基于比较算法的熵守恒
Entropy conservation for comparison-based algorithms
摘要：
基于比较的算法是每个操作的执行完全基于元素之间的一系列比较的结果的算法。基于比较的计算可以通过以下计算模型自然地表示：（a）将数据结构建模为偏序有限集;（b）通过拓扑排序对这些数据进行建模;（c）将计算状态视为此类数据的有限多集;（d）通过它们在状态上的诱导变换来表示计算。在这种观点中，排序算法的抽象规范具有由有限元素集合的任何可能置换给出的输入状态（根据（a）和（b），由离散的偏序集合连同由所有排列给出的其拓扑排序来表示）并且输出状态元素的排序列表（再次根据（a）和（b），由具有唯一拓扑排序的线性有序有限集表示）。熵是衡量“随机性”或“无序性”的一种方法。基于计算模型，我们引入了一个基于比较的算法的熵守恒结果：“获得的数量级与失去的位置级成正比。“直觉上，这个结果与混乱的办公室争论有一定关系，主张一个混乱的办公室，没有东西在正确的地方，但每个物品的位置都是所有者知道的，而不是每个物品都以正确的顺序存储，但所有者再也找不到物品的情况。形式上，我们推广的结果表示类的数据结构通过串并行偏序-一个著名的计算易处理的类。由此产生的熵守恒的“符号”版本将在后续工作中扩展为我们的计算模型核心部分的“操作”版本。
英文摘要：
Comparison-based algorithms are algorithms for which the execution of each operation is solely based on the outcome of a series of comparisons between elements. Comparison-based computations can be naturally represented via the following computational model: (a) model data structures as partially-ordered finite sets; (b) model data on these by topological sorts; (c) considering computation states as finite multisets of such data; (d) represent computations by their induced transformations on states. In this view, an abstract specification of a sorting algorithm has input state given by any possible permutation of a finite set of elements (represented, according to (a) and (b), by a discrete partially-ordered set together with its topological sorts given by all permutations) and output state a sorted list of elements (represented, again according to (a) and (b), by a linearly-ordered finite set with its unique topological sort). Entropy is a measure of "randomness" or "disorder." Based on the computational model, we introduce an entropy conservation result for comparison-based algorithms: "quantitative order gained is proportional to positional order lost." Intuitively, the result bears some relation to the messy office argument advocating a chaotic office where nothing is in the right place yet each item's place is known to the owner, over the case where each item is stored in the right order and yet the owner can no longer locate the items. Formally, we generalize the result to the class of data structures representable via series-parallel partial orders--a well-known computationally tractable class. The resulting "denotational" version of entropy conservation will be extended in follow-up work to an "operational" version for a core part of our computational model.
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