Flushing Without Cascades

Bender, Michael A.; Das, Rathish; Farach-Colton, Martı́n; Johnson, Rob; Kuszmaul, William

doi:10.1137/1.9781611975994.40

Cited by 13 publications

(10 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Second, the fills placed in cups x and y are at most 1: this is because (10)) and f y − f x + ∆ x < ∆ x ≤ 1 (by ( 9)). Third, the fills placed in cups x and y are non-negative: this is because f x − f y + ∆ x > ∆ x ≥ 0 (by (9)) and f y − f x + ∆ y > ∆ x ≥ 0 (by (10)).…”

Section: Claim 52 If Some Cup Inmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Variable-Processor Cup Game

Kuszmaul,

Westover

2020

Preprint

Self Cite

View full text Add to dashboard Cite

The problem of scheduling tasks on p processors so that no task ever gets too far behind is often described as a game with cups and water. In the p-processor cup game on n cups, there are two players, a filler and an emptier, that take turns adding and removing water from a set of n cups. In each turn, the filler adds p units of water to the cups, placing at most 1 unit of water in each cup, and then the emptier selects p cups to remove up to 1 unit of water from. The emptier's goal is to minimize the backlog, which is the height of the fullest cup.The p-processor cup game has been studied in many different settings, dating back to the late 1960's. All of the past work shares one common assumption: that p is fixed. This paper initiates the study of what happens when the number of available processors p varies over time, resulting in what we call the variable-processor cup game.Remarkably, the optimal bounds for the variable-processor cup game differ dramatically from its classical counterpart. Whereas the p-processor cup has optimal backlog Θ(log n), the variable-processor game has optimal backlog Θ(n). Moreover, there is an efficient filling strategy that yields backlog Ω(n 1−ε ) in quasi-polynomial time against any deterministic emptying strategy.We additionally show that straightforward uses of randomization cannot be used to help the emptier. In particular, for any positive constant ∆, and any ∆-greedy-like randomized emptying algorithm A, there is a filling strategy that achieves backlog Ω(n 1−ε ) against A in quasi-polynomial time.

show abstract

Section: Claim 52 If Some Cup Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Variable-Processor Cup Game

Kuszmaul,

Westover

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The cup game was first introduced in the late 1960s [28,29] , and has been studied in many different forms [1, 5-7, 9, 13, 14, 18, 21, 23, 25-30]. The game has found extensive applications in areas such as processor scheduling [1, 5-7, 9, 10, 14, 18, 21, 23-30], network-switch buffer management [4,17,19,32], quality of service guarantees [1,6,26], and data-structure deamortization [2,3,8,13,14,16,20,22,31]. See [23] for a detailed discussion of the related work.…”

Section: Introductionmentioning

confidence: 99%

Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

Kuszmaul

2022

Preprint

View full text Add to dashboard Cite

The bamboo trimming problem considers n bamboo with growth rates h 1 , h 2 , . . . , h n satisfying i h i = 1. During a given unit of time, each bamboo grows by h i , and then the bamboo-trimming algorithm gets to trim one of the bamboo back down to height zero. The goal is to minimize the height of the tallest bamboo, also known as the backlog. The bamboo trimming problem is closely related to many scheduling problems, and can be viewed as a variation of the widely-studied fixed-rate cup game, but with constant-factor resource augmentation.Past work has given sophisticated pinwheel algorithms that achieve the optimal backlog of 2 in the bamboo trimming problem. It has remained an open question, however, whether there exists a simple algorithm with the same guarantee-recent work has devoted considerable theoretical and experimental effort to answering this question. Two algorithms, in particular, have appeared as natural candidates: the Reduce-Max algorithm (which always cuts the tallest bamboo), and the Reduce-Fastest(x) algorithm (which cuts the fastest-growing bamboo out of those that have at least some height x). It is conjectured that Reduce-Max and Reduce-Fastest(1) both achieve backlog 2.This paper improves the bounds for both Reduce-Fastest and Reduce-Max. Among other results, we show that the exact optimal backlog for Reduce-Fastest(x) is x + 1 for all x ≥ 2 (this proves a conjecture of D'Emidio, Di Stefano, and Navarra in the case of x = 2), and we show that Reduce-Fastest(1) does not achieve backlog 2 (this disproves a conjecture of D'Emidio, Di Stefano, and Navarra).Finally, we show that there is a different algorithm, which we call the Deadline-Driven Strategy, that is both very simple and achieves the optimal backlog of 2. This resolves the question as to whether there exists a simple worst-case optimal algorithm for the bamboo-trimming problem.

show abstract

“…Until recently, the applications focused primarily on in-memory data structures, since externalmemory data structures often cannot afford the cost of a buffer overflowing by an ω(1) factor. Recent work shows how to use bounds in tail-size in order to solve this problem, and presents a new technique for applying cup games to external-memory data structures [10]. A key insight is that if a cup game has small tail size, then the water in "overflowed cups" (i.e., cups with fill more than O(1)) can be stored in a small in-memory cache.…”

Section: Introductionmentioning

confidence: 99%

“…The result is that every cup consumes exactly Θ(1) blocks in external memory, meaning that each cup can be read/modified by the data structure in O(1) I/Os. This insight was recently applied to external-memory dictionaries in order to eliminate flushing cascades in write optimized data structures [10].…”

Section: Introductionmentioning

confidence: 99%

How Asymmetry Helps Buffer Management: Achieving Optimal Tail Size in Cup Games

Kuszmaul¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The cup game on n cups is a multi-step game with two players, a filler and an emptier. At each step, the filler distributes 1 unit of water among the cups, and then the emptier selects a single cup to remove (up to) 1 unit of water from.There are several objective functions that the emptier might wish to minimize. One of the strongest guarantees would be to minimize tail size, which is defined to be the number of cups with fill 2 or greater. A simple lower-bound construction shows that the optimal tail size for deterministic emptying algorithms is Θ(n), however.We present a simple randomized emptying algorithm that achieves tail size Õ(log n) with high probability in n for poly n steps. Moreover, we show that this is tight up to doubly logarithmic factors. We also extend our results to the multi-processor cup game, achieving tail size Õ(log n + p) on p processors with high probability in n. We show that the dependence on p is near optimal for any emptying algorithm that achieves polynomial-bounded backlog.A natural question is whether our results can be extended to give unending guarantees, which apply to arbitrarily long games. We give a lower bound construction showing that no monotone memoryless emptying algorithm can achieve an unending guarantee on either tail size or the related objective function of backlog. On the other hand, we show that even a very small (i.e., 1/ poly n) amount of resource augmentation is sufficient to overcome this barrier.

show abstract

Flushing Without Cascades

Cited by 13 publications

References 25 publications

The Variable-Processor Cup Game

The Variable-Processor Cup Game

Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

How Asymmetry Helps Buffer Management: Achieving Optimal Tail Size in Cup Games

Contact Info

Product

Resources

About