Achieving Optimal Backlog in the Vanilla Multi-Processor Cup Game

Abstract

In each step of the p-processor cup game on n cups, a filler distributes up to p units of water among the cups, subject only to the constraint that no cup receives more than 1 unit of water; an emptier then removes up to 1 unit of water from each of p cups. Designing strategies for the emptier that minimize backlog (i.e., the height of the fullest cup) is important for applications in processor scheduling, buffer management in networks, quality of service guarantees, and deamortization.We prove that the greedy algorithm (i.e., the emptyfrom-fullest-cups algorithm) achieves backlog O(log n) for any p ≥ 1. This resolves a long-standing open problem for p > 1, and is asymptotically optimal as long as n ≥ 2p.If the filler is an oblivious adversary, then we prove that there is a randomized emptying algorithm that achieve backlog O(log p + log log n) with probability 1 − 2 − polylog(n) for 2 polylog(n) steps. This is known to be asymptotically optimal when n is sufficiently large relative to p. The analysis of the randomized algorithm can also be reinterpreted as a smoothed analysis of the deterministic greedy algorithm.Previously, the only known bound on backlog for p > 1, and the only known randomized guarantees for any p (including when p = 1), required the use of resource augmentation, meaning that the filler can only distribute at most p(1 − ǫ) units of water in each step, and that the emptier is then permitted to remove 1 + δ units of water from each of p cups, for some ǫ, δ > 0. * MIT CSAIL.

The Variable-Processor Cup Game

“…Starting with the seminal paper of Liu [30], work on the p processor cup game has spanned more than five decades [7,20,8,29,27,33,6,23,30,31,16,10,25,1,15,28]. In addition to processor scheduling [7,20,8,29,27,33,6,23,30,31,1,28,16], applications include network-switch buffer management [21,4,35,19], quality of service guarantees [7,1,28], and data structure deamortization [2,16,15,3,34,22,17,24,9].…”

“…The game has also been studied in many different forms. Researchers have studied the game with a fixed-filling-rate constraint [7,20,8,29,27,33,6,23,30,31], with various forms of resource augmentation [10,25,28,16], with both oblivious and adaptive adversaries [1,7,30,10,25], with smoothed analysis [25,10], with a semi-clairvoyant emptier [28], with competitive analysis [5,18,14], etc.…”

“…For the plain form of the p-processor cup game, the greedy emptying algorithm (i.e., always empty from the fullest cups) is known to be asymptotically optimal [1,10,25], achieving backlog O(log n). The optimal backlog for randomized emptying algorithms remains an open question [16,10,25] and is known to be between Ω(log log n) and O(log log n + log p) [25].…”

“…A priori having variable resources offers neither player a clear advantage. When the number p of processors is fixed, the greedy emptying algorithm (i.e., always empty from the fullest cups), is known to achieve backlog O(log n) [1,10,25] regardless of the value of p. This seems to suggest that, when p varies, the correct backlog should remain O(log n). In fact, when we began this project, we hoped for a straightforward reduction between the two versions of the game.…”

“…When studying randomized emptying strategies, we restrict ourselves to the class of greedy-like emptying strategies. This means that the emptier never chooses to empty from a cup c over another cup c ′ whose fill is more than ω(1) greater than the fill of c. All of the known randomized emptying strategies for the classic p-processor cup game are greedy-like [10,25].…”