Boaz Menuhin | Searcharxiv

arXiv2025

Shuffling Cards When You Are of Very Little Brain: Low Memory Generation of Permutations

How can we generate a permutation of the numbers $1$ through $n$ so that it is hard to guess the next element given the history so far? The twist is that the generator of the permutation (the ``Dealer") has limited memory, while the ``Guesser" has unlimited memory. With unbounded memory (actually $n$ bits suffice), the Dealer can generate a truly random permutation where $\ln n$ is the expected number of correct guesses. Our main results establish tight bounds for the relationship between the guessing probability and the memory $m$ required to generate the permutation. We suggest a method for an $m$-bit Dealer that operates in constant time per turn, and any Guesser can pick correctly only $O(n/m+\log m)$ cards in expectation. The method is fully transparent, requiring no hidden information from the Dealer (i.e., it is "open book" or "whitebox"). We show that this bound is the best possible, even with secret memory. Specifically, for any $m$-bit Dealer, there is a (computationally powerful) guesser that achieves $\Omega(n/m+\log m)$ correct guesses in expectation. We point out that the assumption that the Guesser is computationally powerful is necessary: under cryptographic assumptions, there exists a low-memory Dealer that can fool any computationally bounded guesser. We also give an $O(n)$ bit memory Dealer that generates perfectly random permutations and operates in constant time per turn.

cs.DS↗

arXiv2022

Keep That Card in Mind: Card Guessing with Limited Memory

A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of $n$ cards (labeled $1, ..., n$). For $n$ turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then the card is discarded from the deck. The Guesser receives a point for each correctly guessed card. With perfect memory, a Guesser can keep track of all cards that were played so far and pick at random a card that has not appeared so far, yielding in expectation $\ln n$ correct guesses. With no memory, the best a Guesser can do will result in a single guess in expectation. We consider the case of a memory bounded Guesser that has $m < n$ memory bits. We show that the performance of such a memory bounded Guesser depends much on the behavior of the Dealer. In more detail, we show that there is a gap between the static case, where the Dealer draws cards from a properly shuffled deck or a prearranged one, and the adaptive case, where the Dealer draws cards thoughtfully, in an adversarial manner. Specifically: 1. We show a Guesser with $O(\log^2 n)$ memory bits that scores a near optimal result against any static Dealer. 2. We show that no Guesser with $m$ bits of memory can score better than $O(\sqrt{m})$ correct guesses, thus, no Guesser can score better than $\min \{\sqrt{m}, \ln n\}$, i.e., the above Guesser is optimal. 3. We show an efficient adaptive Dealer against which no Guesser with $m$ memory bits can make more than $\ln m + 2 \ln \log n + O(1)$ correct guesses in expectation. These results are (almost) tight, and we prove them using compression arguments that harness the guessing strategy for encoding.

cs.CC↗