Formal Complexity Analysis of Mobile Problems & Communication and Computation

1. Formal Complexity Analysis of Mobile Problems & Communication and Computation 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. Question: What is the computational complexity of Target Assignment problem? Formal Complexity Analysis 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. Question: What is the computational complexity of Roboflag Drill? Formal Complexity Analysis of Roboflag Drill Problem

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. IISI - Cornell Communication and Computation in Distributed Negotiation Algorithms Carla Gomes, Bart Selman, Carmel Domshlak Sensor Network Problem Sensors { s1 , …, sn}. Targets{ 1 , …, m}. Given a spatial model of the problem domain, and the locations of the targets, determine whether there exists a set of msensor triplets such that: • Sensors within each triplet can communicate one with each other. • All three sensors in the i –th triplet can track the target i . • All the triplets are pairwise sensor-disjoint.

12. 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. From a general model to real-life settings

13. P NP-hard 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

14. P NP-hard 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?

15. pv kv N Level of constraintness pc m kc Level of decomposition (locality) Order of the problem 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. pv kv N Level of constraintness pc m kc Level of decomposition (locality) Order of the problem 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.

17. pv kv N Level of constraintness pc m kc Level of decomposition (locality) Order of the problem 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

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.