An inequality on guessing and its application to sequential decoding

Let (X,Y) be a pair of discrete random variables with X taking one of M possible values, Suppose the value of X is to be determined, given the value of Y, by asking questions of the form “Is X equal to x?” until the answer is “Yes”. Let G(x|y) denote the number of guesses in any such guessing scheme when X=x, Y=y. We prove that E[G(X|Y)^ρ]⩾(1+lnM)^-ρΣ_y [Σ_xP_X,Y(x,y)^1/1+ρ] ^1+ρ for any ρ⩾0. This provides an operational characterization of Renyi´s entropy. Next we apply this inequality to the estimation of the computational complexity of sequential decoding. For this, we regard X as the input, Y as the output of a communication channel. Given Y, the sequential decoding algorithm works essentially by guessing X, one value at a time, until the guess is correct. Thus the computational complexity of sequential decoding, which is a random variable, is given by a guessing function G(X|Y) that is defined by the order in which nodes in the tree code are hypothesized by the decoder. This observation, combined with the above lower bound on moments of G(X|Y), yields lower bounds on moments of computation in sequential decoding. The present approach enables the determination of the (previously known) cutoff rate of sequential decoding in a simple manner; it also yields the (previously unknown) cutoff rate region of sequential decoding for multiaccess channels. These results hold for memoryless channels with finite input alphabets