Distributed graph searching with a sense of direction

Abstract

In this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function ω provides for each vertex v the minimum number of searchers required to guard v, i.e., the fugitive may not pass through v without being detected only if at least ω(v) searchers are present at v. This problem is a generalization of the classical edge searching problem, in which one has ω ≡ 1. We assume that with a graph G to be searched, there is associated a partition (V 1 , . . . , V t ) of its vertex set such that edges are allowed only within each V i and between two consecutive V i 's. We provide an algorithm for distributed monotone connected edge searching of such graphs, where the searchers are initially placed on an arbitrary vertex of G and have no a priori knowledge on G, but they have a sense of direction that lets them recognize whether an edge incident to already explored vertex in V i leads to a vertex in one of V i−1 , V i or V i+1 . Starting from any vertex the algorithm uses at most 3·max i=1,...,t ω(V i )+1 searchers, where ω(V i ) = v∈V i ω(v). We also prove that this algorithm is best possible up to a small additive constant, that is, each distributed searching algorithm in worst case must use 3 · max i=1,...,t ω(V i ) − 1 searchers for some graphs.

Distributed Searching of Partial Grids

“…The above questions related to network topologies can be stated more generally: what properties of the distributed model are crucial for such search for fast and invisible fugitive to be efficient? This work and also a recent one [5] suggest that a 'sense of direction' may be one such factor. Possibly interesting directions may be to analyze the influence of visibility on search scenarios.…”

“…Section 4 describes a distributed algorithm that performs a guaranteed search in partial grids and it is assumed that the algorithm is given an upper bound n on the size of the network. We point out that this algorithm uses a distributed procedure from [5] as a subroutine that is called many times to clear selected parts of a grid, and it can be seen as a generalization from a 'linear' graph structure studied in [5] to a 2-dimensional structure discussed in this work. Also, although both algorithms are conducted via some greedy rules which dictate how a search should 'expand' to unknown parts of the graph, the analysis of our algorithm is different from the one in [5].…”

“…For cleaning all nodes of the i-th expansion of C, provided that G[C + i − 1 ] is clean we will use a procedure from [5] which is more general but for our purposes can be stated as follows.…”

Finding small-width connected path decompositions in polynomial time

“…We also point out that a number of results have been obtained for connected pathwidth or the closely related connected graph (edge) search, including algorithmic and computational ones [2,15,16,21,24,31], monotonicity [25,39], structural properties [3] or distributed algorithms [11,20,27].…”

“…The connectivity constraint for pathwidth is natural and useful in graph searching games [2,20,24]. The connectivity is in some cases implied by potential applications (e.g., security constraints may enforce the clean, or safe, area to be connected) or it is a necessity, like in distributed or online versions of the problem [6,11,27,32].…”