Shellsort的时序证书：射线缺陷与符号-正传递
When More Generators Hurt: Shellsort on Full Product Grids
浏览论文内容
中文总结 AI 辅助
该研究为Shellsort的符号化下界与正表示上界开发了通用证书框架，恢复了相关尺度并得出多项射线类相关的结构结果，为Shellsort的界分析提供了新的工具与结论。
中文摘要 AI 辅助
Shellsort的最优通用下界与经典上界相差一个迭代对数因子。下界使用符号化的、与顺序无关的抵消方法，而上界需要符合趟顺序的正表示。我们为这两种几何结构开发了一个通用的证书框架：前缀傅里叶相位缺陷对最坏情况下的交换次数和比较次数进行下界估计；在第j趟，时序射线商记录了已被前几趟消除的当前间隙的倍数；其截断的、带对权重的空穴质量对该趟的交换次数进行下界估计，加上np开销则可对比较次数进行下界估计。在足够长的活动窗口内，若存在g_j^{(n)}个射线空穴，则符号传递长度最多为4g_j^{(n)}-1，因此近似前缀特征可传递到当前间隙；符号传递长度2可与任意大的射线类共存，尺度局部的Apéry代表元可控制这两个参数。该框架恢复了Pratt的O(n log²n)尺度和稀疏双父尺度，并得出以下结构结果：当p_n - j_0 = o(log n)时，固定前缀后的一致有界全射线类会迫使W_n = Ω(n^{1+1/j_0 - o(1)})；一个仿射族具有普通全局半群类和Θ(n)的导子，但相关射线类为2；平衡三生成元全网格的终端截断射线类至少为exp(Ω((log n)^{2/3}))，这一结果构成了一个证书障碍。
英文摘要
Shellsort repeatedly runs insertion sort with decreasing gaps, so its worst-case cost depends on the gap sequence. Pratt's $2^u3^v$ sequence, one of the few systematic constructions with a proven $O(n\log^2 n)$ bound, includes every product below $n$ of two base numbers, or generators. We ask whether adding more base numbers, and thus more intermediate gaps, can improve this full product grid. We show that it cannot when every product is retained and each base is at most a fixed power of the smallest. With $r$ independent bases (different exponent choices give different products) and $Θ(\log n)$ gaps, the best possible worst-case cost is $n\exp(Θ((\log n)^{1-1/r}))$. Thus two bases give the exponent $\sqrt{\log n}$, whereas three give $(\log n)^{2/3}$: more bases are worse. With a budget of $p$ gaps, matching bounds give the factor $\exp(Θ(\log n/p^{1/r}))$ beyond linear cost. The reason is simple. Few products force the smallest base $m$ to be large, and fullness makes $m$ the next-to-last gap. An input built from reversed blocks is already sorted for every earlier gap, forcing $Ω(nm)$ work in the final pass. Powers of distinct primes give a matching construction. For arbitrary gaps, we count current-gap multiples that earlier gaps cannot form. This gives upper and lower bounds for individual passes. A Fourier argument gives necessary conditions for small total cost, while short nonnegative sums give sufficient conditions. In both settings, useful distances must be available before they are needed.
