环上的分布式消息最优分配
A Distributed Message-Optimal Assignment on Rings
摘要：
考虑一组项目和一组$m$颜色，其中每个项目与一种颜色相关联。还考虑通过环连接的$n$计算代理。每个代理持有物品的子集，并且相同颜色的物品可以由不同的代理持有。我们分析的问题，分配颜色的代理在这样一种方式，（a）每种颜色只分配给一个代理和（b）分配给每个代理的不同颜色的数量是最小的。由于任何颜色分配都需要根据它来分配物品（例如，相同颜色的所有物品都只能由一个代理持有），因此我们将颜色分配的成本定义为给定初始分配时需要移动的物品数量。我们首先表明，这个问题的任何分布式算法需要一个消息的复杂性$Ω(n\cdot m)$，然后我们展示了一个最佳的消息复杂性算法同步环，在多项式时间确定的颜色分配成本最多三倍的最佳。我们还讨论了异步设置的解决方案。最后，我们展示了如何以消息或时间复杂度为代价获得更好的成本解决方案。
英文摘要：
Consider a set of items and a set of $m$ colors, where each item is associated to one color. Consider also $n$ computational agents connected by a ring. Each agent holds a subset of the items and items of the same color can be held by different agents. We analyze the problem of distributively assigning colors to agents in such a way that (a) each color is assigned to one agent only and (b) the number of different colors assigned to each agent is minimum. Since any color assignment requires the items be distributed according to it (e.g. all items of the same color are to be held by only one agent), we define the cost of a color assignment as the amount of items that need to be moved, given an initial allocation. We first show that any distributed algorithm for this problem requires a message complexity of $Ω(n\cdot m)$ and then we exhibit an optimal message complexity algorithm for synchronous rings that in polynomial time determines a color assignment with cost at most three times the optimal. We also discuss solutions for the asynchronous setting. Finally, we show how to get a better cost solution at the expenses of either the message or the time complexity.
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