Information and Computation
Edge-Disjoint Paths in Expander Graphs
SIAM Journal on Computing
Universal traversal sequences with backtracking
Journal of Computer and System Sciences - Complexity 2001
Short Vertex Disjoint Paths and Multiconnectivity in Random Graphs: Reliable Network Computing
ICALP '94 Proceedings of the 21st International Colloquium on Automata, Languages and Programming
A proof of alon's second eigenvalue conjecture
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Randomization and Derandomization in Space-Bounded Computation
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Sense of direction in distributed computing
Theoretical Computer Science - Special issue: Distributed computing
Computing on anonymous networks with sense of direction
Theoretical Computer Science
Short paths in expander graphs
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
The Cover Time of Random Regular Graphs
SIAM Journal on Discrete Mathematics
Graph exploration by a finite automaton
Theoretical Computer Science - Mathematical foundations of computer science 2004
Sampling Regular Graphs and a Peer-to-Peer Network
Combinatorics, Probability and Computing
Expansion properties of a random regular graph after random vertex deletions
European Journal of Combinatorics
Random walks, universal traversal sequences, and the complexity of maze problems
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Memoryless search algorithms in a network with faulty advice
Theoretical Computer Science
Undirected connectivity in log-space
Journal of the ACM (JACM)
Biased Random Walks in Uniform Wireless Networks
IEEE Transactions on Mobile Computing
Impact of local topological information on random walks on finite graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
The impact of edge deletions on the number of errors in networks
OPODIS'11 Proceedings of the 15th international conference on Principles of Distributed Systems
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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 [20, 21]. Each node of the network 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 n-node graph G of maximum degree Δ ≥ 3, reaching a target at a distance of d from the initial location may require an expected time of 2Ω(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 Δ-regular graphs (an important class of expanders). For the ring, R/A is shown to achieve an expected searching time of 2d+kΘ(1) for a worst-case distribution of liars, which is polynomial in both d and k. For random Δ-regular graphs, the expected searching time of the R/A strategy is O(k3 log3 n) a.a.s. The polylogarithmic 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 logn) in random Ω-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.