122 lines
4.8 KiB
TeX
122 lines
4.8 KiB
TeX
\documentclass[11pt,twocolumn]{article}
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\usepackage[margin=0.75in]{geometry}
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\usepackage{times}
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\usepackage{amsmath,amssymb}
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\usepackage{graphicx}
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\usepackage{enumitem}
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\setlist{noitemsep,topsep=0pt}
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\usepackage{titlesec}
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\titlespacing{\section}{0pt}{6pt}{3pt}
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\titlespacing{\subsection}{0pt}{4pt}{2pt}
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\title{\vspace{-15mm}\textbf{The Ubiquity of Space-Time Tradeoffs:\\From Theory to Practice}\vspace{-5mm}}
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\author{Two-Page Summary for Reviewers}
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\date{}
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\begin{document}
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\maketitle
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\vspace{-10mm}
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\section{Core Contribution}
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We demonstrate that Ryan Williams' 2025 theoretical result---TIME[t] $\subseteq$ SPACE[$\sqrt{t \log t}$]---is not merely abstract mathematics, but a fundamental pattern that already governs modern computing systems. Through systematic experiments and analysis of production systems, we bridge the gap between theoretical computer science and practical system design.
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\section{Key Findings}
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\subsection{Experimental Validation}
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We implemented six experimental domains with space-time tradeoffs:
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\begin{itemize}
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\item \textbf{Maze Solving}: Memory-limited DFS uses O($\sqrt{n}$) space vs BFS's O(n), with 5$\times$ time penalty
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\item \textbf{External Sorting}: Checkpointed sort with O($\sqrt{n}$) memory shows 375-627$\times$ slowdown
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\item \textbf{Stream Processing}: Sliding window (O(w) space) is 30$\times$ FASTER than full storage
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\item \textbf{Real LLM (Ollama)}: Context chunking with O($\sqrt{n}$) space shows 18.3$\times$ slowdown
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\end{itemize}
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\textbf{Critical Insight}: Constant factors range from 100$\times$ to 10,000$\times$ due to memory hierarchies (L1/L2/L3/RAM/SSD), far exceeding theoretical predictions but following the $\sqrt{n}$ pattern.
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\subsection{Real-World Systems Analysis}
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\textbf{Databases (PostgreSQL)}
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\begin{itemize}
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\item Buffer pools sized at $\sqrt{\text{database\_size}}$
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\item Query planner: hash joins (O(n) memory) vs nested loops (O(1) memory)
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\item 200$\times$ performance difference aligns with our measurements
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\end{itemize}
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\textbf{Large Language Models}
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\begin{itemize}
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\item Flash Attention: Recomputes attention weights, O(n$^2$) $\rightarrow$ O(n) memory
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\item Enables 10$\times$ longer contexts with 10\% speed penalty
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\item Gradient checkpointing: $\sqrt{n}$ layers stored, 30\% overhead
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\end{itemize}
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\textbf{Distributed Computing}
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\begin{itemize}
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\item MapReduce: Optimal shuffle = $\sqrt{\text{data/node}}$
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\item Spark: Hierarchical aggregation forms $\sqrt{n}$ levels
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\item Memory/network tradeoffs follow Williams' bound
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\end{itemize}
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\subsection{When Tradeoffs Help vs Hurt}
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\begin{minipage}[t]{0.48\columnwidth}
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\textbf{Beneficial:}
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\begin{itemize}
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\item Streaming data
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\item Sequential access
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\item Distributed systems
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\item Fault tolerance
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\end{itemize}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.48\columnwidth}
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\textbf{Detrimental:}
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\begin{itemize}
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\item Interactive apps
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\item Random access
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\item Small datasets
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\item Cache-critical code
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\end{itemize}
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\end{minipage}
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\section{Practical Impact}
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\textbf{Explains Existing Designs}: The size of the database buffer, the ML checkpoint intervals, and the distributed configurations all follow $\sqrt{n}$ patterns discovered by trial and error.
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\textbf{Guides Future Systems}: Provides a mathematical framework for memory allocation and algorithm selection.
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\textbf{Tools for Practitioners}: The interactive dashboard helps developers optimize specific workloads.
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\section{Why This Matters}
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As data grows exponentially while memory grows linearly, understanding space-time tradeoffs becomes critical. Williams' result provides the theoretical foundation; our work shows how to apply it practically despite massive constant factors.
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The pattern $\sqrt{n}$ appears everywhere, from database buffers to neural network training, validating the deep connection between theory and practice.
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\section{Technical Highlights}
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\begin{itemize}
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\item Continuous memory monitoring at 10ms intervals
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\item Cache-aware benchmarking methodology
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\item Theoretical analysis connecting to Williams' bound
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\item Open-source code and reproducible experiments
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\item Interactive visualizations of tradeoffs
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\end{itemize}
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\section{Paper Organization}
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\begin{enumerate}
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\item Introduction with four concrete contributions
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\item Williams' theorem and memory hierarchy background
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\item Experimental methodology with statistical rigor
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\item Results: Maze solving, sorting, streaming, SQLite, LLMs, Ollama
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\item Analysis: Production systems (databases, transformers, distributed)
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\item Practical framework and guidelines
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\item Interactive tools and dashboard
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\end{enumerate}
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\vspace{3mm}
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\noindent\textbf{Bottom Line}: Williams proved what is mathematically possible. We show what is practically achievable and why the gap matters for system design.
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\vspace{3mm}
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\noindent\textit{Full paper includes detailed experiments, system analysis, and interactive tools at \texttt{github.com/sqrtspace}}
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\end{document} |