Although both of these techniques use a quadrangle inequality and
seem similar they are actually quite different and have been used
differently in the literature.
In this talk we show that the Knuth-Yao technique is actually a direct
consequence of total monotonicity. As well as providing new derivations of the Knuth-Yao result, this also permits showing how to solve the Knuth-Yao problem directly using the SMAWK algorithm. Another consequence of this approach is a method for solving online versions of problems with the Knuth-Yao property. The online algorithms given here are asymptotically as fast as the best previously known static ones.