for selecting their colors, and the latter measures the number of constraints which colors of adjacent nodes are subject to. We show that, in the standard LOCAL model for distributed network computing, if l/d > ∆, then conflict coloring can be solved in O( √ ∆) + log * n rounds in n-node graphs with maximum degree ∆, where O ignores the polylog factors in ∆. The dependency in n is optimal, as a consequence of the Ω(log * n) lower bound by [Linial, SIAM J. Comp. 1992] for (∆ + 1)-coloring. An important special case of our result is a significant improvement over the best known algorithm for distributed (∆+ 1)-coloring due to [Barenboim, PODC 2015], which required O(∆ 3/4 ) + log * n rounds. Improvements for other variants of coloring, including (∆ + 1)-list-coloring, (2∆ − 1)-edge-coloring, coloring with forbidden color distances, etc., also follow from our general result on conflict coloring. Likewise, in the framework of centralized local computation algorithms (LCAs), our general result yields an LCA which requires a smaller number of probes than the previously best known algorithm for vertex-coloring, and works for a wide range of coloring problems.

Two identical (anonymous) mobile agents start from arbitrary nodes in an a priori unknown graph and move synchronously from node to node with the goal of meeting. This rendezvous problem has been thoroughly studied, both for anonymous and for labeled agents, along with another basic task, that of exploring graphs by mobile agents. The rendezvous problem is known to be not easier than graph exploration. A well-known recent result on exploration, due to Reingold, states that deterministic exploration of arbitrary graphs can be performed in log-space, i.e., using an agent equipped with O(log n) bits of memory, where n is the size of the graph. In this paper we study the size of memory of mobile agents that permits us to solve the rendezvous A preliminary version of this paper appeared in

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Any organism faces sensory and cognitive limitations which may result in maladaptive decisions. Such limitations are prominent in the context of groups where the relevant information at the individual level may not coincide with collective requirements. Here, we study the navigational decisions exhibited by Paratrechina longicornis ants as they cooperatively transport a large food item. These decisions hinge on the perception of individuals which often restricts them from providing the group with reliable directional information. We find that, to achieve efficient navigation despite partial and even misleading information, these ants employ a locally-blazed trail. This trail significantly deviates from the classical notion of an ant trail: First, instead of systematically marking the full path, ants mark short segments originating at the load. Second, the carrying team constantly loses the guiding trail. We experimentally and theoretically show that the locally-blazed trail optimally and robustly exploits useful knowledge while avoiding the pitfalls of misleading information.DOI: http://dx.doi.org/10.7554/eLife.20185.001

The goal of a hub-based distance labeling scheme for a network G = (V, E) is to assign a small subset S(u) ⊆ V to each node u ∈ V , in such a way that for any pair of nodes u, v, the intersection of hub sets S(u) ∩ S(v) contains a node on the shortest uv-path.The existence of small hub sets, and consequently efficient shortest path processing algorithms, for road networks is an empirical observation. A theoretical explanation for this phenomenon was proposed by Abraham et al. (SODA 2010) through a network parameter they called highway dimension, which captures the size of a hitting set for a collection of shortest paths of length at least r intersecting a given ball of radius 2r. In this work, we revisit this explanation, introducing a more tractable (and directly comparable) parameter based solely on the structure of shortest-path spanning trees, which we call skeleton dimension. We show that skeleton dimension admits an intuitive definition for both directed and undirected graphs, provides a way of computing labels more efficiently than by using highway dimension, and leads to comparable or stronger theoretical bounds on hub set size.

The results of space-like separated measurements are independent of distant measurement settings, a property one might call two-way no-signalling. In contrast, time-like separated measurements are only one-way no-signalling since the past is independent of the future but not vice-versa. For this reason some temporal correlations that are formally identical to non-classical spatial correlations can still be modelled classically. We propose a new formulation of Bell's theorem for temporal correlations, namely we define non-classical temporal correlations as the ones which cannot be simulated by propagating in time the classical information content of a quantum system given by the Holevo bound. We first show that temporal correlations between results of any projective quantum measurements on a qubit can be simulated classically. Then we present a sequence of POVM measurements on a single m-level quantum system that cannot be explained by propagating in time an m-level classical system and using classical computers with unlimited memory.Introduction. The violation of a Bell inequality [1][2][3] demonstrates that the outcomes of an experiment have contradicted a set of well defined classical intuitions. Quantum mechanics allows correlations between spacelike separated parties that have no explanation in terms of a hidden variable model, i.e. they cannot be reproduced with the help of classical computers running preagreed algorithms. However, when correlations are generated in a temporal scenario, by a sequence of time-like separated measurements, it is more difficult to demonstrate their non-classical nature. The causal structure of physics implies only one-way no-signalling, namely the impossibility of sending communication backwards in time. The only bound on forward signalling is the information capacity of the physical system.Here we analyse a single quantum system measured at n points in time and consider to what extent one can prove that the temporal correlations between these measurement outcomes could not be generated by a classical system. We assume an idealization, in which the m-level physical system carries no hidden degrees of freedom. In this case the classical information capacity of the system is log 2 m; known as the Holevo bound [4].The previous approach to demonstrate non-classicality of temporal quantum correlations are so called "temporal Bell inequalities" [5][6][7][8][9][10][11][12][13][14]. One of the problems with this approach, stated in [13], is that the classical assumptions behind the temporal inequalities, which are realism and non-invasiveness, were originally chosen to test the quantumness of a temporal evolution of macroscopic quantum systems [5]. As such, they do not provide a convincing test of quantumness in the case of a single evolving sys-

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A set of mobile robots is deployed on a simple curve of finite length, composed of a finite set of vital segments separated by neutral segments. The robots have to patrol the vital segments by perpetually moving on the curve, without exceeding their maximum speed. The quality of patrolling is measured by the idleness, i.e., the longest time period during which any vital point on the curve is not visited by any robot. Given a configuration of vital segments, our goal is to provide algorithms describing the movement of the robots along the curve so as to minimize the idleness.Our main contribution is a proof that the optimal solution to the patrolling problem is attained either by the cyclic strategy, in which all the robots move in one direction around the curve, or by the partition strategy, in which the curve is partitioned into sections which are patrolled separately by individual robots. These two fundamental types of strategies were studied in the past in the robotics community in different theoretical and experimental settings. However, to our knowledge, this is the first theoretical analysis proving optimality in such a general scenario. Throughout the paper we assume that all robots have the same maximum speed. In fact, the claim is known to be invalid when this assumption does not hold, cf. [Czyzowicz et al., Proc. ESA 2011].