[论文解读] Balanced allocation on graphs

Krishnaram Kenthapadi, Rina Panigrahy‎|arXiv (Cornell University)|Jan 22, 2006

Algorithms and Data Compression参考文献 20被引用 41

一句话总结

该论文研究了通过图结构约束的 n 个球分配至 n 个桶的问题，其中每个球从底层图中由边连接的两个桶中进行选择。研究表明，对于几乎正则图（度数为 n^Ω(1/log log n)），最大负载保持为 O(log log n)，并且通过使用大小为 d/2 的对齐组，仅需两次随机选择即可实现 O(log log n / d) 的最大负载，优于标准 d 选策略，当冲突通过策略性方式解决时表现更优。

ABSTRACT

It is well known that if n balls are inserted into n bins, with high probability, the bin with maximum load contains (1 + o(1)) log n/log log n balls. Azar, Broder, Karlin, and Upfal [1] showed that instead of choosing one bin, if d ≥ 2 bins are chosen at random and the ball in serted into the least loaded of the d bins, the maximum load reduces drastically to log log n/log d + O(1). In this paper, we study the two choice balls and bins process when balls are not allowed to choose any two random bins, but only bins that are connected by an edge in an underlying graph. We show that for n balls and n bins, if the graph is almost regular with degree ne, where e is not too small, the previous bounds on the maximum load continue to hold. Precisely, the maximum load is log log n + O(1/e) + O(1). So even if the graph has degree nΩ(1/log log n), the maximum load is O(log log n). For general Δ-regular graphs, we show that the maximum load is log log n + O(log n/log (Δ/log4n)) + O(1) and also provide an almost matching lower bound of log log n + log n/log (Δ log n). Further this does not hold for non-regular graphs even if the minimum degree is high.Vocking [29] showed that the maximum bin size with d choice load balancing can be further improved to O(log log n/d) by breaking ties to the left. This requires d random bin choices. We show that such bounds can be achieved by making only two random accesses and querying d/2 contiguous bins in each access. By grouping a sequence of n bins into 2n/d groups, each of d/2 consecutive bins, if each ball chooses two groups at random and inserts the new ball into the least-loaded bin in the lesser loaded group, then the maximum load is O(log log n/d) with high probability. Furthermore, it also turns out that this partitioning into aligned groups of size d/2 is also essential in achieving this bound, that is, instead of choosing two aligned groups, if we simply choose random but possibly unaligned random sets of d/2 consecutive bins, then the maximum load jumps to Ω(log log n/log d) even if the two sets are always chosen to be disjoint.

研究动机与目标

• 理解图结构如何影响负载均衡，当球只能选择通过边连接的桶时。
• 确定在何种图条件下最大负载仍接近最优的 d 选策略边界。
• 研究对齐的桶组是否能在仅使用两次随机选择的情况下维持低最大负载。
• 比较在图约束下，对齐与非对齐随机桶选择的性能差异。
• 为各种图正则性条件建立最大负载的紧致上下界。

提出的方法

• 在底层图上建模球桶过程，其中每个球选择两个相邻桶进行放置。
• 分析在‘两选’规则下的最大负载：将球放置于两个所选桶中负载较轻的一个。
• 引入分组策略：将 n 个桶划分为 2n/d 个大小为 d/2 的连续对齐组。
• 对于每个球，随机选择两个组，并将球放置于这两个组中负载最轻的桶内。
• 使用概率分析和集中不等式推导最大负载的上界。
• 通过构造对抗性场景，证明非对齐选择会导致更高负载，从而建立下界。

实验结果

研究问题

• RQ1 当球被限制仅能选择图中相邻桶时，最大负载是多少？其与标准 d 选模型相比如何？
• RQ2 是否仅使用两次随机选择即可实现 O(log log n / d) 的最大负载，而非 d 次？
• RQ3 桶组的对齐性如何影响图约束分配中的最大负载？
• RQ4 底层图的哪些结构性质（如正则性、最小度）是维持低最大负载所必需的？
• RQ5 对齐与非对齐随机组选择之间是否存在根本性的性能差异？

主要发现

• 对于度数为 n^Ω(1/log log n) 的几乎正则图，最大负载为 O(log log n)，与标准 d 选模型中的最佳已知界一致。
• 当桶被划分为大小为 d/2 的对齐组，并随机选择两个组时，最大负载以高概率为 O(log log n / d)。
• 对齐分组策略至关重要：非对齐的 d/2 个桶的随机选择会导致最大负载为 Ω(log log n / log d)，显著更差。
• 对于 Δ-正则图，最大负载被限制在 log log n + O(log n / log(Δ / log⁴n)) + O(1) 之内，显示出度与负载之间的权衡。
• 建立了下界 log log n + log n / log(Δ log n)，表明对正则图而言，上界几乎紧致。
• 即使最小度较高，非正则图仍无法实现低最大负载，表明正则性对性能至关重要。