A Distributed Approach to the Evasion Problem
Abstract
:1. Introduction
 We adapt the algorithm from [2] to a distributed computing setting, where each mobile sensor only needs to track their neighbors.
 We propose an algorithm for enumerating possible evasion paths up to homotopy.
2. Materials and Methods
2.1. Previous Algorithm for the Evasion Problem
2.1.1. Network Assumptions and Problem Formulation
2.1.2. Adams’ Algorithm
Algorithm 1 Adams’ Evasion Detection algorithm. This algorithm was described in [2]. 
Input: A collection N of coordinates $({x}_{i},{y}_{i})$ of sensing nodes with coverage radius r. Output: True if evasion is possible, False otherwise.

2.2. Distributing the Algorithm
Algorithm 2 Distributed Evasion Detection. 

2.2.1. Tracked Information
 Its own location
 Locations of its neighbors and the neighbors’ neighbors
 A Voronoi diagram of these locations
 A Delaunay complex of these locations
 An $\alpha $complex generated by the Delaunay complex and the sensing range $\alpha $
 For each region in the (planar) $\alpha $complex graph,
 
 Whether that region is labelled False or True
 
 What the nodes along the boundary of that region are
 A list of neighbors $DN\left(v\right)$ in the Delaunay graph
 A list of neighbors $N\left(v\right)$ in the $\alpha $complex
 A list of twohop neighbors in the $\alpha $complex
2.2.2. Initial Phase
2.2.3. Location Update
2.2.4. EventDriven Messages
2.2.5. Neighbor Changes
2.2.6. Edge Changes
2.2.7. Region Changes
2.2.8. Routing Protocol
2.2.9. Examples
2.3. Two Hop Neighbors Suffice
2.4. Enumerating Evasion Paths
2.4.1. Backwards Propagation
Algorithm 3 Evasion paths enumeration through backward propagation. 
Given: A full history of an Evasion Detection using either Algorithm 1 or Algorithm 2.

2.4.2. Forwards Propagation
Algorithm 4 Evasion paths enumeration through forward propagation. 

2.4.3. Examples
Time step  Evasion path list 
1  $\left[X\right]$ 
2  $[X,A]$, $[X,B]$ 
3  $[X,A,A]$, $[X,A,{A}^{\prime}]$, $[X,B,B]$, $[X,B,{B}^{\prime}]$ 
4  $[X,A,{A}^{\prime},{A}^{\prime}]$, $[X,B,B,B]$ 
Time step  Evasion path list 
4  $\left[Y\right]$, $\left[{Y}^{\prime}\right]$ 
3  $[A,Y]$, $[B,Y]$, $[{A}^{\prime},{Y}^{\prime}]$, $[{B}^{\prime},{Y}^{\prime}]$ 
2  $[A,A,Y]$, $[B,B,Y]$, $[A,{A}^{\prime},{Y}^{\prime}]$, $[B,{B}^{\prime},{Y}^{\prime}]$ 
1  $[X,A,A,Y]$, $[X,B,B,Y]$, $[X,A,{A}^{\prime},{Y}^{\prime}]$, $[X,B,{B}^{\prime},{Y}^{\prime}]$ 
3. Discussion
Author Contributions
Funding
Conflicts of Interest
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Symbols  Events 

Del$\alpha $  a Delaunayedge becomes an $\alpha $edge. 
$\alpha $Del  an $\alpha $edge becomes a Delaunayedge. 
$\Delta $+  a triangle is added (Voronoi vertex is within $\u03f5$ of all vertices). 
$\Delta $  a triangle disappears. 
Nbr+  a new Delaunay neighbor appears. 
Nbr  a Delaunay neighbor disappears. 
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Khryashchev, D.; Chu, J.; VejdemoJohansson, M.; Ji, P. A Distributed Approach to the Evasion Problem. Algorithms 2020, 13, 149. https://doi.org/10.3390/a13060149
Khryashchev D, Chu J, VejdemoJohansson M, Ji P. A Distributed Approach to the Evasion Problem. Algorithms. 2020; 13(6):149. https://doi.org/10.3390/a13060149
Chicago/Turabian StyleKhryashchev, Denis, Jie Chu, Mikael VejdemoJohansson, and Ping Ji. 2020. "A Distributed Approach to the Evasion Problem" Algorithms 13, no. 6: 149. https://doi.org/10.3390/a13060149