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Consider a team of mobile software agents deployed to capture a (possibly hostile) intruder in a network. All agents, including the intruder move along the network links; the intruder could be arbitrarily fast, and aware of the positions of all the agents. The problem is to design the agents' strategy for capturing the intruder. The main efficiency parameter is the size of the team. This is an instance of the well known graph-searching problem whose many variants have been extensively studied in the literature. In all existing solutions, and in all the variants of the problem, it is assumed that agents can be removed from their current location and placed in another network site arbitrarily and at any time. As a consequence, the existing optimal strategies cannot be employed in situations for which agents cannot access the network at any point, or cannot "jump" across the network, or cannot reach an arbitrary point of the network via an internal travel through insecure zones. This motivates the contiguous search problem in which agents cannot be removed from the network, and clear links must form a connected sub-network at any time, providing safety of movements. This new problem is NP-complete in general. We study it for tree networks, and we consider its more general version, the weighted case, which arises naturally when considering networks whose nodes and links are of different nature and thus require a different number of agents to be explored. We give a linear-time algorithm that computes, for any tree T , the minimum number of agents to capture the intruder, and the corresponding search strategy. Beside its optimality in time, our algorithm is naturally distributed:

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A central theme in distributed network algorithms concerns understanding and coping with the issue of locality . Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems . In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD( t ) (for local decision ) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD( t , p , q ), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p , and rejects incorrect ones with probability at least q . We show that p 2 + q = 1 is a threshold for the containment of LD( t ) in BPLD( t , p , q ). More precisely, we show that there exists a language that does not belong to LD( t ) for any t = o ( n ) but does belong to BPLD( 0 , p , q ) for any p , q ∈ (0,1) such that p 2 + q ≤ 1. On the other hand, we show that, restricted to hereditary languages, BPLD( t , p , q )=LD( O ( t )), for any function t , and any p , q ∈ (0,1) such that p 2 + q > 1. In addition, we investigate the impact of nondeterminism on local decision, and establish several structural results inspired by classical computational complexity theory. Specifically, we show that nondeterminism does help, but that this help is limited, as there exist languages that cannot be decided locally nondeterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with nondeterminism that enables to decide all languages in constant time . Finally, we introduce the notion of local reduction, and establish a couple of completeness results.

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.

Abstract. Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is (D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound (n 2 ) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

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