Master 2
Leader Election in a Sensor Networks (Exercises)
Franck Petit
∗We will investigate the Leader Election Problem in a sensor network. We will come up with the following question: “Given a set of sensors scattered on the plane where no two sensors are located at the same position, what are the (minimal) geometric conditions to be able to deterministically agree on a single sensorL, called the leader?”
The sensors are assumed to be uniform and anonymous, i.e., they all execute the same program using no local parameter (such that a visual identity) allowing to differentiate any of them. However, we assume that each sensor is a computational unit having the ability to determine the positions of then sensors within an infinite decimal precision. We assume no kind of communication medium. Each sensor has its own localx-y Cartesian coordinate system defined by two coordinate axes (xand y), together with their orientations, identified as the positive and negative sides of the axes.
Question 1. Given n≥2 sensors scattered on the plane, is it always possible to elect a leader Lif the sensors share the same coordinate system? If the answer is “yes”, then provide a distributed algorithm and prove it.
Otherwise, show a counter-example. Deduce a theorem from the answer.
Question 2. Same questions as for Question 1 assuming that the sensors share the same oriented direction and the same handedness (chirality).
Question 3. Same questions as for Question 1 assuming that the sensors share the oriented direction of a common axis only.
Question 4. With the same settings as for Question 3, can we refine the result with respect ton? If the answer is “yes”, then provide a distributed algorithm whenever it make sense. Otherwise, prove your result.
Question 5. Same questions as in Question 1 assuming that the sensors have no sense of direction (i.e., they share no common axis) but share the same handedness.
Question 6. Same questions as in Question 1 assuming that the sensors have no sense of direction and no chirality.
Question 7. Same questions as in Question 6 (with the same settings) assuming thatn <4.
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