Time-Space Tradeoffs for the Memory Game

TL;DR

This paper establishes fundamental lower bounds on the time and space complexity for strategies to solve the Memory game, revealing inherent tradeoffs and extending previous theoretical models.

Contribution

It proves new time-space tradeoff lower bounds for Memory game algorithms, including a tight bound in a restricted model and a broader conjecture for general strategies.

Findings

Proves that ST = Ω(n^2 log n) in a comparison-only model.

Establishes that ST^2 = Ω(n^3) in a general computational model.

Conjectures a stronger bound ST = Ω(n^2) for all models.

Abstract

A single-player game of Memory is played with $n$ distinct pairs of cards, with the cards in each pair bearing identical pictures. The cards are laid face-down. A move consists of revealing two cards, chosen adaptively. If these cards match, i.e., they bear the same picture, they are removed from play; otherwise, they are turned back to face down. The object of the game is to clear all cards while minimizing the number of moves. Past works have thoroughly studied the expected number of moves required, assuming optimal play by a player has that has perfect memory. In this work, we study the Memory game in a space-bounded setting. We prove two time-space tradeoff lower bounds on algorithms (strategies for the player) that clear all cards in $T$ moves while using at most $S$ bits of memory. First, in a simple model where the pictures on the cards may only be compared for equality, we prove that $ST = \Omega(n^2 \log n)$. This is tight: it is easy to achieve $ST = O(n^2 \log n)$ essentially everywhere on this tradeoff curve. Second, in a more general model that allows arbitrary computations, we prove that $ST^2 = \Omega(n^3)$. We prove this latter tradeoff by modeling strategies as branching programs and extending a classic counting argument of Borodin and Cook with a novel probabilistic argument. We conjecture that the stronger tradeoff $ST = \widetilde{\Omega}(n^2)$ in fact holds even in this general model.