Efficient dynamic programming using quadrangle inequalities

Abstract

Dynamic programming is one of several widely used problem-solving techniques in computer science and operation research. In applying this technique, one always seeks to find speed-up by taking advantage of special properties of the problem at hand. However, in the current state of art, ad hoc approaches for speeding up seem to be characteristic; few general criteria are known. In this paper we give a quadrangle inequality condition for rendering speed-up. This condition is easily checked, and can be applied to several apparently different problems. For example, it follows immediately from our general condition that the construction of optimal binary search trees may be speeded up from 0{n3) steps to 0(n2), a result that was first obtained by Knuth using a different and rather complicated argument.