# Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

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## Abstract

**:**

## 1. Introduction

- They pass a single time over the decomposition in a bottom-up fashion;
- they use $O(f\left(s\right)\xb7{log}^{O\left(1\right)}\phantom{\rule{0.166667em}{0ex}}n)$ space; and
- they do not modify the decomposition, including re-arranging it.

## 2. Preliminaries

**Definition 1**(Treedepth)

**.**

## 3. Myhill–Nerode Families

**Definition 2**(Dynamic programming TM)

**.**

**Definition 3**(Myhill–Nerode family)

**.**

- 1.
- For every $({}^{\circ}H,q)\in \mathcal{H}$ it holds that $\left|{}^{\circ}H\right|=\left|\mathcal{H}\right|\xb7{log}^{O\left(1\right)}\phantom{\rule{-0.166667em}{0ex}}\left|\mathcal{H}\right|$ and $q={2}^{\left|\mathcal{H}\right|\xb7{log}^{O\left(1\right)}\phantom{\rule{-0.166667em}{0ex}}\left|\mathcal{H}\right|}$.
- 2.
- For every subset $\mathcal{I}\subseteq \mathcal{H}$ there exists an s-boundaried graph ${}^{\circ}{G}_{\mathcal{I}}\in {}^{\circ}{\mathcal{G}}_{s}$ with $|{}^{\circ}{G}_{\mathcal{I}}|=|\mathcal{H}|\xb7{log}^{O\left(1\right)}\phantom{\rule{-0.166667em}{0ex}}\left|\mathcal{H}\right|$ and an integer ${p}_{\mathcal{I}}$ such that for every $({}^{\circ}H,q)\in \mathcal{H}$ it holds that$$({}^{\circ}{G}_{\mathcal{I}}\oplus {}^{\circ}H,{p}_{\mathcal{I}}+q)\notin {\mathsf{\Pi}}_{\mathrm{DP}}\iff ({}^{\circ}H,q)\in \mathcal{I}.$$

**Lemma**

**1.**

**Proof.**

## 4. Space Lower Bounds for Dynamic Programming

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Dominating Set Using $\mathit{O}({\mathit{d}}^{\mathbf{3}}log\mathit{d}+\mathit{d}log\mathit{n})$ Space

**Theorem**

**4.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

Algorithm 1: Computing dominating sets with very little space. |

## 6. Fast Dominating Set Using $\mathit{O}({\mathbf{2}}^{\mathit{d}}\mathit{d}log\mathit{d}+\mathit{d}log\mathit{n})$ Space

**Definition 4**(Convolution)

**.**

Algorithm 2: Computing dominating sets with ${O}^{\ast}({2}^{d})$ space. |

**Theorem**

**5.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

## 7. Conclusions and Future Work

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The gadget ${}^{\circ}{\mathsf{\Gamma}}_{W}$ for ${\mathcal{D}}_{W}=\{{D}_{1},{D}_{2},{D}_{3}\}$. Padding-vertices are not included.

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**MDPI and ACS Style**

Chen, L.-H.; Reidl, F.; Rossmanith, P.; Sánchez Villaamil, F. Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming. *Algorithms* **2018**, *11*, 98.
https://doi.org/10.3390/a11070098

**AMA Style**

Chen L-H, Reidl F, Rossmanith P, Sánchez Villaamil F. Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming. *Algorithms*. 2018; 11(7):98.
https://doi.org/10.3390/a11070098

**Chicago/Turabian Style**

Chen, Li-Hsuan, Felix Reidl, Peter Rossmanith, and Fernando Sánchez Villaamil. 2018. "Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming" *Algorithms* 11, no. 7: 98.
https://doi.org/10.3390/a11070098