The concave least-weight subsequence problem revisited
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
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
Sequence Alignment Algorithms for Run-Length-Encoded Strings
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Approximate dynamic programming using halfspace queries and multiscale Monge decomposition
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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 paper 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 showing how 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 paper allow construction of an optimal binary search tree in an online fashion (adding a node to the left or right of the current nodes at each step) in O(n) time per step.We conclude by discussing how the general technique described here is also applicable to later extensions of the Knuth-Yao result, such as those developed by Borchers and Gupta.