排序可以指数加速纯动态规划
Sorting Can Exponentially Speed Up Pure Dynamic Programming
摘要：
许多离散的最小化问题，包括各种版本的最短路径问题，可以有效地解决动态规划（DP）算法是“纯”的，因为他们只执行基本的操作，如min，max，+，但没有条件分支通过if-then-else在其递归方程。已知任何求解无向n-顶点图上最小权生成树问题的纯（min，+）DP算法至少需要执行$2^{Ω(\sqrt{n})}$运算。我们表明，这个问题可以解决一个纯粹的（最小，最大，+）DP算法只执行$O(n^3)$操作。该算法本质上是一个（min，max）算法：加法运算仅用于输出最终值。最小和最大运算的存在意味着现在DP算法可以排序：这解释了论文的标题。
英文摘要：
Many discrete minimization problems, including various versions of the shortest path problem, can be efficiently solved by dynamic programming (DP) algorithms that are "pure" in that they only perform basic operations, as min, max, +, but no conditional branchings via if-then-else in their recursion equations. It is known that any pure (min,+) DP algorithm solving the minimum weight spanning tree problem on undirected n-vertex graphs must perform at least $2^{Ω(\sqrt{n})}$ operations. We show that this problem can be solved by a pure (min,max,+) DP algorithm performing only $O(n^3)$ operations. The algorithm is essentially a (min,max) algorithm: addition operations are only used to output the final values. The presence of both min and max operations means that now DP algorithms can sort: this explains the title of the paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
