Contention Resolution, With and Without a Global Clock
In the Contention Resolution problem $n$ parties each wish to have exclusive use of a shared resource for one unit of time. The problem has been studied since the early 1970s, under a variety of assumptions on feedback given to the parties, how the parties wake up, knowledge of $n$, and so on. The most consistent assumption is that parties do not have access to a global clock, only their local time since wake-up. This is surprising because the assumption of a global clock is both technologically realistic and algorithmically interesting. It enriches the problem, and opens the door to entirely new techniques. Our primary results are: [1] We design a new Contention Resolution protocol that guarantees latency $$O\left(\left(n\log\log n\log^{(3)} n\log^{(4)} n\cdots \log^{(\log^* n)} n\right)\cdot 2^{\log^* n}\right) \le n(\log\log n)^{1+o(1)}$$ in expectation and with high probability. This already establishes at least a roughly $\log n$ complexity gap between randomized protocols in GlobalClock and LocalClock. [2] Prior analyses of randomized ContentionResolution protocols in LocalClock guaranteed a certain latency with high probability, i.e., with probability $1-1/\text{poly}(n)$. We observe that it is just as natural to measure expected latency, and prove a $\log n$-factor complexity gap between the two objectives for memoryless protocols. The In-Expectation complexity is $Θ(n \log n/\log\log n)$ whereas the With-High-Probability latency is $Θ(n\log^2 n/\log\log n)$. Three of these four upper and lower bounds are new. [3] Given the complexity separation above, one would naturally want a ContentionResolution protocol that is optimal under both the In-Expectation and With-High-Probability metrics. This is impossible! It is even impossible to achieve In-Expectation latency $o(n\log^2 n/(\log\log n)^2)$ and With-High-Probability latency $n\log^{O(1)} n$ simultaneously.