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Ellis-PIC - Point of Interest Coverage Coverage Chris Ellis...

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Unformatted text preview: Point of Interest Coverage Coverage Chris Ellis COT6410 10/6/08 Points of Interest (POI) Points Surveillance of locations or objects is Surveillance useful for study or intelligence gathering useful POI can be located in hostile/dangerous POI environments environments Autonomous tracking agents can be used Tracking Agents Tracking Remote or AI controlled Mobile (land or air) Equipped with cameras Equipped or sensor devices or Safer than deploying Safer humans to hostile environments environments (http://www.european-security.com/imgbiblio/warrior_01s.jpg) (www.iirobotics.com/catalog/images/rovio-main.jpg) Concerns Concerns What if there are more POI than trackers? How do we optimally place trackers to How cover the POI? cover P P P P P P P P P P P P P P P Formalizing the Problem Formalizing Represent POI as vertices in a weighted, Represent fully connected graph fully Assign Euclidean distance between two Assign POI as weight of edge between corresponding vertices corresponding Choose a fixed integer R as radius of Choose vision from Tracker vision Restrict Tracker placement to existing Restrict vertices vertices Problem Declaration Problem Optimization Problem Given: a graph G = (V,E) and integer R Problem: What is the minimum number of Problem: trackers with vision range R required to cover all vertices v ∈ V? A vertex is “covered” if a tracker is placed on vertex it, or there is an edge with weight < R connected to another vertex with a tracker located on it located Problem Declaration Problem Decision Problem (POIC) Given: a graph G and integers R and K Problem: Can all vertices v ∈ V be covered by K Problem: trackers with vision range R? trackers POIC ∈ NP POIC Oracle produces a set of vertices S ⊆ V ∋ |S| = K Oracle For each vertex v ∈ V if v ∈ S or v is incident to some For vertex in S with edge weight less than R, mark that vertex as covered vertex If any vertex v ∈ V is not covered, return “No”, If otherwise return “Yes” otherwise Problem Reduction Problem Reduce to Vertex Cover Convert an arbitrary instance of VC to Convert POIC POIC Given an instance of VC Given a graph G = (V,E) and integer K graph Construct an instance of POIC Construct a graph G’ = (V’,E’) and integer K’, R graph Problem Reduction Problem Let V’ = V Let K’ = K Let R = 2 For each edge e = (vi,vj) iin E construct an n edge e’ = (vi,vj) with weight 1 edge For each edge in the compliment of G For construct an edge with weight eleventyconstruct billion Caveat Caveat “True” instances of POIC are based on True” real-world locations real-world Vertices and edges conform to the Vertices constraints on points in 2D or 3D Euclidean space Euclidean My definition of POIC does not Proof of Correctness Proof One to one mapping from v ∈ V to v’ ∈ V’ One vi covers vk (VC sense) iff vi’ covers vk’ iff (POIC sense) (POIC vi is covered by vk (VC sense) iff vi’ iis s iff covered by vk’ (POIC sense) covered VC of G exists iff POIC of G’ exists ...
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