A Distributed Approach to the Evasion Problem

Abstract

The Evasion Problem is the question of whether—given a collection of sensors and a particular movement pattern over time—it is possible to stay undetected within the domain over the same stretch of time. It has been studied using topological techniques since 2006—with sufficient conditions for non-existence of an Evasion Path provided by de Silva and Ghrist; sufficient and necessary conditions with extended sensor capabilities provided by Adams and Carlsson; and sufficient and necessary conditions using sheaf theory by Krishnan and Ghrist. In this paper, we propose three algorithms for the Evasion Problem: one distributed algorithm extension of Adams’ approach for evasion path detection, and two different approaches to evasion path enumeration.

Efficient Evader Detection in Mobile Sensor Networks

“…Given a fat graph, we have an involution α that acts on the half-edges by reversing orientation, as well as a bijection σ that returns the next half-edges about a vertex in a counter-clockwise direction. In Figure 2 (2,14,12,17,19). The other three boundary cycles in this example are (4,20,21,22,18,10,15), (6,16,8), and (1, 3, 5, 7, 9, 11, 13); see Figure 2 (right).…”

“…For example, the input required to compute the example in Section 6 of [13] is enough information to instead directly compute the Reeb graph of the uncovered region. The papers [2][3][4] raise and partially address the problem of computing the entire space of evasion paths, which in [19] is implemented for planar sensors that can measure cyclic orderings of and local distances to nearby sensors. Though [19] assumes that each sensor knows its positional coordinates, this assumption can be slightly weakened: if instead each sensor only knows which sensors it overlaps with, and the exact distances to overlapping sensors, then this is enough information to compute alpha complexes.…”

“…• Pair consisting of a 2-simplex and a free edge is removed. Just as in adding a simplex pair, the labelling for this case can be done by removing the 1-simplex and then removing the While [19] treats the boundary cycle label update as a state-based algorithm, we observe that this is not necessary. This was initially surprising to us, given that the proof of [3,Theorem 3] is heavily reliant on its case-based organization.…”

“…The boundary cycles of a fat graph are the orbits of the composition ϕ = σ • α. In our example, we compute that ϕ(2) = 14, ϕ(14) = 12, ϕ(12) = 17, ϕ(17) = 19, and ϕ(19) = 2, so one of the boundary cycles is the orbit(2,14,12,17,19). The other three boundary cycles in this example are(4,20,21,22,18,10,15),(6,16,8), and (1,3,5,7,9,11,13); see Figure2 (right).We note that each boundary cycle corresponds to a connected component of the complement of the graph.…”