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

Authors:
Wolfgang Bein;Mordecai J. Golin;Lawrence L. Larmore;Yan Zhang
Affiliations:
University of Nevada, Las Vegas, NV;Hong Kong University of Science and Technology, Kowloon, Hong Kong;University of Nevada, Las Vegas, NV;Hong Kong University of Science and Technology, Kowloon, Hong Kong
Venue:
ACM Transactions on Algorithms (TALG)
Year:
2009

Citing 13
Cited 1

The concave least-weight subsequence problem revisited

Journal of Algorithms
On an efficient dynamic programming technique of F. F. Yao

Journal of Algorithms
Constructing trees in parallel

SPAA '89 Proceedings of the first annual ACM symposium on Parallel algorithms and architectures
Dynamic programming with convexity, concavity and sparsity

Theoretical Computer Science - Selected papers of the Combinatorial Pattern Matching School
Extending the quadrangle inequality to speed-up dynamic programming

Information Processing Letters
Perspectives of Monge properties in optimization

Discrete Applied Mathematics
On-line dynamic programming with applications to the prediction of RNA secondary structure

SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Efficient dynamic programming using quadrangle inequalities

STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Efficient Algorithms for Optimal Stream Merging for Media-on-Demand

SIAM Journal on Computing
Efficient Alphabet Partitioning Algorithms for Low-Complexity Entropy Coding

DCC '05 Proceedings of the Data Compression Conference
The algebraic Monge property and path problems

Discrete Applied Mathematics
Notes on searching in multidimensional monotone arrays

SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Monge strikes again: optimal placement of web proxies in the internet

Operations Research Letters

Approximation algorithms for speeding up dynamic programming and denoising aCGH data

Journal of Experimental Algorithmics (JEA)

Quantified Score

Hi-index	0.00

Visualization

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.