Locating a target with an agent guided by unreliable local advice

Abstract

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.

The Impact of Edge Deletions on the Number of Errors in Networks

“…A series of papers [HKK04,HKKK08,HIKN10] tackle the problem of locating a target (node, resource, data, ...) in presence of liars.…”

“…The main performance measure is the number of edge traversals during a request. Several algorithms, either generic or dedicated to some topologies, and bounds are presented in [HKK04,HKKK08,HIKN10] and are typically of the form O(d + k O(1) ) (for path,grids, expanders,. .…”

Slowdown for the geodesic-biased random walk

“…On the other hand, when no vertices are excited, i.e., X = ∅, the geodesic-biased walk reduces to the simple random walk on G, and an old result of Lawler [11] gives a uniform polynomial bound (see also [1,6]) for the expected hitting time of E[τ a (b, ∅)] = O(n 3 ). Many of the existing results in the literature [8,7,5] show that the expected hitting time of a fixed target in the geodesic-biased walk, for various graphs G and random choices of the set X of excited vertices, is significantly smaller than Lawler's uniform bound. Motivated by this, we shall investigate how much the geodesic-bias can decrease the hitting time of a fixed target.…”

The impact of dynamic events on the number of errors in networks

“…In the following, for a given target t, we informally refer to a liar as a node containing bad information about the location of t. A series of papers [11,10,9] tackles the problem of locating a target (node, resource, data, ...) in presence of liars.…”

“…The main performance measure is the number of edge traversals during a request. Several algorithms, either generic or dedicated to some topologies, and bounds are presented in [11,10,9] and are typically of the form O d + k O (1) (for path, grids, expanders, . .…”