Efficient Algorithms for the Bee-Identification Problem
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The bee-identification problem, formally defined by Tandon, Tan and Varshney (2019), requires the receiver to identify "bees" using a set of unordered noisy measurements. In this previous work, Tandon, Tan, and Varshney studied error exponents and showed that decoding the measurements jointly results in a significantly smaller error exponent. In this work, we study algorithms related to this joint decoder. First, we demonstrate how to perform joint decoding efficiently. By reducing to the problem of finding perfect matching and minimum-cost matchings, we obtain joint decoders that run in time quadratic and cubic in the number of "bees" for the binary erasure (BEC) and binary symmetric channels (BSC), respectively. Next, by studying the matching algorithms in the context of channel coding, we further reduce the running times by using classical tools like peeling decoders and list-decoders. In particular, we show that our identifier algorithms when used with Reed-Muller codes terminate in almost linear and quadratic time for BEC and BSC, respectively. Finally, for explicit codebooks, we study when these joint decoders fail to identify the "bees" correctly. Specifically, we provide practical methods of estimating the probability of erroneous identification for given codebooks.
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