Achieving optimal backlog in multi-processor cup games

Abstract

The single-and multi-processor cup games can be used to model natural problems in areas such as processor scheduling, deamortization, and buffer management.At the beginning of the single-processor cup game, n cups sit in a row, initially empty. In each step of the game, a filler distributes 1 unit of water among the cups, and then an emptier selects a cup and removes 1 + ε units from that cup. The goal of the emptier is to minimize the amount of water in the fullest cup, also known as the backlog. It is known that the greedy algorithm (i.e., empty the fullest cup) achieves backlog O(log n), and that no deterministic algorithm can do better.We show that the performance of the greedy algorithm can be greatly improved with a small amount of randomization: After any step i, and for any k ≥ Ω(log ε −1 ), the emptier achieves backlog at most O(k) with probability at least 1 − O(2 −2 k ). Our algorithm, which we call the smoothed greedy algorithm, can also be interpreted as a one-shot smoothed analysis of the standard greedy algorithm.Whereas bounds for the single-processor cup game have been known for more than fifteen years, proving nontrivial bounds on backlog for the multi-processor extension has remained open. We present a simple analysis of the greedy algorithm for the multi-processor cup game, establishing a backlog of O(ε −1 log n), as long as δ, the game's other speed-augmentation constant, is at least 1/poly(n).Turning to randomized algorithms, we encounter an unexpected phenomenon: When the number of processors p is large, the backlog after each step drops to constant with large probability. Specifically, we show that if δ and ε satisfy reasonable constraints, then there exists an algorithm that bounds the backlog after a given step by three or less with probability at least 1 − O(exp(−Ω(ε 2 p)). We further extend the guarantees of our randomized algorithm to consider larger backlogs.When ε is constant, we prove that our results are asymptotically optimal, in the sense that no algorithms can achieve better bounds, up to constant factors in the backlog and in p. Moreover, we prove robustness results, demonstrating that our randomized algorithms continue to behave well even when placed in bad starting states.The small amount of resource augmentation used by the smoothed greedy algorithm makes it robust to the setting in which cups begin in a bad initial starting state. In particular, we show that if b units of water are maliciously placed into cups at the beginning of the game, then for steps i > b ε , the b units of water have no affect on the guarantees given by the algorithm.The multi-processor cup game. The multi-processor version of the same scheduling question has proven to be much harder. In each step of the multi-processor cup game, the filler distributes some amount of water proportional to p, the number of processors, among the cups (i.e., threads), and the emptier picks p cups and removes a unit of water from each. If the filler is unrestricted in their placement of the water, then they can ensure an...

Achieving Optimal Backlog in the Vanilla Multi-Processor Cup Game

“…For fixed p, and as n grows large, this becomes backlog O(log log n), which is known to be asymptotically optimal. As noted by Bender et al [9], any analysis of the smoothed greedy algorithm also doubles as a smoothed analysis for the deterministic greedy algorithm. This is because the smoothed greedy algorithm works by first randomly perturbing the initial starting state of the game, and then following a variant of the standard deterministic greedy algorithm on the perturbed game.…”

“…Recently, Bender et al [9] showed that as long as δ ≥ 1 poly(n) , the greedy emptying algorithm achieves backlog O( 1 ǫ log n). When ǫ ≥ Ω(1), this results in an asymptotically optimal backlog of O(log n).…”

“…Dietz and Raman also give a matching lower bound construction that achieves backlog Ω(log log n) with probability 1 poly(n) . Recently, Bender et al [9] presented several randomized algorithms that work with far less resource augmentation. For the single-processor game, they introduced the smoothed greedy algorithm, which achieves backlog O(log log n) with probability 1 − 1 poly(n) , as long as ǫ ≥ 1 polylog(n) .…”

“…For the single-processor game, they introduced the smoothed greedy algorithm, which achieves backlog O(log log n) with probability 1 − 1 poly(n) , as long as ǫ ≥ 1 polylog(n) . Bender et al [9] also gave a more intricate algorithm for the multi-processor cup game. Their algorithm achieves backlog O( 1 ǫ log log n) with probability 1 − 1 poly(n) , as long as ǫ, δ ≥ 1 poly(p) satisfy certain natural constraints.…”

“…The algorithms of Dietz and Raman [16] and Bender et al [9] both rely heavily on the use of resource augmentation to achieve their bounds. Designing randomized algorithms for either the single-processor or multi-processor cup game that do not rely on resource augmentation remains an open question.…”