Extending the Quadrangle Inequality to Speed-Up Dynamic Programming

Proving that a function satisfies the quadrangle inequality is a powerful and elegant way to show that a dynamic programming algorithm to compute that function can be sped up by a factor of the input size. In this paper we consider two problems that do no fit in the usual general cases of functions that satisfy the quadrangle inequality but for which the proof of the quadrangle inequality still carries through: the multi-peg Tower of Hanoi problem with weighted disks and the problem of constructing a Rectilinear Steiner Minimal Arborescence (RSMA) on a slide. We prove the quadrangle inequality holds for a generalized function that unifies the two problems. This speeds up algorithms for these problems from 0(n3p) to O(n2p) and from O(n3) to 0(n2) respectively.