On the I/O Complexity of the CYK Algorithm and of a Family of Related DP Algorithms

TL;DR

This paper establishes tight lower bounds on the I/O complexity of various dynamic programming algorithms, including CYK, and discusses algorithms that match these bounds, highlighting the impact of memory size and recomputation.

Contribution

It provides the first asymptotically tight I/O lower bounds for a broad class of DP algorithms, including CYK, and refines methods to analyze their complexity considering memory constraints.

Findings

I/O lower bound of Ω(n^3 / (√M B)) for DP algorithms without recomputation

Matching upper bounds are discussed for several algorithms

Lower bound for CYK algorithm considering grammar size

Abstract

Asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity of a class of dynamic programming algorithms including matrix chain multiplication, optimal polygon triangulation, and the construction of optimal binary search trees. Assuming no recomputation of intermediate values, we establish an $\Omega\left(\frac{n^3}{\sqrt{M}B}\right)$ I/O lower bound, where $n$ denotes the size of the input and $M$ denotes the size of the available fast memory (cache). When recomputation is allowed, we show the same bound holds for $M < cn$, where $c$ is a positive constant. In the case where $M \ge 2n$, we show an $\Omega\left(n/B\right)$ I/O lower bound. We also discuss algorithms for which the number of executed I/O operations matches asymptotically each of the presented lower bounds, which are thus asymptotically tight. Additionally, we refine our general method to obtain a lower bound for the I/O complexity of the Cocke-Younger-Kasami algorithm, where the size of the grammar impacts the I/O complexity. An upper bound with asymptotically matching performance in many cases is also provided.