Variations on the Guessing Problem

Guessing with Distributed Encoders

“…With distributed encoders, however, task encoding [9] and guessing no longer have the same asymptotics; see Remark 3. Lower and upper bounds for guessing with a helper, i.e., an encoder that does not observe X, but has access to a random variable that is correlated with X, can be found in [5].…”

“…Thus, in guessing a sequence of independent and identically distributed (IID) random variables, a description rate of approximately H 1/(1+ρ) (X) bits per symbol is needed to drive the ρth moment of the number of guesses to one as the sequence length tends to infinity [4,5] (see Section 2 for more related work). In this paper, we generalize the single-encoder setting from Figure 1 to the setting with distributed encoders depicted in Figure 2, which is the analog of Slepian-Wolf coding [6] for guessing: A source generates a sequence of pairs {(X i , Y i )} n i=1 over a finite alphabet X × Y.…”

“…Despite the resemblance between (10)- (12) and the Slepian-Wolf region, there is an important difference: while Slepian-Wolf coding allows separate encoding with the same sum rate as with joint encoding, this is not necessarily true in our setting: Remark 2. Although the sum rate constraint (12) is the same as in single-source guessing [5], separate encoding of X n and Y n may require a larger sum rate than joint encoding of X n and Y n .…”

Guessing with a Bit of Help

“…Nonetheless, as other problems employing a helper (e.g., source coding [ 17 , 18 ]), it is more realistic to impose communication constraints and to assume that Bob can only send a compressed description of to Alice. This setting was recently addressed by Graczyk and Lapidoth [ 19 , 20 ], who considered the case where Bob encodes at a positive rate using bits before sending this description to Alice. They then characterized the best possible guessing-moments attained by Alice for general distributions as a function of the rate R .…”

“…Related Work. As mentioned above, Graczyk and Lapidoth [ 19 , 20 ] considered the same guessing question if Bob can communicate with Alice at some positive rate R , i.e., can use bits to describe . This setup facilitates the use of large-deviation-based information-theoretic techniques, which allowed the authors to characterize the optimal reduction in the guessing-moments as a function of R to the first order in the exponent.…”