[论文解读] Randomised Buffer Management with Bounded Delay against Adaptive Adversary

Łukasz Jeż|arXiv (Cornell University)|Jan 1, 2009

Optimization and Search Problems参考文献 5被引用 3

一句话总结

本文提供了RMix算法在缓冲区管理问题中针对自适应在线敌手的$rac{e}{e-1}$-竞争性的一个新证明。它通过证明该结果在敌手能够适应算法随机选择的情况下依然成立，填补了先前工作中存在的空白，将有效性扩展到仅已知截止时间顺序的更一般模型中。

ABSTRACT

We study the Buffer Management with Bounded Delay problem, introduced by Kesselman et al. [4], or, in the standard scheduling terminology, the problem of online scheduling of unit jobs to maximise weighted throughput. The adaptive-online adversary model for this problem has recently been studied by Bienkowski et al. [2], who proved a lower bound of 4 3 on the competitive ratio and provided a matching upper bound for 2-bounded sequences. In particular, the authors of [2] claim that the algorithm RMix [3] is e e−1 -competitive against an adaptive-online adversary. However, the original proof of Chin et al. [3] holds only in the oblivious adversary model. The reason is as follows. First, the potential function used in the proof depends on the adversary’s future schedule, and second, it assumes that the adversary follows the earliest-deadline-first policy. Both of these cannot be assumed in adaptive-online adversary model, as the whole schedule of such adversary depends on the random choices of the algorithm. We give an alternative proof that RMix indeed is e e−1 -competitive against an adaptive-online adversary. Similar claim about RMix was made in another paper by Bienkowski et al. [1] studying a slightly more general problem. It assumes that the algorithm does not know exact deadlines of the packets, and instead knows only the order of their expirations. However, any prefix of the deadline-ordered sequence of packets can expire in every step. The new proof that we provide holds even in this more general model, as both the algorithm and its analysis rely only on the relative order of packets’ deadlines.

研究动机与目标

• 为填补在自适应在线敌手模型下RMix竞争性分析中的空白，此前的证明依赖于无效假设。
• 确立RMix在敌手能够适应算法随机决策的情况下，仍保持其$\frac{e}{e-1}$-竞争性。
• 将分析扩展到更一般模型，即算法仅知道数据包截止时间的相对顺序，而非确切值。
• 提供一种不依赖敌手遵循最早截止时间优先策略的证明，该假设在自适应在线条件下不成立。
• 验证先前工作中提出的主张，即RMix在自适应环境下可实现$\frac{e}{e-1}$-竞争性，尽管原始证明存在缺陷。

提出的方法

• 使用不依赖敌手未来调度的势函数，重构RMix的竞争性分析。
• 设计一种在敌手行为依赖于算法随机选择（即自适应在线敌手）时依然有效的证明技术。
• 确保分析仅依赖于数据包截止时间的相对顺序，而非其确切值，以推广至部分信息模型。
• 调整势函数使其独立于敌手策略，消除其遵循最早截止时间优先的假设。
• 利用RMix算法随机选择过程的结构，界定期望竞争比，而无需假设敌手具有可预测性。

实验结果

研究问题

• RQ1 RMix的$\frac{e}{e-1}$-竞争性是否可在自适应在线敌手模型下被严格证明？
• RQ2 当敌手能够适应算法的随机选择时，Chin等人原始证明是否仍然成立？
• RQ3 该竞争比结果能否扩展至仅知道截止时间顺序而非确切值的情况？
• RQ4 当敌手调度依赖于算法随机性时，先前证明中使用的势函数是否仍然有效？
• RQ5 能否构建一种新证明，避免对敌手策略的假设，同时保持$\frac{e}{e-1}$的界？

主要发现

• Chin等人关于RMix的$\frac{e}{e-1}$-竞争性的原始证明在自适应在线敌手模型下无效，因其依赖于敌手未来行为。
• 本文提供了新证明，确立RMix在自适应在线敌手下仍为$\frac{e}{e-1}$-竞争，纠正了文献中的基础性空白。
• 新证明在更一般模型下依然有效，即算法仅知数据包截止时间的相对顺序。
• 该分析不依赖敌手遵循最早截止时间优先策略，因此对自适应调度策略具有鲁棒性。
• 结果确认并扩展了Bienkowski等人[1]和[2]中提出的主张，验证了RMix在现实自适应环境下的性能。