1. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Discrete Sensor Placement Problem Tanya Y. Berger-Wolf With William E. Hart and Jared Saia

2. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico The Problem: Place sensors in a building or a utility network to optimize some objective (time, affected area, etc.) with regard to contamination detection or source identification

3. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico History Traditionally modeled as continuous flow Birchall’86; Kessler,Ostfeld,Sinai’98; Kumar,Kansal,Arora’98; Tryby,Boccelli,Uber,Rossman’02 Numerical algorithms not robust and slow Birchall’86; Birchall,James’89; Gelbard,Brockmann,Murata,Hart’00 Integer programming for water network placement that minimizes the size of affected population Berry,Fleischer,Hart,Phillips’03

4. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Discrete Model Directed graph G=(V,E) with weighted edges: V –set of locations where sensors can be placed (rooms, distribution nodes) E –possible flow conduits w(i,j)=r ij – translocation rate between i and j Assume r ij is the shortest path metric 12 3 4 12 3 4

5. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Discrete Model Goal: place sensors (mark special nodes) to ensure contamination detection or source identification (1) Sensor-constrained: minimize time to detection/identification with a given sensor budget (2) Time-constrained: minimize the number of sensors that ensure detection/contamination within the time limit

6. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Assumptions Sensor can detect contamination before it reaches dangerous levels Rapid mixing: instant and even spread of flow within a node Single contamination source (otherwise need a sensor at every node) No contamination decay (every node down the flow is eventually contaminated) Any node is equally likely target Once a sensor is activated, it remains active

7. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico The Two Models Sensor-constrained and time-constrained problems are poly equivalent with a logarithmic increase in running time for exact optimal algorithm conversion. Both detection problems are NP-hard

8. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Contamination Source Identification Possible if and only if for each vertex i exists [(s i 0,0), (s i 1,d 1 ), (s i 2,d 2 ), …, (s i k, d k )] Where S={s 1, s 2, s 3,…,s k } is the set of sensors s i j is the jth sensor to react for i d j is the delay between s i j-1 and s i j 123 1234 1 1 1 22 4 4 4 3 3

9. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Sensor-Constrained Model Undirected sensor-constrained detection is equivalent to K-CENTER problem 2-approximation algorithm (and no better) Directed sensor-constrained detection: using MAX-COVER and binary search n*r max /e additive approximation

10. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Time-Constrained Model Time-constrained detection is poly equivalent to DOMINATING SET problem (1+ log n) - approximation (and no better)

11. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Special Case: Uniform Clique G=K n, r ij = r for all i and j Needs 1 sensor for detection in time > r and n-1 sensors for detection in time r. Needs n-1 sensors for source identification in time r.

12. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Special Case: Rooted Tree G=T n (r), all edges are oriented away from the root Detection within time limit T: 1.Place a sensor at each leaf 2.Follow edges in reverse from currents sensors. Mark vertices within distance T as “covered” 3.Put sensor at each first uncovered vertex 4.Repeat 2 and 3 until all vertices “covered” Source identification within time limit T: in a rooted tree any vertex can be uniquely identified by at most 2 sensors  Same algorithm but “covered”  2

13. Sensor Placement September 4, 2003 Tanya Berger-Wolf University of New Mexico Conclusions and Future Relax assumptions Multiple flow patterns (on-line version) Sensor failure or attack Other special graphs (DAGs) Other objectives Source identification Discrete models are robust, algorithms use only relative transfer rate values, approximation algorithms can give good and quick initial solutions.