1. Formal Complexity Analysis of Mobile Problems & Communication and Computation in Distributed Sensor Networks in Distributed Sensor Networks Carla P. Gomes Cornell University

2. Formal Complexity Analysis of Mobile Problems (joint work with Matt Earl and Raff D’Andrea)

3. Input: Set of attackers initial location velocity direction One or more defenders initial location velocity direction Question: Can the defenders intersect all the attackers in a given time? Target Assignment Problem

4. Formal Complexity Analysis of Target Assignment Problem Question: What is the computational complexity of Target Assignment problem?

5. - NP-hard problem Reduction from Euclidean TSP Formal Complexity Analysis of Target Assignment Problem

6. Approximations This problem is approximable within a constant factor of the optimal solution, in polynomial time; it admits a PTAS (polynomial time approximation scheme that allows us to be arbitrarily close to the optimum; polynomial in the length of the input but not polynomial in the performance ratio) Formal Complexity Analysis of Target Assignment Problem

7. Input: Set of attackers initial location velocity (constant) direction (constant) One defender initial location velocity (constant) direction – piecewise linear Goal area Question: Can the defender intersect all the attackers before they reach the goal area? RoboFlag Drill Base

8. Formal Complexity Analysis of Roboflag Drill Problem Question: What is the computational complexity of Roboflag Drill?

9. NP-hard for the general problem Polynomial for some classes (e.g., if the attackers move in parallel and if they are equidistant from the goal area) Fixed number of attackers: Fixed Parameter Complexity Class RoboFlag Drill Base (conjectures)

10. Communication and Computation in Distributed Sensor Networks (joint work with Carmel Domshlak and Bart Selman)

11. Communication and Computation in Distributed Negotiation Algorithms Carla Gomes, Bart Selman, Carmel Domshlak Sensor Network Problem Sensors { s 1, …, s n }. Targets {  1, …,  m }. Given a spatial model of the problem domain, and the locations of the targets, determine whether there exists a set of m sensor triplets such that: 1.Sensors within each triplet can communicate one with each other. 2.All three sensors in the i –th triplet can track the target  i. 3.All the triplets are pairwise sensor-disjoint. IISI - Cornell

12. From a general model to real-life settings Spatial Modeling •Sensor model –Possible locations on the terrain. •Communication model –Communication abilities of the sensors as a function of basic sensor spec and the terrain conditions. •Visibility model –Tracking abilities of the sensors as a function of target parameters, basic sensor spec and the terrain conditions.

13. From a general model to real-life settings Spatial Modeling Sensor model –Possible locations on the terrain. Communication model –Communication abilities of the sensors as a function of basic sensor spec and the terrain conditions. Visibility model –Tracking abilities of the sensors as a function of target parameters, basic sensor spec and the terrain conditions. Complexity analysis of computation and communication of negotiation protocols on problems modelled as above. –Formal analysis –Empirical analysis PNP-hard

14. From a general model to real-life settings Spatial Modeling Sensor model –Possible locations on the terrain. Communication model –Communication abilities of the sensors as a function of basic sensor spec and the terrain conditions. Visibility model –Tracking abilities of the sensors as a function of target parameters, basic sensor spec and the terrain conditions. Complexity analysis of computation and communication of negotiation protocols on problems modelled as above. –Formal analysis –Empirical analysis Temporal model of moving targets –Analysis of alternative (complete) renegotiation schemes. –Can we renegotiate in real-life settings? PNP-hard

15. N m kckc kvkv pcpc pvpv Order of the problem Level of decomposition (locality) Level of constraintness Results Spatial Modeling A Grid-based sensor network model has been developed. The locality of sensor communicability and target visibility is modeled via controlled parameters. The constraintness of communicability and visibility is modeled via probability distributions w.r.t. the locality parameters.

16. Results Spatial Modeling A Grid-based sensor network model developed. The locality of sensor communicability and target visibility is modeled via controlled parameters. The constrainedness of communicability and visibility is modeled via probability distributions w.r.t. the locality parameters. Complexity analysis of computation and communication of negotiation protocols. Formal analysis covering all the subclasses of the problem –Identified polynomial algorithms for tractable cases (e.g., (1) when visibility is restricted to small window and (2) communication is locally complete (local graph is complete)). –Non-trivial NP-completeness proofs for intractable cases. Comprehensive empirical analysis. N m kckc kvkv pcpc pvpv Order of the problem Level of decomposition (locality) Level of constraintness

17. Results Spatial Modeling A Grid-based sensor network model has been developed. The locality of sensor communicability and target visibility is modeled via controlled parameters. The constrainedness of communicability and visibility is modeled via probability distributions w.r.t. the locality parameters. Complexity analysis of computation and communication of negotiation protocols. Formal analysis covering all the subclasses of the problem –Polynomial algorithms for tractable cases –Non-trivial NP-completeness proofs for intractable cases. Comprehensive empirical analysis. Temporal model of moving targets Several algorithms for dynamic renegotiation have been analysed in the scope of a specially designed evaluation framework. Complete renegotiation has been shown to be practically feasible. Mean time to solve: Renegotiation – 0.059 sec Negotiation from scratch – 0.084 sec N m kckc kvkv pcpc pvpv Order of the problem Level of decomposition (locality) Level of constraintness

18. Phase Transition in SensorDNP Sharp transition in solvability at critical level of resources (Pc – probability of communication; Pv – probability of visibility)

19. Summary  Formal Complexity Analysis of Mobile Problems  Distributed Sensor Networks Complexity analysis Phase transition phenomena with corresponding peak in complexity for distributed sensor networks; Controlled randomization can increase performance of negotiation protocols dramatically.