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We consider query-based data acquisition and the corresponding informationrecovery problem, where the goal is to recover $k$ binary variables(information bits) from parity measurements of those variables. The queries andthe corresponding parity measurements are designed using the encoding rule ofFountain codes. By using Fountain codes, we can design potentially limitlessnumber of queries, and corresponding parity measurements, and guarantee thatthe original $k$ information bits can be recovered with high probability fromany sufficiently large set of measurements of size $n$. In the query design,the average number of information bits that is associated with one paritymeasurement is called query difficulty ($\bar{d}$) and the minimum number ofmeasurements required to recover the $k$ information bits for a fixed $\bar{d}$is called sample complexity ($n$). We analyze the fundamental trade-offsbetween the query difficulty and the sample complexity, and show that thesample complexity of $n=c\max\{k,(k\log k)/\bar{d}\}$ for some constant $c>0$is necessary and sufficient to recover $k$ information bits with highprobability as $k\to\infty$.

Fundamental Limits on Data Acquisition: Trade-offs between Sample Complexity and Query Difficulty