Object Tracking Using Multiset Color Coding

We consider the tracking problem of an object that can randomly appear on a line of a fixed length. We want to determine the position of the object and rely on sensors that can detect objects with a predefined range and report that to a remote observer. Sensors are equipped with a transmitter that can transmit at a limited data rate. Once a sensor is triggered, it transmits its own ID to inform. A straightforward protocol is to label each sensor with different ID. However, this would require a large number of unique IDs and many bits to represent. As a result, higher data rate is required. We propose a newly defined protocol using multiset color coding with optimal design or efficiency in reusing a much smaller number of IDs for the whole system. We only require the minimal number of bits for labeling each sensor. We derive the factor of reduction and show its significance. We present some optimal constructions for the required multiset color coding sequence. Besides, we derive the general upper and lower bounds for the maximum length (size) of the system. Numerical examples have also demonstrated the effectiveness and improvement by the proposed new method.