Abstract. We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely constantly connected dynamic graphs. This problem has already been studied in the case where the agent knows the dynamics of the graph and the underlying graph is a ring of n vertices [5]. In this paper, we consider the same problem and we suppose that the underlying graph is a cactus graph (a connected graph in which any two simple cycles have at most one vertex in common). We propose an algorithm that allows the agent to explore these dynamic graphs in at most 2 O( √ log n) n time units. We show that the lower bound of the algorithm is 2 Ω( √ log n) n time units.

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We study the problem of finding a destination node t by a mobile agent in an unreliable network having the structure of an unweighted graph, in a model first proposed by Hanusse et al. [21,20]. Each node is able to give advice concerning the next node to visit so as to go closer to the target t. Unfortunately, exactly k of the nodes, called liars, give advice which is incorrect. It is known that for an nnode graph G of maximum degree ∆ ≥ 3, reaching a target at a distance of d from the initial location may require an expected time of 2 Ω(min{d,k}) , for any d, k = O(log n), even when G is a tree. This paper focuses on strategies which efficiently solve the search problem in scenarios in which, at each node, the agent may only choose between following the local advice, or randomly selecting an incident edge. The strategy which we put forward, called R/A, makes use of a timer (step counter) to alternate between phases of ignoring advice (R) and following advice (A) for a certain number of steps. No knowledge of parameters n, d, or k is required, and the agent need not know by which edge it entered the node of its current location. The performance of this strategy is studied for two classes of regular graphs with extremal values of expansion, namely, for rings and for random ∆-regular graphs (an important class of expanders). For the ring, R/A is shown to achieve an expected searching time of 2d + k Θ(1) for a worstcase distribution of liars, which is polynomial in both d and k. For random ∆-regular graphs, the expected searching time of the R/A strategy is O(k 3 log 3 n) a.a.s. The polylog- * A full version of this paper is available online [19] Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. PODC'10, July 25-28, 2010, Zurich, Switzerland. Copyright 2010 ACM 978-1-60558-888-9/10/07 ...$10.00.arithmic factor with respect to n cannot be dropped from this bound; in fact, we show that a lower time bound of Ω(log n) steps holds for all d, k = Ω(log log n) in random ∆-regular graphs a.a.s. and applies even to strategies which make use of some knowledge of the environment.Finally, we study oblivious strategies which do not use any memory (in particular, with no timer). Such strategies are essentially a form of a random walk, possibly biased by local advice. We show that such biased random walks sometimes achieve drastically worse performance than the R/A strategy. In particular, on the ring, no biased random walk can have a searching time which is polynomial in d and k.

Abstract. We present a distributed asynchronous algorithm that, for every undirected weighted n-node graph G, constructs name-independent routing tables for G. The size of each table isÕ( √ n ), whereas the length of any route is stretched by a factor of at most 7 w.r.t. the shortest path. At any step, the memory space of each node isÕ( √ n ). The algorithm terminates in time O(D), where D is the hop-diameter of G. In synchronous scenarios and with uniform weights, it consumes O(m √ n + n 3/2 min {D, √ n }) messages, where m is the number of edges of G. In the realistic case of sparse networks of poly-logarithmic diameter, the communication complexity of our scheme, that isÕ(n 3/2 ), improves by a factor of √ n the communication complexity of any shortest-path routing scheme on the same family of networks. This factor is provable thanks to a new lower bound of independent interest.

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling L all gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree la-A preliminary version of this paper has appeared in the proceedings of the 7th International Workshop on Distributed Computing (IWDC), Kharagpur, India, December 27-30, 2005, as part of Cohen, R. et al.: Labeling schemes for tree representation. 2 Algorithmica (2009) 53: 1-15beling L up labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M up (T ) and S up (T ) for L up and M all (T ) and S all (T ) for L all . The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0, . . . , deg(v) − 1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for M up (T ), S up (T ) and S all (T ) but NP-hard for M all (T ) (even for 3-regular planar graphs). We show that for every graph G and port labeling there exists a spanning tree T for which S up (T ) = O(n log log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port labeling. We conclude by discussing some applications for our tree representation schemes.

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling L all gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling L up labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M up (T ) and S up (T ) for L up and M all (T ) and S all (T ) for L all . The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0, . . . , deg(v) − 1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for M up (T ), S up (T ) and S all (T ) but NP-hard for M all (T ) (even for 3-regular planar graphs). We show that for every graph G and port labeling there exists a spanning tree T for which S up (T ) = O(n log log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port labeling. We conclude by discussing some applications for our tree representation schemes.

In this paper, we propose a general scheme, called Algorithm STlC, to compute spanning-tree-like data structures on arbitrary networks. STlC is self-stabilizing and silent and, despite its generality, is also efficient. It is written in the locally shared memory model with composite atomicity assuming the distributed unfair daemon, the weakest scheduling assumption of the model. Its stabilization time is in O (n maxCC) rounds, where n maxCC is the maximum number of processes in a connected component. We also exhibit polynomial upper bounds on its stabilization time in steps and process moves holding for large classes of instantiations of Algorithm STlC. We illustrate the versatility of our approach by proposing several such instantiations that efficiently solve classical problems such as leader election, as well as, unconstrained and shortest-path spanning tree constructions.