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 $${\omega }$$ω provides for each vertex $$v$$v the minimum number of searchers required to guard $$v$$v, i.e., the fugitive may not pass through $$v$$v without being detected only if at least $${\omega }(v)$$ω(v) searchers are present at $$v$$v. This problem is a generalization of the classical edge searching problem, in which one has $${\omega }\equiv 1$$ω≡1. We assume that with a graph $$G$$G to be searched, there is associated a partition $$(V_1,\ldots ,V_t)$$(V1,…,Vt) of its vertex set such that edges are allowed only within each $$V_i$$Vi and between two consecutive $$V_i$$Vi’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 $$G$$G and have no a priori knowledge on $$G$$G, but they have a sense of direction that lets them recognize whether an edge incident to already explored vertex in $$V_i$$Vi leads to a vertex in one of $$V_{i-1}, V_i$$Vi-1,Vi or $$V_{i+1}$$Vi+1. Starting from any vertex the algorithm uses at most $$3\cdot \max _{i=1,\ldots ,t}{\omega }(V_i)+1$$3·maxi=1,…,tω(Vi)+1 searchers, where $${\omega }(V_i) = \sum _{v\in V_i}{\omega }(v)$$ω(Vi)=∑v∈Viω(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 $$3\cdot \max _{i=1,\ldots ,t}{\omega }(V_i)-1$$3·maxi=1,…,tω(Vi)-1 searchers for some graphs.