Generalized majorization is not a total order, but a preorder. Second, we show that greedy-solvable problems can be formalized as optimization problems in which the objective is monotone with respect to this exchange ordering. We outline greedy algorithms for such problems, including those that exploit additional properties of the objective, such as convexity. Examples from the literature illustrate how diverse well-known applications of greed can be expressed. We show that the space of all binary Huffman codes for a finite alphabet defines a lattice , ordered by the imbalance of the code trees.
Submodular functions are discrete analogues of convex functions.
These results give perspective on Huffman coding, and suggest new approaches to coding as optimization over a lattice. D. Stott Parker, Prasad Ram Majorization is an important partial order on multisets. Majorization with respect to a partial order is shown to be a special case of majorization as exchangeability. In this special case, the resulting majorization order always defines a distributive lattice that is isomorphic to the standard real vector lattice. In classical majorization theory, the semigroup of doubly-stochastic matrices plays a crucial role, both in the definition of majorization and in the modeling of exchange.