  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 $2^{\Omega(\sqrt{n})}$ 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.
