Mutual Exclusion is a fundamental problem in distributed computing, and the problem of proving upper and lower bounds on the RMR complexity of this problem has been extensively studied. Here, we give matching lower and upper bounds on how RMR complexity trades off with space. Two implications of our results are that constant RMR complexity is impossible with subpolynomial space and subpolynomial RMR complexity is impossible with constant space for cache-coherent multiprocessors, regardless of how strong the hardware synchronization operations are. To prove these results we show that the complexity of mutual exclusion, which can be "messy" to analyze because of system details such as asynchrony and cache coherence, is captured precisely by a simple and purely combinatorial game that we design.
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