The Knuth-Yao quadrangle-inequality speedup is a consequence of total monotonicity

Abstract

There exist several general techniques in the literature for speeding up naive implementations of dynamic programming. Two of the best known are the Knuth-Yao quadrangle inequality speedup and the SMAWK algorithm for finding the row-minima of totally monotone matrices. 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 article 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 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. For example, the Knuth-Yao technique speeds up the standard dynamic program for finding the optimal binary search tree of n elements from Θ(n³) down to O(n²), and the results in this article allow construction of an optimal binary search tree in an online fashion (adding a node to the left or the right of the current nodes at each step) in O(n) time per step.

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Index Terms

• Computing methodologies

  • Symbolic and algebraic manipulation

    • Symbolic and algebraic algorithms

• Information systems

  • Information storage systems

    • Record storage systems

      • Directory structures

        • B-trees

• Mathematics of computing

  • Discrete mathematics

    • Graph theory

      • Trees

• Theory of computation

  • Design and analysis of algorithms